WEDNESDAY marks the one-year countdown to the deadline for making a PPI claim – and with millions of policies having been taken out, the clock is ticking if you want to make a complaint.
The deadline for PPI complaints is August 29 2019. As many as 64 million PPI policies have been sold in the UK, mostly in the 20 years after 1990 – but some stretching as far back as the '70s.
Some £31.5 billion has already been paid back between January 2011 and May 2018 to customers who complained about the way they were sold PPI. If you had a PPI policy you weren't happy about you can complain yourself for free, and it might be easier than you think.
PPI or payment protection insurance was designed to cover repayments if you found you were unable to make them yourself for particular reasons – perhaps due to being made redundant, having an accident or being ill.
What financial products might I have had that PPI has been added to?
PPI was tacked onto products such as personal and business loans, credit cards, store cards, mortgages, catalogue credit and other types of credit, such as car finance and hire purchase agreements for household items.
So why did PPI become controversial?
PPI was found to have often been mis-sold, to people who didn't need it because it was unsuitable, felt under pressure to sign up, or in some cases didn't even realise they had a PPI policy because it was simply added without them asking.
Are there other circumstances in which I could make a PPI claim?
Yes – importantly, you may now find you're entitled to money back just for having the policy – even if you don't think it was mis-sold to you.
There's a new reason you can complain – and it's if the bank or other provider earned a high level of commission from the sale of PPI but you weren't told when you bought it. Providers were often paid commission by the insurer, as a reward for the sale of PPI.
The money for this commission would come out of the payments you made for the policy. A high level of commission typically means it was more than half of what you paid for your PPI policy.
These rules mean it's also possible to complain even if you've had a previous complaint about the mis-selling of PPI rejected.
You don't need to know or explain how much commission was paid for your PPI policy.
How can I check if I had PPI?
If the policy was taken out a very long time ago, or you aren't quite sure, you may need to do a bit of detective work – but there is help out there with this.
Dig out any old paperwork which could help – but if you haven't got this you can still ask the firm. The Financial Conduct Authority has page on its website that helps people track down the details of firms (fca.org.uk/ppi/how-to-complain/search-for-provider).
What free help is available for me to make a complaint?
Websites like MoneySavingExpert.com and Resolver.co.uk have free tools which can help take the hassle out of complaining.
Citizens Advice, Which?, the Money Advice Service, the Financial Conduct Authority and the Financial Ombudsman Service also have lots of information on their websites.


Manasquan junior attackman Canyon Birch is the 2018 SSN Player of the Year after a historic season where he scored 123 goals and tallied 175 points. (Photo by Paula Lopez/palimages.com).
Abbes scored 35 goals and added 19 assists to total 54 points as a sophomore. He was a Second Team All-Class A North selection by the coaches.
With linemate Canyon Birch garnering the lion's share of defenses' attention and headlines, Anderson quietly had an excellent season with 48 goals and 57 assists (tied for the Shore Conference high) and was one of just four players in the Shore to eclipse the 100-point plateau. He was a Second Team All-Shore pick last season and should again be one of the top point producers in the area.
Ard's 36 goals as a sophomore were tops for Middletown South. He added seven assists and was a Second Team All-Class A North selection by the coaches.
The reigning Shore Sports Network Player of the Year and First Team All-State selection is ready to continue his assault on the record book and lead Manasquan's quest for a Group 1 title. Birch set a state record with 175 points last season, including a Shore Conference-record 123 goals plus 52 assists in a season for the ages. Birch, who is committed to Penn State, is tied with Southern's Dylan Jinks for the most career goals in Shore Conference history and needs just 43 points to break the career points record of 431 held by CBA's Tommy Duerr. He is also within striking distance of the state career records of 362 goals and 468 points held by legendary A.L. Johnson midfielder Matt Poskay.
After two years as an understudy, Doel takes over as the top attackman for the Patriots. He deposited 22 goals last season while adding a team-high 32 assists for 54 total points. He was a Second Team All-Class A North selection by the coaches.
Doyle scored nine goals with three assists in limited action last season on a veteran Wall squad. He'll step to the forefront this season and is expected to be a top playmaker for a high-high-scoring Crimson Knights offense.
Gosse netted 41 goals last season to lead the attack for the Lancers.
Johnson scored 28 goals with seven assists and was a Second Team All-Class A South pick as a sophomore last season.
Laverty led the Jaguars in scoring as a sophomore with 29 goals and 12 assists for 41 points. At 6-foot-3 and 225 pounds, he brings a unique physical presence and will be a tough matchup for opposing defensemen.
Lowe was 13th in the Shore in points last season with 65 on 40 goals and 25 assists and returns to lead the Spartans' offense under new head coach Kevin Preston. He was a Second Team All-Class B North pick last year.
Although it seems like CBA lost its entire team to graduation, the cupboard is far from bare for head coach Dave Santos. Macrae leads the way after pacing the Colts with 39 goals last season while adding 15 assists. He was a Second Team All-Shore pick last season.
Martin was a second-line attackman for the Colts as a freshman but still found a way to post 27 goals and 14 assists. CBA lost 19 seniors to graduation so its incumbent on players like Martin to produce with increased playing time. A big season could be on the horizon.
Mulligan joined a loaded Warriors' team as a freshman and contributed immediately with 37 goals and 23 assists for 60 points. He was a Second Team All-Class B South pick last season and along with seniors Canyon Birch and Ryan Anderson comprise one of New Jersey's most explosive attack lines.
Expectations are very high for Peters, who had six goals, six assists and 19 ground balls as a complementary player during his freshman year. He'll play midfielder and attack for Wall and could be the breakout of the year in the Shore.
Russo was a First Team All-Class A South selection last season when he tallied 25 goals and 20 assists for 45 points.
Schellin was top 20 in the Shore in scoring last season with 39 goals and 21 assists for 60 points.
Vitale had a huge season for the Lions last year with 58 goals and 24 assists for 82 points, which was good for ninth in the Shore. The Second Team All-Class B South pick eclipsed the 100-goal mark last May and is committed to McDaniel College.
Waltonowski was a top-20 scorer in the Shore last season with 23 goals and 37 assists for 60 points and teams with Dylan Vitale to form one of the top scoring duos in Class B South.
Zenkert's 64 goals were the fourth-most in the Shore Conference last season. He added 16 assists to finish with 80 points, which was 10th in the Shore.


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PALATINE, IL, (JUNE 4, 2018) -- JOURNEYS | The Road Home, a homeless service organization committed to individually assessing and serving the homeless and near-homeless in north/northwest suburban Cook County, is organizing its 11th annual 5K fun run/walk to take place on Saturday, June 16 at 8:30 a.m. in the Palatine Towne Square. This year, Tavares Chambliss, 36, of Palatine, better known as American Ninja Warrior's "Neon Ninja," will join ranks with staff and race registrants as a superhero for the homeless.
Chambliss, who grew up in Wheeling and attended Harper College, is no stranger to the harrowing realities of homelessness often amplified by a number of fortified barriers to stability, including poverty, domestic violence and a shortage of affordable housing. Prior to partnering with JOURNEYS in the effort to raise awareness and crucial funding for life-sustaining and life-changing programs, Chambliss and his mother periodically distributed homemade care packages in backpacks containing food and essential hygiene supplies to the homeless on the West side of Chicago.
Joining JOURNEYS in the fight to end the invisble plight of the suburban homeless throughout his own hometown and surrounding communities struck Chambliss as the distinctive next step in his adventure all the way to the top of famed Mt. Midoriyama on the national hit TV show American Ninja Warrior.
Chambliss, an accomplished athlete, personal trainer and acclaimed Neon Ninja makes tackling challenging obstacles look effortless and easy. For those facing homelessness or the risk of losing their homes due to eviction or foreclosure, tackling great obstacles with poise and prowess is not always possible, and many more families today are struggling to triumph in the fight against suburban poverty.
Like Chambliss, JOURNEYS believes in empowering people to reach their full potential by breaking barriers and promoting healthier, happier lifestyles in the clients they serve-- more than half of whom are women and children experiencing a housing crisis. The agency connects people to critical services including emergency shelter, affordable housing, showers, clothing, ongoing food and welfare assistance, long-term case management and transportation vouchers, among a multitude of other resources.
According to the Brookings Institute, there are now more people living in poverty in suburban contexts outside of major metropolitan areas than there are people living in poverty in big cities, including Chicago.
This past winter, demand for JOURNEYS' supportive services outpaced the available space in the agency's network of emergency PADS shelters, necessitating the implementation of a lottery system for entry into the the shelters for the first time in its 29-year history. The agency welcomed the highest number of individuals and families during this period than ever before and also housed a record 164 people, making this year one for the record books.
Chambliss is no stranger to setting records, stretching limits, beating odds and exceeding remarkable goals and the agency looks forward to partnering with him to make their superhero 5K the biggest and best to date. Chambliss will be leading participants in a warmup and commencing the race. He will also award first-prize medals and take photos with participants and might bring along other familiar faces from the hit Ninja Warrior franchise.
Participants have the option of completing a 5K or one-mile event. The 5K race is a USATF certified timed race. Tap House Grill in Palatine will host a post-race party complete with free food and drinks for racers.
For more information on JOURNEYS or this event, please visit www.journeystheroadhome.org.
JOURNEYS | The Road Home is a non-profit agency that provides shelter, social services and housing to the homeless and those at risk of becoming homeless in 37 towns in north/northwest suburban Cook County. The mission of JOURNEYS is to individually assess and serve its clients, while broadening community awareness and involvement. The agency consists of the PADS Shelter Program, the HOPE Center Supportive Services Program, and the Pathways Housing Readiness Program focusing on intervention and assistance for those in need. In 2016-2017, we provided over 26,000 clinical services to 847 people across the north and northwest suburbs of Cook County.


text:
When the Flying Burrito Brothers introduced me to Bluegrass music
Last week, I wrote a post about my appreciation for Bluegrass music (see here) and that appreciation all began with the LP…
"Last of the Red Hot Burritos"
The Flying Burrito Brothers
Producer: Jim Dickson, A&M Records, Release date: May, 1972, Length: 34:45
The Flying Burrito Brothers had already crashed and burned as a band by the time this album was released in 1972. Ex-Byrds, Gram Parsons and Chris Hillman, had formed the group back in 1968, hoping to pursue their dream of melding country music, R&B, and rock-and-roll. Their debut LP, "The Gilded Palace of Sin," remains a pioneering classic (see my review here), but Parsons' descent into drug and alcohol addiction quickly spelled trouble. Their sophomore release, "Burrito Deluxe," was uneven, and their Parsons-less, eponymous third album, although pleasant, was uninspiring.
After four years, with zero commercial success, Hillman had had enough and decided to pull the plug. To fulfill their contractual obligations, the band released this live album. I liked it a lot and it ranked right up there as one of my most-played LPs back in my high school heydays. At its end, when this album was recorded, the band consisted of Hillman on bass and lead vocals, Rick Roberts on rhythm guitar and vocals, Al Perkins on pedal steel guitar, Kenny Wertz on guitar and banjo and vocals, and another ex-Byrd, Michael Clarke on drums. For the short Bluegrass set, Byron Berline (fiddle) and Roger Bush (upright bass) came out on stage and helped out.
"Christine's Tune (aka Devil in Disguise)" (Chris Hillman, Gram Parsons) – 3:54 – A great tune from the band's debut album with plenty of tasty pedal steel licks from Perkins.
"Six Days on the Road" (Earl Green, Carl Montgomery) – 3:03 – The boys do their rendition of this classic truck driving ditty. The song had been a #2 country hit for Dave Dudley in 1963 and was a staple in Burrito concerts.
"My Uncle" (Chris Hillman, Gram Parsons) – 2:20 – Another song from the band's debut; a lament about being drafted during the Vietnam War.
"Dixie Breakdown" (Jimmie Lunceford, Don Reno) – 2:17 – With this tune and the following two, Hillman and the Burritos add Berline and Bush to the mix and do it up Bluegrass style. This is a classic 1958 bluegrass song from Don Reno. Hillman contributes an excellent mandolin solo.
"Don't Let Your Deal Go Down" (Louise Certain, Gladys Stacey Flatt, Jerry Organ, Wayne Walker) – 2:20 – Roger Bush does a nice job on lead vocals on this 1925 classic. It doesn't get much "rootsier" than this.
"Orange Blossom Special" (Ervin T. Rouse) – 3:39 – Byron Berline and his fiddle shine on this 1938 classic. I thoroughly enjoyed these three Bluegrass numbers and it's clear from the live recording that the audience did as well.
"Ain't That a Lot of Love" (Homer Banks, Deanie Parker) – 3:20 – The Burritos deftly put a country spin on this 1966 Stax-Memphis, driving R&B soul tune. Yeah, Al Perkins could play.
"High Fashion Queen" (Chris Hillman, Gram Parsons) – 3:22 – From their "Burritos Deluxe" album.
"Don't Fight It" (Wilson Pickett, Steve Cropper) – 2:56 – Another countrification of an R&B tune, this one written by Wilson Pickett in 1965.
"Hot Burrito #2" (Chris Ethridge, Gram Parsons) – 4:35 – One of the best songs from the band's debut LP.
"Losing Game" (James Carr, Dennis Weaver) – 2:59 – Hillman struggles a bit with his vocals on this R&B song written by Memphis soul artist, James Carr, in 1967.
Postscript: Back when album covers were considered works of art, this LP cover was a simply a thing of beauty. Credit goes to famous album cover illustrator, Joe Garnett. Being the completist that I am, I'm going to have to review the Burrito's 2nd and 3rd studio LPs somewhere down the road.
Posted in Music, The ByrdsTagged Music
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9 thoughts on "When the Flying Burrito Brothers introduced me to Bluegrass music"
I have probably heard that band before. I grew up listening to "The Ozark Mountain Daredevils" vinyl records with my Mom. They are more southern rock, but a lot of their stuff is bluegrass. Always liked bluegrass music. God bless!
Thanks, Ryan! I really did enjoy "Jackie Blue" from the OMDs back in 1975.
I love bluegrass, I have a big family in Mississippi that I've only met once that travel and play concerts. It's really neat.
Thanks, sister. I appreciate your mentioning your family bluegrass connection. It's definitely a foot-tapping, smiley-face kind of music. When a band is tight and all pulling in the same direction at a fast clip, it's an amazing thing to hear.
I learn of this band from you! What a name! The song "My Uncle" sounds like something I want to listen to. Waiting for my house internet to be back to normal, using my phone's data plan right now.
I think you would actually find "My Uncle" interesting as an example of the rising anti-Vietnam sentiment of the youth culture in 1968 when the song was written. Gram Parsons, who wrote the song after receiving his draft notice in the mail, was the absolute last person the U.S. Army would have wanted in its ranks.
Wow I really got to listen to this tonight when hopefully my home internet works! Don't want to burn my data with my phone.
Here's a link to the version that was on the live album I just reviewed:

redpajama_set_name: RedPajamaCommonCrawl


Almost three months passed since the last update. Not impressive, we admit. But the Machine, dear machine has not stopped running at all.
We are starting to work on a 7″ release, but since we are busy folks, to put it mildly, the release date (or even the release year) are still very uncertain :-). But whenever sketches start showing up we will start sharing it.
Meanwhile 'Killing something that's already dead' are still alive and kicking. Regular post and tweets from all over the world keep popping up on the web and since the release in mid January tracks from the album has been streamed on average 20 times pr day from Spotify and Bandcamp alone. By our standards that is amazing and we are the most greatful guys ever!
And in addition there might be a remix of 'Get out' in the making. We will keep you posted whenever that is ready.


0%67°43°Cloudy with a high of 67 °F (19.4 °C). Winds variable at 3 to 8 mph (4.8 to 12.9 kph).Night - Partly cloudy. Winds variable. The overnight low will be 45 °F (7.2 °C).
45%75°51°Scattered showers today with a high of 75 °F (23.9 °C) and a low of 51 °F (10.6 °C). There is a 45% chance of precipitation.
20%70°42°Thunderstorms today with a high of 70 °F (21.1 °C) and a low of 42 °F (5.6 °C).
15%74°53°Partly cloudy today with a high of 74 °F (23.3 °C) and a low of 53 °F (11.7 °C).
Today - Cloudy with a high of 67 °F (19.4 °C). Winds variable at 3 to 8 mph (4.8 to 12.9 kph).


text:
\section{Introduction}
\label{sec:intro}

Let $W$ be a finite Coxeter group, generated by a set of simple
reflections $S$ with length function $\ell \colon W \to \mathbb{N}\cup\{0\}$.  Each
subset $J \subseteq S$ generates a
so-called standard parabolic
subgroup $W_J = \Span{J}$ of $W$.
Conjugates of standard parabolic subgroups are called parabolic
subgroups.  These subgroups are themselves Coxeter groups and
therefore play an important role in the structure theory of finite
Coxeter groups.  A well-known property of the cosets of a standard
parabolic subgroup $W_J$ in $W$ is that each coset contains a unique
element of minimal length.  The subgroup $W_J$ hence
possesses a
distinguished right transversal $X_J$, consisting of the
minimal length coset representatives.
Due to a theorem of Howlett~\cite{Howlett1980} and later work of Brink
and Howlett~\cite{BrinkHowlett1999}, it is known that and how the
normalizer $N_W(W_J)$ of the parabolic subgroup $W_J$ is a semidirect
product of $W_J$ and a subgroup $N_J$ consisting of precisely those
minimal length coset representatives $x \in X_J$ which leave $J$ as
subset of $W$ invariant in the conjugation action of $W$ on its
subsets, i.e., $N_J = \{x \in X_J: J^x = J\}$.

In this note we show that most centralizers of elements in $W$ enjoy a
similar semidirect product decomposition.  Pfeiffer and
R\"ohrle~\cite{PfeifferRoehrle2005} have shown, based on
Richardson's~\cite{Richardson1982} characterization of involutions as
central longest elements of parabolic subgroups of $W$, that if $w \in
W$ is an involution then its centralizer in $W$ coincides with the
normalizer of a parabolic subgroup, and as such is a semidirect
product.  This note can be regarded as a generalization of the result
for involutions to all elements of~$W$.  Our results effectively
reduce questions regarding the conjugacy classes of elements in a
finite Coxeter group $W$ to the cuspidal conjugacy classes, that is
those conjugacy classes which are disjoint from any proper parabolic
subgroup of~$W$.  Cuspidal conjugacy classes of elements of $W$ play a
central role in the algorithmic approach to the conjugacy classes of
finite Coxeter groups in Chapter~3 of the book by Geck and
Pfeiffer~\cite{GePf2000}.  We refer the reader to this book as a
general introduction to the theory of finite Coxeter groups.

We will call certain conjugacy classes of elements of a finite Coxeter
group $W$ \emph{non-compliant}; see Definition~\ref{def:non-compliant}.
  Without exception, these are
conjugacy classes of $W$ which nontrivially intersect a parabolic
subgroup of type $D_n$ with $n > 4$.  Hence, if $W$ has no parabolic
subgroups of type $D$, part~(ii) of the following theorem applies
without restrictions. We can now formulate our main theorem as follows.

\begin{Theorem} \label{thm:1} Let $W$ be a finite Coxeter group and
  let $w \in W$.  Let $V$ be the smallest parabolic subgroup of $W$
  that contains $w$.  Then the following hold.
\begin{enumerate}
\item The centralizer $C_V(w) = C_W(w) \cap V$ is a normal
  subgroup of the centralizer $C_W(w)$ with quotient $C_W(w)/C_V(w)$
  isomorphic to the normalizer quotient $N_W(V)/V$.
\item The centralizer $C_W(w)$ splits over $C_V(w)$ with complement
  isomorphic to $N_W(V)/V$ unless $w$ lies in a non-compliant conjugacy
  class of elements of~$W$.
\end{enumerate}
\end{Theorem}

The parabolic subgroup $V$ in the theorem is well-defined as the
intersection of all parabolic subgroups of $W$ that contain $w$, due
to the fact that intersections of parabolic subgroups are parabolic
subgroups, see Theorem~\ref{2.1.12} below.  For the proof of
the theorem, we will assume that $w$ has \emph{minimal length} in its
conjugacy class in $W$.  Then $V$ is the \emph{standard} parabolic
subgroup $W_J$ of $W$, where $J = J(w)$, the set of generators
occurring in a reduced expression for $w$.  The proof of part~(i) is
carried out in Section~\ref{sec:centralizers}.  Part~(ii) of the
theorem is established case by case in
Section~\ref{sec:cases}.  The results for the exceptional types of
Coxeter groups have been obtained with the help of computer programs
using the GAP~\cite{GAP} package CHEVIE~\cite{chevie}.  These programs
are available through the second author's ZigZag~\cite{zigzag} package.
In Section~\ref{sec:application}, we use Theorem~\ref{thm:1} to
provide a new short proof of a theorem of Solomon, and then discuss
its relation to MacMahon master theorem~\cite[page 98]{macmahon}.

\section{Preliminaries.}
\label{sec:prelim}

In this section we recall some results about distinguished coset
representatives and conjugacy classes in a finite Coxeter group $W$,
generated by a set of simple reflections $S$ and with length function
$\ell$.

For $w \in W$, we set $J(w) = \{s_1, \dots, s_l\} \subseteq S$, if $w
= s_1 \dots s_l$ is a reduced expression, i.e., if $l = \ell(w)$.
As a consequence of Matsumoto's theorem, $J(w)$ does not depend
on the choice of a reduced expression for~$w$.

For $w \in W$, let
\begin{align*}
  \mathcal{D}(w) = \{s \in S : \ell(sw) < \ell(w)\}
\end{align*}
be its \emph{descent set}, and
let
\begin{align*}
   \mathcal{A}(w) = \{s \in S : \ell(sw) > \ell(w)\} = S \setminus \mathcal{D}(w)
\end{align*}
be its \emph{ascent set}.
The set
\begin{align*}
  X_J = \{ w \in W : J \subseteq \mathcal{A}(w) \}
\end{align*}
is a right transversal for $W_J$ in $W$, consisting of the
elements of minimal length in each coset.  For each element $w \in W$
there are unique elements $u \in W_J$ and $x \in X_J$ such that $w = u
\cdot x$.  Here the explicit multiplication dot indicates that the
product $ux$ is \emph{reduced}, i.e., that $\ell(ux) = \ell(u) +
\ell(x)$.  An immediate consequence is the following lemma.

\begin{Lemma}[\protect{\cite[Lemma 2.1.14]{GePf2000}}]
  \label{2.1.14}
  Let $J \subseteq S$.  Then $\ell(w^x) \geq \ell(w)$ for all $w \in
  W_J$, $x \in X_J$.
\end{Lemma}

We denote the longest element of $W$ by $w_0$.
For $J \subseteq  S$, we denote by $w_J$ the  longest element of the
parabolic subgroup $W_J$.

\begin{Lemma}
  \label{1.5.2}
  Let $w \in W$ and let $J = \mathcal{D}(w)$.  Then $w = w_J \cdot x$ for some
  $x \in X_J$.
\end{Lemma}

\begin{proof}
  This follows from \cite[Lemma 1.5.2]{GePf2000} and
\cite[Proposition 2.1.1]{GePf2000}
\end{proof}

For $J,K\subseteq  S$ define  $X_{JK} = X_J \cap  X_K^{-1}$.  Then
$X_{JK}$ is  a set of  minimal length double coset  representatives of
$W_J$ and $W_K$ in $W$.

\begin{Theorem}[\protect{\cite[Theorem 2.1.12]{GePf2000}}]
  \label{2.1.12}
  Let $J, K \subseteq S$ and  let $x \in X_{JK}$. Then $W_J^x \cap W_K
  = W_L$, where $L = J^x \cap K$.
\end{Theorem}

\begin{Theorem}[\protect{\cite[Theorem 2.3.3]{GePf2000}}]
  \label{2.3.3}
  Suppose $J, K$ are conjugate subsets of $S$ and that $x \in X_J$ is
  such that $J^x = K$.  If $s \in \mathcal{D}(x)$ then $x = d \cdot y$, where
  $d = w_J w_L$ for $L= J \cup \{s\}$, and $y \in X_L$.
\end{Theorem}

For the conjugacy classes of $W$, we are particularly interested in
elements of minimal length.  These elements have useful properties,
such as the following.

\begin{Proposition}[\protect{\cite[Corollary 3.1.11]{GePf2000}}]
  \label{3.1.11}
  Let $C$ be a conjugcay class of $W$ and let $w, w'$ be elements of
  minimal length in $C$.  Then $J(w') = J(w)^x$ for some $x \in
  X_{J(w), J(w')}$.
\end{Proposition}

A conjugacy class $C$ of elements of $W$ is called a \emph{cuspidal class} if $C \cap W_J = \emptyset$ for all proper subsets $J$ of $S$.
Cuspidal classes never fuse in the following sense.

\begin{Theorem}[\protect{\cite[Theorem 3.2.11]{GePf2000}}]
  \label{3.2.11}
  Let $J \subseteq  S$ and let $w \in W_J$ be  such that the conjugacy
  class $C_J$ of $w$ in $W_J$ is cuspidal in $W_J$.  Then
\[
C_J = C \cap W_J,
\]
where $C$ is the conjugacy class of $w$ in $W$.
\end{Theorem}

If $w$ is of minimal length in its conjugacy class, then it is also of
minimal length in its conjugacy class in the Coxeter group $W_{J(w)}$,
which by \cite[Proposition 3.2.12]{GePf2000} is a cuspidal class of
$W_{J(w)}$

Below, we review some basic facts about Coxeter groups of classical
type, that is of type $A$, $B$ or $D$.  For a more detailed review of
the combinatorics of the conjugacy classes of finite Coxeter groups of
classical type we refer the reader to the
description~\cite{Pfeiffer1994} of the implementation of the character
tables of these groups in GAP.

\subsection{Type \texorpdfstring{$A$}{A}.}
\label{sec:intro-a}

Suppose $W$ is a Coxeter group of type
$A_{n-1}$.  Then $W$ is isomorphic to the symmetric group $\mathfrak{S}_n$ on
the $n$ points $[n] = \{1, \dots, n\}$, with Coxeter generators $s_i =
(i, i+1)$, $i = 1, \dots, n-1$.  The \emph{cycle type} of a
permutation $w \in W$ is the partition of $n$, which contains a part
$l$ for each $l$-cycle of $w$, where fixed points count as $1$-cycles.
Since any two elements of $w$ are conjugate in $W$ if and only if they
have the same cycle type, the conjugacy classes of elements of $W$ are
naturally parametrized by the partitions of $n$.

Here, it will be convenient to write partitions as weakly \emph{increasing}
sequences.  Given a partition $\lambda =
(\lambda_1, \dots, \lambda_t)$ of $n$ (that is a sequence of positive
integers $\lambda_1 \leq \dots \leq \lambda_t$ with $\lambda_1 + \dots
+ \lambda_t = n$), there is a corresponding parabolic subgroup $W_J =
\mathfrak{S}_{\lambda_1} \times \dots \times \mathfrak{S}_{\lambda_t}$ containing an
element $w$ with cycle type $\lambda$.  A particular element of
minimal length in this conjugacy class is
the product $w_{\lambda}$ of $t$ disjoint cycles
consisting of $\lambda_i$ successive points, for $i = 1, \dots, t$.
For example, a minimal length representative of the conjugacy class
of elements with cycle structure
$1124$  in $\mathfrak{S}_8$ is $w_{1124} = (1)(2)(3,4)(5,6,7,8) = (3,4)(5,6,7,8)$.

Note that $w_{\lambda}$ is a Coxeter element of $W_J$,
the product
\begin{align} \label{eq:wa}
  w_{\lambda} = \prod_{s_i \in J} s_i
\end{align}
(in \emph{decreasing} order) of all $s_i \in J$. For example,
$J(w_{1124}) = \{s_3, s_5, s_6, s_7\}$, and $w_{1124} = s_7 s_6 s_5
s_3$.

\subsection{Type \texorpdfstring{$B$}{B}.}
\label{sec:intro-b}

Suppose $W$ is a Coxeter group of type $B_{n}$.
Then $W$ is isomorphic to the group of permutations on $\{-n,\dots,-1,0,1, \dots, n\}$
satisfying $w(-i) = - w(i)$. Alternatively, we can represent this group
as the group of signed permutations, i.e. injective maps from $[n]$ to
$[n] \cup -[n]$ with precisely one of $i$ and $-i$ in the image. Since the elements
are permutations, we can write them in cyclic form. We have two types of cycles:
cycles which do not contain $i$ and $-i$ for any $i$, and cycles in which $i$ is
an element if and only if $-i$ is an element. Cycles of the first type come in
natural pairs, and instead of $(i_1,i_2,\dots,i_k)(-i_1,-i_2,\dots,-i_k)$, we write
$(i_1,i_2,\dots,i_k)$ and call it a positive cycle. Cycles of the second type are
of the form $(i_1, i_2 \dots,i_k,-i_1,-i_2,\dots,-i_k)$. We shorten that to
$(i_1,i_2,\ldots,i_k)^-$ and call it a negative cycle. For example, the permutation
$$-4 \mapsto -2, \: -3 \mapsto 1, \: -2 \mapsto 4, \: -1 \mapsto 3, \: 0 \mapsto 0, \: 1 \mapsto -3, \: 2 \mapsto -4, \: 3 \mapsto -1, \: 4 \mapsto 2$$
is written as $(1,-3)(2,-4)^-$. In this notation, every
signed permutation looks like an ordinary permutation in cyclic form, except that every element
and every cycle can have a minus sign. Note that we can change the sign of all elements
in a cycle without changing the signed permutation. The Coxeter generators are
$t_1 = (1)^-$ and $s_i = (i,i+1)$, $i = 1, \dots, n-1$.
We also set $t_i = (i)^-$, for $i > 1$.

An element $w \in W(B_n)$ can also be represented in the form of a
signed permutation matrix.  This is an $n \times n$ matrix which acts
on the standard basis $\{e_1, \dots, e_n\}$ of $\mathbb{R}^n$ in the same way
as the permutation $w$ acts on the points $[n] = \{1, \dots, n\}$, i.e., for $i \in [n]$, it
maps $e_i$ to $e_{\Size{w(i)}}$ or its negative, depending on whether
  $w(i)$ is positive or negative.  We will briefly use this matrix
  representation of $W(B_n)$ in Section~\ref{sec:n-c}.

Since conjugation on a signed permutation in cyclic form works in the same way as
with usual permutations (if we conjugate with $w$, an element $i$ of any cycle is
replaced by $w(i)$), two signed permutations are conjugate if and only if
they have the same number of negative cycles of every length, and the
same number of positive cycles of every length. The \emph{cycle type} of a permutation $w \in W$ is a double partition
$\lambda = (\lambda^+,\lambda^-)$ with $\Size{\lambda^+} + \Size{\lambda^-} = n$,
so that $\lambda^+$ contains a part $l$ for each
positive $l$-cycle of $w$, and $\lambda^-$ contains a part $l$ for each
negative $l$-cycle of $w$. Two elements of $W$ are conjugate in $W$ if and only if they have the
same cycle type, and therefore the conjugacy classes of elements of $W$ are
naturally parametrized by the double partitions of $n$. For example, the conjugacy
class of $(1,5,-2)(4,7)(3)^-(6,-8)^-$ is $(21,32)$.

Take $J \subseteq \{t_1,s_1,s_2,\ldots,s_{n-1}\}$ and $w \in W_J$. If $s_i \notin J$, the elements
$i+1,\ldots,n$ appear in positive cycles of $w$ with all positive elements. Therefore, if we are given a double partition
$(\lambda^+,\lambda^-)$ of $n$, $\lambda^+ = (\lambda_1^+, \dots, \lambda_t^+)$,
$\lambda^- = (\lambda_1^-, \dots, \lambda_s^-)$, the smallest parabolic
subgroup $W_J$ that contains an element of cycle type $(\lambda^+,\lambda^-)$ is of the form $W(B_{\Size{\lambda^-}}) \times \mathfrak{S}_{\lambda_1^+}
\times \dots \times \mathfrak{S}_{\lambda_t^+}$.
According to the description \cite[3.4.2]{GePf2000}
of conjugacy classes of $W$,
there is a minimal length representative
$w_{\lambda}$
of the corresponding conjugacy class of the following form. The negative cycles
contain $1,\dots,\Size{\lambda^-}$, and the positive cycles contain $\Size{\lambda^-}+1,\dots,n$;
furthermore, each cycle contains only consecutive numbers in increasing order. For example, a minimal length
representative of the conjugacy class corresponding to $\lambda = (112,23)$ is
$w_{\lambda} = (1,2)^-(3,4,5)^-(6)(7)(8,9)$.

\subsection{Type \texorpdfstring{$D$}{D}.}
\label{sec:intro-d}

Suppose $W$ is a Coxeter group of type $D_{n}$.  Then $W$ is
isomorphic to the group of permutations on $\{-n,\dots,-1,0,1, \dots,
n\}$ satisfying $w(-i) = - w(i)$ and with an even number of $i>0$
satisfying $w(i)<0$. Alternatively, we can represent this group as the
group of signed permutations with an even number of $i$ mapping to
$-[n]$. These are precisely the signed permutations with an even number
of negative cycles. The Coxeter generators are $u = (1, -2)$ and $s_i
= (i,i+1)$, $i = 1, \dots, n-1$.  The \emph{cycle type} of a
permutation $w \in W$ is a double partition $(\lambda^+,\lambda^-)$
with $\Size{\lambda^+} + \Size{\lambda^-} = n$, so that $\lambda^+$ contains a
part $l$ for each positive $l$-cycle of $w$, and $\lambda^-$ contains
a part $l$ for each negative $l$-cycle of $w$. If two elements of $W$
are conjugate, they have the same cycle type. Having the same cycle type,
however, is not a sufficient condition for conjugacy. For example, $u$ and $s_1$
have the same cycle type but are not conjugate. It is easy to see that if
they have the same cycle type $(\lambda^+,\lambda^-)$ and $\Size{\lambda^-} > 0$ or
$\lambda^+$ contains an odd part, they are conjugate. If they have the
same cycle type $(\lambda^+,\emptyset)$, where $\lambda^+$ contains
only even parts, they are conjugate if and only if the number of
negative numbers in their cycle decomposition has the same parity.

We call a partition \emph{even} if it consists of even parts only.
The conjugacy classes of elements of $W$ are naturally parametrized by
double partitions of $n$, where $\lambda^-$ has an even number of
parts, with two classes when $\lambda^- = \emptyset$ and $\lambda^+$
is even.  Given a double partition $(\lambda^+,\lambda^-)$ of $n$,
$\lambda^+ = (\lambda_1^+, \dots, \lambda_t^+)$, $\lambda^- =
(\lambda_1^-, \dots, \lambda_s^-)$, $s$ even, the corresponding
parabolic subgroup $W_J$ is of the form $W(D_{\Size{\lambda^-}}) \times
\mathfrak{S}_{\lambda_1^+} \times \dots \times \mathfrak{S}_{\lambda_t^+}$.
Furthermore, there is a minimal length representative $w_{\lambda}$ of the
corresponding conjugacy class of the following form. The negative
cycles contain $1,\dots,\Size{\lambda^-}$, the positive cycles contain
$\Size{\lambda^-}+1,\dots,n$, and each cycle contains only
consecutive numbers in increasing order. If $\lambda^- = \emptyset$
and $\lambda^+$ has only even parts, then there is an extra representative
$w_{\lambda}'$ with the first positive cycle starting with $-1$ instead of $1$.  For
example, for $\lambda = (112,23)$, we have $w_{\lambda} = (1,2)^-(3,4,5)^-(6)(7)(8,9)$, and
for $(224,\emptyset)$, we have $w_{\lambda} = (1,2)(3,4)(5,6,7,8)$ and
$w_{\lambda}' = (-1,2)(3,4)(5,6,7,8)$.

\section{Centralizers.}
\label{sec:centralizers}

In this section we prove a general theorem about the structure of
element centralizers in finite Coxeter groups.  It is shown to be a
consequence of Theorem~\ref{3.2.11}, which in the book~\cite{GePf2000}
 has
been established by a careful case-by-case analysis.  Without loss of
generality, we may assume that $w \in W$ is an element of minimal
length in its conjugacy class in $W$.

\begin{Theorem} \label{thm:2}
  Suppose $w \in W$ has minimal length in its
  conjugacy class in $W$ and let $J = J(w)$. Then $C_W(w)W_J = N_W(W_J)$.
\end{Theorem}

Clearly, part (i) of Theorem~\ref{thm:1} follows from this result.

\begin{proof}
Denote by $C$ the conjugacy class of $w$ in $W$ and by $C_J$ its
conjugacy class in $W_J$, which is cuspidal in $W_J$.
By Theorem~\ref{3.2.11}, $C_J = C \cap W_J$, which implies
that for every $x\in N_W(W_J)$ there exists an element $u \in W_J$ with
$w^{x} = w^u$. So $xu^{-1}\in C_W(w)$, i.e. $x\in C_W(w)W_J$, and hence
$N_W(W_J)\subseteq C_W(w)W_J$.

Now it only remains to show that $C_W(w)\subseteq N_W(W_J)$.  Let $y
\in C_W(w)$ and write it as $y = u x v^{-1}$ for $u, v \in W_J$ and a
double coset representative $x \in X_{JJ}$.  From $w \in C_J \cap
C_J^y$ it then follows that $w^v \in C_J \cap C_J^x \subseteq W_J \cap
W_J^x = W_{J \cap J^x}$, by Theorem~\ref{2.1.12}, and since
$C_J$ is a cuspidal class in $W_J$, we must have $J \cap J^x = J$,
whence $x \in N_J \subseteq N_W(W_J)$ and thus $y = u x v^{-1} \in
N_W(W_J)$.
\end{proof}

The following additional results are of independent interest
and will be used in the proof of Theorem~\ref{thm:3} below.

\begin{Proposition} \label{prop:1}
  Suppose $w \in W$ has minimal length in its conjugacy class in $W$
  and let $J = J(w)$.  Then the following hold.
\begin{enumerate}
\item The conjugacy class of $w$ in $W$ is a disjoint union of
  conjugates of the conjugacy class of $w$ in $W_J$.
\item If $a \in C_W(w)$ and $x \in X_J$ are such that $C_{W_J}(w) a
  \subseteq W_J x$, then $x \in N_J$.
\item If $x \in X_J$ is such that $\ell(w^x) = \ell(w)$, then $J(w^x) =
  J^x$.
\item If $v \in W$ is such that $\ell(w^v) = \ell(w)$, then $J(w^x) =
  J^x$, where $v = u \cdot x$ with $u \in W_J$ and $x \in X_J$.
\end{enumerate}
\end{Proposition}
\begin{proof}
  (i) and (ii) follow from the proof of Theorem~\ref{thm:2}.

  (iii) Let $K = J(w^x)$.  By Proposition~\ref{3.1.11}, there exists
  an element $y \in X_{KJ}$ such that $K^y = J$.  Hence $w^{xy} \in
  W_J$ and $\ell(w^{xy}) = \ell(w^x) = \ell(w)$ and (since $C \cap W_J
  = C_J$)  $w^{xy} = w^u$ for some $u \in W_J$.  Moreover, $X_K =
  y X_J$. Hence $u^{-1}xy$ centralizes $w^u$, and if we write $u^{-1}xy
  = a \cdot z$ for $a \in W_J$ and $z \in X_J$ then, by (ii), $z \in
  N_J$.  It follows that $z y^{-1} \in X_{JK}$ is the unique
  minimal length representative of the coset $W_Jx$, hence $x = z
  y^{-1}$ and $J^x = J(w^x)$.

  (iv) We have $w^v = (w^u)^x$. Conjugation with $x$ does not decrease the length
  (Lemma~\ref{2.1.14}), so $\ell(w) = \ell(w^v) \geq \ell(w^u)$ and
  therefore $\ell(w^u) = \ell(w)$. By (iii), with $w$ replaced by $w^u$, we have
  $J(w^v) = J(w^u)^x$, and $J(w^u) = J(w)$, which finishes the proof.
\end{proof}

\section{Complements.}
\label{sec:cases}

In this section we prove part (ii) of Theorem~\ref{thm:1} for each type of
irreducible finite Coxeter group, case by case.
We start with the general observation that part (ii) of the theorem is
straightforward in the following situations.

\begin{Lemma}
  \label{la:cori}
  Let $w \in W$ be an element of minimal length in its conjugacy class
  in $W$, and let $J = J(w)$.  If $w$ is cuspidal in $W$ or if
  $C_{W_J}(w) = W_J$ then $N_J$ is a complement of $C_{W_J}(w)$ in $C_W(w)$.
\end{Lemma}

\begin{proof}
  If $w$ is cuspidal then $W_J = W$ and both quotients $N_W(W_J)/W_J$
  and $C_W(w)/C_{W_J}(w)$ are trivial.

  If $C_{W_J}(w) = W_J$ then
$w = w_J$ and
$C_W(w) = N_W(W_J)$~\cite[Proposition 2.2]{PfeifferRoehrle2005}.
\end{proof}

Our general strategy in search of a centralizer complement for $w$ will
be to identify a complement $M$ of $W_J$ in its normalizer that
centralizes $w$.  More precisely, we have the following consequence
of Theorem~\ref{thm:2}.

\begin{Proposition}
  \label{pro:M}
  Let $w \in W$ be of minimal length in its conjugacy class, let $J =
  J(w)$ and suppose that the normalizer complement $N_J$ is generated
  by elements $x_1, \dots, x_r$.  Let $u_1, \dots, u_r \in W_J$ be
  such that $u_i x_i \in C_W(w)$, $i = 1, \dots, r$, and set $M =
  \langle u_1 x_1, \dots, u_r x_r\rangle$.  Then $M$ is a complement
  of $C_{W_J}(w)$ in $C_W(w)$ provided that $M \cap W_J = 1$.
\end{Proposition}

\begin{proof}
  Clearly, $W_J M = W_J N_J = N_W(W_J)$.  From $M \cap W_J = 1$ it
  then follows that $M$ is a complement of $W_J$ in its normalizer.
  Moreover, $M$ is a subgroup of $C_W(w)$ since each of its generators
  centralizes~$w$.  From Theorem~\ref{thm:2} it then follows that $M$
  is a complement of $C_{W_J}(w)$ in $C_W(w)$.
\end{proof}

\subsection{Type \texorpdfstring{$A$}{A}.}
\label{sec:a}

Let $\lambda = (1^{a_1}, 2^{a_2}, \dots, n^{a_n}) $ be a partition of $ n$,
let $w_{\lambda}$ be as in (\ref{eq:wa}) and let $J = J(w_{\lambda})$.
Then $W_J$ is a direct product
\begin{align*}
  W_J = \mathfrak{S}_1^{a_1} \times \mathfrak{S}_2^{a_2} \times \dots \times \mathfrak{S}_n^{a_n}
\end{align*}
of symmetric groups, and its normalizer
\begin{align*}
  N_W(W_J) = \mathfrak{S}_1 \wr \mathfrak{S}_{a_1} \times
\mathfrak{S}_2 \wr \mathfrak{S}_{a_2} \times
\dots \times \mathfrak{S}_n \wr \mathfrak{S}_{a_n}
\end{align*}
is a direct product of wreath products of symmetric with symmetric groups.  In a similar way,
the centralizer
\begin{align*}
  C_W(w_{\lambda}) = C_1 \wr \mathfrak{S}_{a_1} \times C_2 \wr \mathfrak{S}_{a_2} \times \dots \times C_n \wr \mathfrak{S}_{a_n}
\end{align*}
is a direct product of wreath products of cyclic with symmetric groups,
and the centralizer
\begin{align*}
  C_{W_J}(w_{\lambda}) = C_1^{a_1} \times C_2^{a_2} \times \dots \times C_n^{a_n}.
\end{align*}
is a direct product of cyclic groups.  Clearly, the quotients
$N_W(W_J)/W_J$ and $C_W(w_{\lambda})/C_{W_J}(w_{\lambda})$ are both
isomorphic to $\mathfrak{S}_{a_1} \times \mathfrak{S}_{a_2} \times \dots \times
\mathfrak{S}_{a_n}$.  In order to show that the particular normalizer
complement $N_J$ is also a complement of $C_{W_J}(w_{\lambda})$ in
$C_W(w_{\lambda})$, we introduce some notation.  Let us define as
\begin{align} \label{eq:sss}
s(o,m)
= (s_{o+1} s_{o+2} \dotsm s_{o+2m-1})^m
= (o+1,o+2, \dots, o+2m)^m
\end{align}
the permutation that swaps, after an offset $o$, two adjacent blocks
of $m$ points $\{o+1, \dots, o+m\}$ and $\{o+m+1, \dots, o+2m\}$.
For example, $s(2,3) = (s_3 s_4 s_5 s_6 s_7)^3 = (3, 4, 5, 6, 7, 8)^3 = (3,6)(4,7)(5,8)$.  And $s_i = s(i-1, 1)$.
Then
\begin{align*}
  N_J = \mathfrak{S}_{a_1} \times \mathfrak{S}_{a_2} \times \dots  \times \mathfrak{S}_{a_n}
\end{align*}
is a direct product of symmetric groups
$\mathfrak{S}_{a_m}$, with Coxeter generators
$s(o_m, m)$,
$s(o_m{+}m,m)$,    \dots,
$s(o_m{+}(a_m{-}2)m, m)$,
and offsets
\begin{align} \label{eq:om-a}
  o_m = a_1 + 2 a_2 + \dots + (m-1) a_{m-1},
\end{align}
for those $m \in \{ 1, \dots, n\}$ with $a_m > 0$.

\begin{Proposition}
  \label{pro:a}
  Let $\lambda $ be a partition of $ n$, let $w_{\lambda}$ be the permutation
  with cycle structure $\lambda$ from (\ref{eq:wa})  and let $J =
  J(w_{\lambda})$ be the corresponding subset of $S$.
Then $N_J$ is a complement
  of $C_{W_J}(w_{\lambda})$ in $C_W(w_{\lambda})$.
\end{Proposition}

\begin{proof}
  It suffices to consider the case $\lambda = (m^a)$ since all of
  $W_J$, $N_W(W_J)$, $C_W(w_{\lambda})$, $C_{W_J}(w_{\lambda})$, and
  $N_J$ are subgroups of the direct product
  \begin{align*}
    \mathfrak{S}_{1a_1} \times \mathfrak{S}_{2a_2} \times \dots \times \mathfrak{S}_{n a_n}
  \end{align*}
  inside $\mathfrak{S}_n$ and one can argue componentwise.

  If $\lambda = (m^a)$, then $N_J$ is isomorphic to $\mathfrak{S}_a$, with
  $a-1$ Coxeter generators $s(o,m)$, $s(m,m)$, \dots, $s((a{-}2)m,m)$,
  permuting the blocks of $m$ points
  \begin{center}
    $\{1, \dots, m\}$, $\{m{+}1, \dots, 2m\}$, \ldots,
    $\{(a{-}1)m{+}1,\dots, am\}$.
  \end{center}
  Clearly
  \begin{align*}
    w_{\lambda} = (1, \dots, m)(m{+}1, \dots, 2m) \dotsm
    ((a{-}1)m{+}1,\dots, am)
  \end{align*}
  is centralized by $N_J$.
  The claim now follows from Theorem~\ref{thm:2}.
\end{proof}

\subsection{Type \texorpdfstring{$B$}{B}.}
\label{sec:b}

Let $\lambda $ be a double partition of $ n$ with $\lambda^+ = (1^{a_1}, 2^{a_2}, \dots, n^{a_n})$ and $\lambda^- = (1^{b_1}, 2^{b_2}, \dots, n^{b_n})$,
let $w_{\lambda}$ be as in Section~\ref{sec:intro-b},
and let $J = J(w_{\lambda})$ be the corresponding subset of $S$.
Then $W_J$ is a direct product
\begin{align*}
  W_J = W(B_{\Size{\lambda^-}}) \times \mathfrak{S}_1^{a_1} \times \mathfrak{S}_2^{a_2} \times \dots \times \mathfrak{S}_n^{a_n}
\end{align*}
and its normalizer
\begin{align*}
  N_W(W_J) = W(B_{\Size{\lambda^-}}) \times \mathfrak{S}_1 \wr W(B_{a_1})
\times \mathfrak{S}_2 \wr W(B_{a_2}) \times \dots
\times \mathfrak{S}_n \wr W(B_{a_n})
\end{align*}
is a direct product of $W(B_{\Size{\lambda^-}})$
with wreath products of symmetric groups and
Coxeter groups of type $B$.
In a similar way, the centralizer
\begin{align*}
  C_W(w_{\lambda}) =
C_{W(B_{\Size{\lambda^-}})}(w_{\lambda})
\times C_1 \wr W(B_{a_1}) \times C_2 \wr W(B_{a_2}) \times \dots \times C_n \wr
  W(B_{a_n})
\end{align*}
is a direct product of $C_{W(B_{\Size{\lambda^-}})}(w_{\lambda})$ and  wreath products, and the centralizer
\begin{align*}
C_{W_J}(w_{\lambda}) =
  C_{W(B_{\Size{\lambda^-}})}(w_{\lambda}) \times
  C_1^{a_1} \times C_2^{a_2} \times \dots \times C_n^{a_n}
\end{align*}
is a direct product of $C_{W(B_{\Size{\lambda^-}})}(w_{\lambda})$
and cyclic groups.
Clearly, the quotients
$N_W(W_J)/W_J$ and $C_W(w_{\lambda})/C_{W_J}(w_{\lambda})$ are both
isomorphic to $W(B_{a_1}) \times W(B_{a_2}) \times \dots \times
W(B_{a_n})$.  In order to show that a variant of the particular normalizer
complement $N_J$ is a complement of $C_{W_J}(w_{\lambda})$ in
$C_W(w_{\lambda})$, we introduce some more notation.

Denote by $r(o,m)$ the  permutation
defined by
\begin{align*}
  x.r(o,m) =
  \begin{cases}
    2o+m+1-x, & \text{if } o+1 \leq x \leq o+m, \\
    x, & \text{otherwise.}
  \end{cases}
\end{align*}
In this way, $r(o,m) = (o+1,o+m)(o{+}2, o{+}m{-}1) \dotsm $ reverses the range
$\{o+1, \dots, o+m\}$ and thus is the longest element of the
symmetric group $\mathfrak{S}_{\{o+1, \dots, o+m\}}$
with Coxeter generators $s_{o+1}, \dots, s_{o+m-1}$.
For example, $r(2,5) = (3,7)(4,6)$.

Moreover, denote
\begin{align*}
  t(o, m) = (o{+}1)^- (o{+}2)^- \dotsm (o{+}m)^-,
\end{align*}
which acts as $-1$ on the
points $\{o+1, o+2, \dots, o+m\}$ and as identity everywhere else.

If $\lambda^+ = (m^a)$ and $\lambda^- = \emptyset$, then $W_J$ is a
direct product of $a$ copies of $\mathfrak{S}_m$ and $N_J$ is isomorphic to
$W(B_a)$, with Coxeter generators
\begin{center}
  $r(0,m)\, t(0,m)$ and $s(0,m)$, $s(m,m)$, \dots, $s((a{-}2)m,m)$.
\end{center}
In general, if $\lambda^+ = (1^{a_1}, 2^{a_2}, \dots, n^{a_n})$, then
$W_J$ is a direct product of $W(B_{\Size{\lambda^-}})$ and direct products
of isomorphic symmetric groups $\mathfrak{S}_m$ and $N_J$ is a direct product
of groups $W(B_{a_m})$, with Coxeter generators
\begin{center}
  $r(o_m,m)\, t(o_m, m)$ and $s(o_m,m)$,
$s(o_m + m,m)$, \dots, $s(o_m + (a_m{-}2)m,m)$
\end{center}
and offsets
\begin{align} \label{eq:om-b}
  o_m = \Size{\lambda^-} + a_1 + 2 a_2 + \dots + (m-1)a_{m-1},
\end{align}
for those $m \in \{ 1, \dots, n\}$ with $a_m > 0$.  Unfortunately,
this group $N_J$ usually does not centralize $w_{\lambda}$.  However,
if we define a group $N_{\lambda}$ as the subgroup of $W$ generated by
the same elements as $N_J$, with $t(o_m,m)$ in place of $r(o_m,m)\,
t(o_m,m)$, then $N_{\lambda}$ is a centralizing complement.

\begin{Proposition}
  \label{pro:b}
  Let $\lambda $ be a double partition of $ n$, let $w_{\lambda}$ be
  as in Section~\ref{sec:intro-b}, and let $J = J(w_{\lambda})$ be the corresponding subset
  of $S$.  Then
\begin{align*}
  N_{\lambda} =
\Span{
t(o_m, m), \,
s(o_m{+}km, m)
\mid k=0,\dots,a_m{-}2,\,
 m = 1, \dots, n,\,
a_m > 0}
\end{align*}
is a complement of
  $C_{W_J}(w_{\lambda})$ in $C_W(w_{\lambda})$.
\end{Proposition}

\begin{proof}
  Clearly, $N_{\lambda}$ centralizes $w_{\lambda}$ since its
  generators $t(o_m,m)$ and $s(o_m{+}km, m)$ do.  The statement now
  follows with Proposition~\ref{pro:M} from the fact that
  $N_{\lambda}$ is a complement of $W_J$ in its normalizer in~$W$.
\end{proof}

\subsection{Type \texorpdfstring{$D$}{D}.}
\label{sec:d}

The case of Coxeter groups of type $D_n$ is best dealt with by
comparing it to the situation in type $B_n$.  Throughout this section,
we assume $n \geq 4$, denote by $W$ the Coxeter group of type $B_n$
with Coxeter generators $S$, as described in
Section~\ref{sec:intro-b}, and by $W^+$ the Coxeter group of type
$D_n$ with Coxeter generators $S^+$, consisting of the signed
permutations with an even number of negative cycles as described in
Section~\ref{sec:intro-d}.

The following properties are easy to establish and we leave
their proofs to the reader.

\begin{Lemma}
\label{la:d}
  Let $w \in W$ be an element of cycle type $\lambda = (\lambda^+, \lambda^-)$
such that $\lambda^-$ has an even number of parts and that
$w$ has minimal length in its conjugacy class in $W$.
Also let $J = J(w)$.  Then the following hold.
\begin{enumerate}
\item $w$ has minimal length in its class in $W^+$.
\item If $\lambda^- = \emptyset$ and $\lambda^+$ is even then
  $C_{W^+}(w) = C_W(w)$, otherwise $C_{W^+}(w)$ has index $2$ in
  $C_W(w)$.
\item If $J^+ = S^+ \cap W_J$ then $W^+_{J^+}$ is the smallest
  parabolic subgroup of $W^+$ containing $w$.
\item
If $\lambda^- = \emptyset$ then $J^+ = J$, otherwise
$W^+_{J^+} = W_J \cap W^+$ is a subgroup of index $2$
in $W_J$
\item $C_{W^+_{J^+}}(w) = C_{W_J}(w) \cap W^+$ is a subgroup of index
$2$ in  $C_{W_J}(w)$ unless $\lambda^- = \emptyset$.
\item If $\lambda^+$ is not even and $\lambda^- = \emptyset$ then
  $N^+_{J^+} = N_J \cap W^+$ is a subgroup of index $2$ in $N_J$,
  otherwise $N_J \cong N^+_{J^+}$.
\end{enumerate}
\end{Lemma}

The  parabolic subgroup $W^+_{J^+}$ is of the form
$D_{\Size{\lambda^-}} \times \mathfrak{S}_{\lambda_1^+} \times \dots \times \mathfrak{S}_{\lambda_t^+}$,
where $D_m$ is the subgroup of $W^+$ generated by $\{u, s_1, \dots, s_{m-1}\}$,
for $m =  2, \dots, n$.

\begin{Definition}
  \label{def:non-compliant}
  We call a double partition $\lambda = (\lambda^+, \lambda^-)$ a
  \emph{non-compliant double partition} if  $\lambda^+$
  consists of a single odd part $m$ and $\lambda^-$ is a nonempty even
  partition of even length.

  We call a conjugacy class $C$ of $W$ a \emph{non-compliant class},
  if, for some odd $n > 4$, there is a non-compliant double partition
  $\lambda$ of $n$ and a parabolic subgroup $W_M$ of $W$ which has an
  irreducible component $W_K$ of type $D_n$, such that $C$ contains an
  element of $W_M$ whose projection on $W_K$ has cycle type $\lambda$.
\end{Definition}

For example, the elements of $W = W(D_5)$ with cycle type $(1, 22)$
form a non-compliant class.  For another example, the elements of $W =
W(D_7)$ of cycle type $(21, 22)$ form a non-compliant class, as some
of them lie in a parabolic subgroup $W_M$ of type $D_5 \times A_1$,
with $D_5$-part of cycle type $(1,22)$.

\begin{Lemma}
  An element $w \in W(D_n)$ of cycle type $\lambda = (\lambda^+,
  \lambda^-)$ lies in a non-compliant class if and only if $\lambda^+$
  is not even and $\lambda^-$ is nonempty and even.
\end{Lemma}

The next result shows that, in a Coxeter group $W^+$ of type $D_n$,
the centralizer $C_{W^+}(w)$ splits over $C_{W^+_J}(w)$, unless the
class of $w \in W^+$ is non-compliant.  Here, we write $J^+(w)
\subseteq S^+$ for the set of generators occurring in a reduced
expression of $w$ when considered as an element of $W^+$, in order to
distinguish it from the set $J(w) \subseteq S$
of generators in a reduced expression of $w \in W$.

\begin{Proposition}
  \label{pro:d}
  Let $\lambda = (\lambda^+,\lambda^-) $ be a double partition of $ n$ be such that
  $\ell(\lambda^-)$ is even.
Let $w_{\lambda}$  and $N_{\lambda}$ be as in Proposition~\ref{pro:b}
  and let $J^+ = J^+(w_{\lambda})$ be the corresponding subset of $S^+$.
  Then the following hold.
  \begin{enumerate}
  \item If $\lambda^+$ is even then $N_{\lambda}$ is a complement of
  $C_{W^+_{J^+}}(w_{\lambda})$ in $C_{W^+}(w_{\lambda})$.
\item If $\lambda^+$ is not even and $\lambda^- = \emptyset$ then
  $N_{\lambda} \cap W^+$ is a subgroup of index $2$ in $N_{\lambda}$
  and a complement of $C_{W^+_{J^+}}(w_{\lambda})$ in
  $C_{W^+}(w_{\lambda})$.
\item If $\lambda^+ = (1^{a_1}, \dots, n^{a_n})$ and $\lambda^- =
  (\lambda^-_1, \dots, \lambda^-_s)$ is not even then there is an
  index $j \leq s$ such that $k = \lambda^-_1 + \dots + \lambda^-_j$
  is odd, and the subgroup
\begin{align*}
  N^+_{\lambda} =
\Span{
t(0, k)^m\, t(o_m, m), \,
s(o_m{+}im, m)
\mid i=0,\dots,a_m{-}2,\,
 m = 1, \dots, n,\,
a_m > 0}
\end{align*}
is a complement of
  $C_{W^+_{J^+}}(w_{\lambda})$ in $C_{W^+}(w_{\lambda})$.
  \end{enumerate}
\end{Proposition}

Note that $t(0,k)^m = 1$ if $m$ is even
and $t(0,k)^m = t(0,k)$ if $m$ is odd.

\begin{proof}
Let $J = J(w_{\lambda})$ be the subset of $S$ corresponding to $\lambda$.
In all three cases it suffices to find a complement $N^*$
of $W_J$ in its normalizer in $W$ that centralizes $w_{\lambda}$
such that $\Size{N^* \cap W^+} = \Size{N^+_{J^+}}$.
For then $N^* \cap W^+_{J^+} = 1$ and
$N_{W^+}(W^+_{J^+}) \subseteq N_W(W_J) = W_J N^*$
imply that $N^* \cap W^+$ is a
complement of
$W^+_{J^+}$ in its normalizer in $W^+$ that
centralizes $w_{\lambda}$, and the claim follows with
Proposition~\ref{pro:M}.

(i)
If $\lambda^+$ is even then
$N_{\lambda}$ is contained in $W^+$  and $N^* = N_{\lambda}$ will do.

(ii)
If $\lambda^- = \emptyset$ and $\lambda^+$ is not even
then $J^+ = J$ but $N_{J^+}$ is subgroup of index $2$ in $N_J$
and $N^* = N_{\lambda} \cap W^+$ will do.

(iii) If $\lambda^-$ is not even
then $N^+_{\lambda}$ is a
complement of $W_J$ in its normalizer in
$W$ that is contained in  $W^+$
and centralizes $w_{\lambda}$,
whence $N^* = N^+_{\lambda}$ will do.
\end{proof}

\subsection{Type \texorpdfstring{$I$}{I}.}
\label{sec:i}

Suppose $W$ is a Coxeter group of type $I_2(m)$.  Then $W$ is the
group generated by generators $s_1$ and $s_2$ satisfying $(s_1s_2)^m =
(s_2s_1)^m$. Each element of $W$ is either cuspidal or an
involution. Hence the theorem for this type follows from
Lemma~\ref{la:cori}.

\subsection{Exceptional Types.}
\label{sec:exceptional}

Although in type $A$ each conjugacy class contains an element $w$ such
that the normalizer complement $N_J$ is also a centralizer complement,
this cannot be expected in general to be the case.  However, from the
preceding examples one sees that it is frequently possible to
construct from $N_J$ an isomorphic copy $N_J^*$ which is a centralizer
complement.  In each of the above examples, $N_J^*$ is obtained from
$N_J$ by replacing generators $x_i$ of $N_J$ by products $w_L x_i$ for
suitable subsets $L \subseteq J$.

Based on this observation,
we formulate an algorithm, which in practice  always
finds a centralizer complement, except for elements of non-compliant
classes.

\bigskip
\noindent
\textbf{Algorithm} CentralizerComplement.
\nopagebreak

\smallskip
\noindent
\textbf{Input:} A finite Coxeter group $W$ and an element $w$
of minimal length in its conjugacy class in $W$.

\smallskip
\noindent
\textbf{Output:} a centralizer complement for $w$, or fail if none exists.
\begin{enumerate} \renewcommand{\labelenumi}{\arabic{enumi}.}
\item set $J \gets J(w)$.
\item find involutions $x_1, \dots, x_r$ generating the normalizer
complement $N_J$.
\item for each element $v$ of minimal length in the $W_J$-conjugacy
  class of $w$ do the following:
\begin{itemize}
\item let $u \in W_J$ be such that $v^u = w$;
\item  for each $i = 1, \dots, r$, set
\begin{align*}
  Y_i \gets
  \begin{cases}
    \{x_i\}, & \text{if } v^{x_i} = v, \\
\{w_L x_i: L \subseteq J,\, x_i^{w_L} = x_i,\, v^{w_L} = v^{x_i}\}, & \text{otherwise.}
  \end{cases}
\end{align*}
\item if there are elements $y_i \in Y_i$, $i = 1, \dots, r$, such that
$M = \langle y_1, \dots, y_r\rangle$ satisfies
 $M \cap W_J = 1$
then return $M^u$.
\end{itemize}
\item return fail (if we ever get here).
\end{enumerate}

Note that, by Proposition~\ref{pro:M}, any group $M$ found in this way
is necessarily a complement of the centralizer of $w$ in~$W_J$.

For $W$ irreducible of exceptional type, the algorithm produces a centralizer
complement in all but seven cases.  Each case corresponds to a
non-compliant class from the following table.  In this table we list,
for each non-compliant class $C$ of $W$, its position $i$ in CHEVIE's
list of conjugacy classes of $W$, its name, a reduced expression for a
representative $w$ of minimal length, the set $J(w)$, a set $M
\supseteq J(w)$, the type of $W_M$ exhibiting a direct factor of type
$D_{2l+1}$, and the label $\lambda$ of the conjugacy class of
$W(D_{2l+1})$ containing the projection of $w$.
\begin{align*}
\begin{array}{cccccccc}
  \hline
W & i & \text{name} & w \in C & J(w) & M & \text{type} & \lambda \\
  \hline
  \hline
E_6 & 7 & D_4(a_1) & {}_{342345} & 2345 & 12345 & D_5 & (1,22) \\
  \hline
  \hline
E_7 & 9 & D_4(a_1) & {}_{425423} & 2345 & 12345 & D_5 & (1,22) \\
    & 42 & D_4(a_1)+A_1 & {}_{4254237} & 23457 & 123457  & D_5 \times A_1 & (1,22) \\
  \hline
  \hline
E_8 & 16 & D_4(a_1) & {}_{242345} & 2345 & 12345 & D_5 & (1,22) \\
    & 44 & D_6(a_1) & {}_{24234567} & 234567 & 2345678 & D_7 & (1,42) \\
    & 53 & D_4(a_1)+A_2 & {}_{34234578} & 234578 & 1234578 & D_5 \times A_2 & (1,22) \\
    & 73 & D_4(a_1)+A_1 & {}_{3542348} & 23458 & 123458 & D_5 \times A_1 & (1,22) \\
  \hline
\end{array}
\end{align*}

In the next section we show that in all of these cases, and indeed
whenever $w$ lies in a non-compliant class, no complement exists.

\subsection{Non-compliance.}
\label{sec:n-c}

The class of $W(D_5)$ with label $\lambda = (1, 22)$ contains the element
\begin{align*}
  w_{\lambda}
= ts_1 s_2s_1ts_1s_2s_3
= (ts_1 t)s_2(ts_1t)s_1s_2s_3
= us_2us_1s_2s_3,
\end{align*}
which lies in the parabolic subgroup $W_J$ of
type $D_4$.  Its centralizer $C_{W_J}(w)$ in $W(D_4)$ has order $16$
and its centralizer $C_W(w)$ in $W(D_5)$ has order $32$.  However,
$C_{W_J}(w)$ has no complement in $C_W(w)$, since the coset $C_W(w)
\setminus C_{W_J}(w)$ contains no element of order $2$ (as a
straightforward computation in GAP will confirm).

The next result shows that this is indeed always the case, when the
cycle type of $w \in W(D_n)$ is a non-compliant double partition
of~$n$.

\begin{Proposition} \label{prop:non-compliant}
  Suppose that $W$ is of type $D_n$
  and let $w \in W$ be an element of minimal length in its conjugacy
  class with $J(w) = J$.  If the cycle type of $w \in W$ is a
  non-compliant double partition of $n$ then the centralizer
  $C_{W_J}(w)$ has no complement in $C_W(w)$.
\end{Proposition}

\begin{proof}
Recall from Section~\ref{sec:intro-b} that elements of $W(B_n)$ can be represented as signed
permutation matrices, i.e., matrices with
exactly one non-zero entry $1$ or $-1$ in each row and column.
Such an element  lies in $W(D_n)$ if and only if its matrix has
an even number of entries $-1$,
and it is an involution if and only if the matrix is symmetric.

Now suppose that $n = m + k$ is odd and that $w \in W = W(D_n)$ is an
element of minimal length in a conjugacy class with cycle type
$\lambda = (\lambda^+, \lambda^-)$ where the partition $\lambda^+$
consists of a single odd part $m$ and $\lambda^-$ is a nontrivial
partition of an even number $k$, consisting of an even number of even
parts.  Then $W_J$ for $J = J(w)$ has type $D_k \times A_{m-1}$ and,
by the description of $N_J$ in Section~\ref{sec:b} and
Lemma~\ref{la:d}(vi), its normalizer $N_W(W_J)$ has a complement of
order $2$ (and of type $B_1$), generated by the quotient $w_J w_0$.

We may assume that $J = S \setminus \{s_{k+1}\}$, so that, as signed
permutation on the set $\{1, \dots, n\}$, the element $w$ induces an
even number of negative cycles on the $k$ points $\{1, \dots, k\}$ and
a positive $m$-cycle on the $m$ points $\{k+1, \dots, n\}$.  The
centralizer $C_W(w)$ cannot move points from outside the $m$-cycle
into the $m$-cycle and thus consists of block diagonal matrices
\begin{align*}
\mathop{\mathrm{diag}}(A, B) = \left[
\begin{array}{cc}
  A & 0 \\ 0 & B
\end{array}
\right],
\end{align*}
of a $k \times k$ matrix $A$ and
an $m \times m$-matrix $B$,
which modulo $2$ have the same number of
entries $-1$
since $C_W(w)$ is a subgroup of $W(D_n)$.
Moreover, for each element $\mathop{\mathrm{diag}}(A, B)$ in $C_W(w)$
the number of entries $-1$ on the diagonal of $A$,
is even, since with every point in $\{1, \dots, k\}$
being mapped to its negative, the entire
cycle which contains it must be negated.

The centralizer of $w$ in $W_J$ consists precisely
of those elements $\mathop{\mathrm{diag}}(A, B) \in C_W(w)$ which have an
even number of entries $-1$ in both $A$ and $B$,
since $A$ is the matrix  of an element in $W(D_k)$.

Let $u$ be an involution in $C_W(w)$.  Then its matrix $\mathop{\mathrm{diag}}(A, B)$
is symmetric, and an even number of entries $-1$ on the diagonal of
$A$ implies that both $A$ and $B$ have an even number of entries $-1$
and thus $u \in C_{W_J}(w)$.

It follows that $C_{W_J}(w)$ has no complement in $C_W(w)$.
\end{proof}

More generally, if $w$ lies in a non-compliant class of
a finite Coxeter group $W$, then
its centralizer has no complement.

\begin{Theorem} \label{thm:non-compliant}
  Let $W$ be a finite Coxeter
  group.  Suppose $w$ is an element of minimal length in a
  non-compliant conjugacy class of $W$ with $J(w) = J$.  Then the
  centralizer $C_{W_J}(w)$ has no complement in $C_W(w)$.
\end{Theorem}

\begin{proof}
  Suppose first that $W$ is a direct product $W_1 \times W_2$ of
  nontrivial standard parabolic subgroups $W_1$ and $W_2$, that $w =
  w_1 w_2$ with $w_1 \in W_1$ and $w_2 \in W_2$, and that $w_1$ lies
  in a non-compliant class of $W_1$.  Then $W_J = W_{J_1} \times
  W_{J_2}$ for certain subsets $J_1 \subseteq W_1 \cap S$ and $J_2
  \subseteq W_2 \cap S$.  If $C_{W_{J_1}}(w_1)$ has no complement in
  $C_{W_1}(w_1)$ then $C_{W_J}(w)$ cannot have a complement in $C_W(w)
  = C_{W_1}(w_1) \times C(w_2)$.

  Next,  suppose that
$w \in W_L$ for some
 $L \subseteq S$ and that
$w$ lies in a non-compliant
class of $W_L$.
  Suppose $N$ is a complement
of $C_{W_J}(w)$ in $C_W(w)$, that is
$C_W(w) = C_{W_J}(w) \rtimes N$.
Then the centralizer of $w$ in  $W_L$,
\begin{align*}
  C_{W_L}(w)
= C_W(w) \cap W_L
= (C_{W_J}(w) \rtimes N) \cap W_L
= C_{W_J}(w) \rtimes (N \cap W_L),
\end{align*}
in contradiction to our assumption that $w$ lies in a
non-compliant class.

The theorem now follows from Definition~\ref{def:non-compliant} and
Proposition~\ref{prop:non-compliant}.
\end{proof}

\section{Applications.}
\label{sec:application}

In this section we first use Theorem \ref{thm:1} to prove a result about
minimal length representatives of conjugacy classes. Then we show how it
implies the celebrated Solomon's character formula. Finally, we
discuss the interpretation of Solomon's theorem as a Coxeter group
analogue of MacMahon master theorem.

\begin{Theorem}
  \label{thm:3}
  Assume $w$ has minimal length in its conjugacy class in $W$. Then the
  following hold for any $v \in W$:
  \begin{enumerate}
  \item $J(w^v) = \mathcal{D}(v^{-1}) \iff v = w_{J(w)}$;
  \item $J(w^v) = \mathcal{A}(v^{-1}) \iff v = w_{J(w)} w_0$.
  \end{enumerate}
\end{Theorem}

\begin{proof}
  Let $L = \mathcal{D}(v^{-1})$.  Then $v^{-1} = w_L \cdot x$ for some $x \in
  X_L$, by Lemma~\ref{1.5.2}.  Clearly $\ell((w^v)^{w_L}) =
  \ell(w^v)$, since $J(w^v) = L$.  By Lemma~\ref{2.1.14}, conjugation
  by the coset representative $x$ does not decrease the length, hence
  $\ell(w^v) = \ell((w^v)^{w_L}) \leq \ell((w^v)^{w_L x}) = \ell(w)$
  and it follows that $w^v$ has minimal length in its conjugacy class
  as well. By Proposition~\ref{3.1.11}, $L = J(w^v)$ and $J(w)$ are
  conjugate subsets of~$S$.  Proposition \ref{prop:1} (iv) says
  more precisely that $x$ is a conjugating element, i.e., $L^x = J(w)$.

  Assume that $\ell(x) > 0$ and let $s \in \mathcal{D}(x)$. Then $s \notin L$,
  since $x \in X_L$; denote $L \cup \{s\}$ by $M$. By
  Theorem~\ref{2.3.3}, $x$ is a reduced product $x = d \cdot y$ with
  $y \in X_M$ and $d = w_L w_M$, the longest coset representative of
  $W_L$ in $W_M$. It follows that $v^{-1} = w_L \cdot x = w_M \cdot
  y$, whence $M \subseteq \mathcal{D}(v^{-1}) = L \subsetneq M$.  The
  contradiction shows that $x = 1$, and therefore $v = w_L$ and $L =
  J(w)$.

  (ii)   Note  that   $J(w^{w_0})  =   J(w)^{w_0}$,   $\mathcal{D}(x^{w_0})  =
  \mathcal{D}(x)^{w_0}$, and $\mathcal{A}(x)  = \mathcal{D}(x w_0)$ for all  $x \in W$, whence
  $\mathcal{D}(x)^{w_0} = \mathcal{A}(w_0 x)$.  Therefore, it follows from (i) that
  \begin{align*}
    J(w^{x w_0}) = J(w^x)^{w_0} = \mathcal{D}(x^{-1})^{w_0} = \mathcal{A}((x
    w_0)^{-1}) \iff x = w_{J(w)},
  \end{align*}
  as desired, for $v = x w_0$.
\end{proof}

The following formula, first proved by Solomon~\cite{Solomon1966}
in 1966, is an easy consequence of the previous result.

\begin{Theorem}[Solomon's theorem]
  For $J \subseteq S$, let $\pi_J$ denote the permutation character of
  the action of $W$ on the cosets of $W_J$ defined by $\pi_J(w) =
  \Size{\mathop{\mathrm{Fix}}_{W/W_J}(w)}$, and let $\epsilon$ be the sign character of
  $W$, defined by $\epsilon(w) = (-1)^{\ell(w)}$ for $w \in W$. Then
  \begin{align*}
    \sum_{J \subseteq
    S} (-1)^{\Size{J}} \pi_J = \epsilon.
  \end{align*}
\end{Theorem}

\begin{proof}
  The formula follows if we can show that $\sum_{J \subseteq S} (-1)^{\Size{J}} \pi_J(w) =
  \epsilon(w)$ for all $w \in W$.
  We have $W_J x w = W_J x \iff x w x^{-1} \in W_J$, so
  \begin{align*}
    \sum_{J \subseteq S} (-1)^{\Size{J}} \pi_J(w)
    = \sum_{J \subseteq S} (-1)^{\Size{J}} \sum_{\substack{x \in X_J\\ xwx^{-1} \in W_J}} 1
    = \sum_{x \in W} \sum_{J(xwx^{-1}) \subseteq J \subseteq \mathcal{A}(x)} (-1)^{\Size{J}},
\end{align*}
where we reversed the order of summation and used the facts that $x
\in X_J \iff J \subseteq \mathcal{A}(x)$ and $xw x^{-1} \in W_J \iff J(x w x^{-1})
\subseteq J$.  The Binomial Theorem implies that
\begin{align*}
  \sum_{A \subseteq J \subseteq B} (-1)^{\Size{J}} = (-1)^{\Size{A}} \sum_{I \subseteq B \setminus A} (-1)^{\Size{I}} =
  \begin{cases}
    (-1)^{\Size{A}} & \text{if } A = B, \\
    0 &  \text{otherwise.}
  \end{cases}
\end{align*}
But then
\begin{align*}
 \sum_{J \subseteq S} (-1)^{\Size{J}} \pi_J(w) = \sum_{\substack{x \in W \\ J(xwx^{-1}) =\mathcal{A}(x)}} (-1)^{\Size{\mathcal{A}(x)}}.
\end{align*}
Since $\sum_{J \subseteq S} (-1)^{\Size{J}} \pi_J$ is a class function, it is enough to choose
 $w$ with minimal length in its conjugacy class. By Theorem~\ref{thm:3}(ii), this sum then
 consists only of the one term for $x^{-1} = w_{J(w)} w_0$, and we have
\begin{align*}
\sum_{J \subseteq S} (-1)^{\Size{J}} \pi_J(w) = (-1)^{\Size{\mathcal{A}(w_0 w_{J(w)})}}.
\end{align*}
But $\Size{\mathcal{A}(w_0 w_{J(w)})}
= \Size{\mathcal{D}(w_{J(w)})^{w_0}}
= \Size{\mathcal{D}(w_{J(w)})} =
\Size{J(w)}$ and then the claim follows from  the fact that $(-1)^{\Size{J(w)}} =
(-1)^{\ell(w)}$~\cite[Exercise 3.17]{GePf2000}.
\end{proof}

Solomon proves this formula generically for all types of finite
Coxeter groups, and he has published three different versions of the proof.  His
original proof~\cite{Solomon1966} depends on an application the Hopf
trace formula to the Coxeter complex of the finite Coxeter group $W$,
a later proof (of a more general statement) uses a decomposition of
the group algebra of $W$.  The third version of the
proof~\cite{Solomon1976} is based on properties of a homomorphism of
the descent algebra of $W$ into the character ring of~$W$ (see also
\cite[Exercise 3.15]{GePf2000}).  None of these proofs have the
combinatorial flavor of the above proof.

Finally, let us explain how to interpret Solomon's theorem as a generalization
of a special case of the celebrated MacMahon master theorem. For a connection with a
different result due to MacMahon, see \cite[\S 6]{Solomon1968}.

MacMahon's master theorem states that for a matrix $X = (x_{ij})_{n\times n}$,
the functions
\begin{equation} \label{macmahon}
 \frac 1{\det(\mathrm{Id}-X)},
\end{equation}
where $\mathrm{Id}$ denotes the $n \times n$-identity matrix, and
\begin{align*}
  \sum_{w} x_{v_1 w_1} x_{v_2 w_2} \cdots x_{v_m w_m},
\end{align*}
where $w = w_1 w_2 \cdots w_m$
runs over all words in $\{1,2,\ldots,n\}$ and $v = v_1 v_2 \cdots v_m$ is the
weakly increasing rearrangement of $w$, are equal. In particular, for
any permutation $w \in S_n$, the coefficient of $x_{1w(1)}\cdots x_{n
w(n)}$ in \eqref{macmahon} is equal to $1$. For $I \subseteq [n]$, denote by
$X_I$ the submatrix $(x_{ij})_{i,j \in I}$.  We have
\begin{multline*}
  \frac 1{\det(\mathrm{Id}-X)} = \frac 1{\sum_{I \subseteq [n]} (-1)^{\Size{I}} \det
X_I} = \frac 1{1 - \sum_{\emptyset \neq I \subseteq [n]} (-1)^{\Size{I}-1}
\det X_I} \\
= \sum_{k \geq 0} \Bigl( \sum_{\emptyset \neq I \subseteq
[n]} (-1)^{\Size{I}-1} \det X_I\Bigr)^k = \sum_{k \geq 0} \sum
(-1)^{\Size{I_1}-1 + \ldots + \Size{I_k}-1} \det X_{I_1} \cdots \det X_{I_k},
\end{multline*}
where the last sum runs over all $k$-tuples $(I_1,\ldots,I_k)$ of
non-empty subsets of $[n]$. Since we are interested in the coefficient
of $x_{1w(1)}\cdots x_{n w(n)}$ (in which all indices are represented,
and each index is represented only twice, once as a first index and
once as a second index), we can limit the sum to ordered set
partitions $(I_1,\ldots,I_k)$ of the set $[n]$. Note that we have
$(-1)^{\Size{I_1}-1 + \ldots + \Size{I_k}-1} = (-1)^{n-k}$.

Recall that the symmetric group $S_n$ is a Coxeter group $W$ of type $A_{n-1}$ with Coxeter generators
$S = \{s_1,\ldots,s_{n-1}\}$, $s_i = (i,i+1)$. Choose a composition $\lambda \vdash n$.
By Merris-Watkins formula~\cite{MW85} (and not hard to prove independently),
the coefficient of $x_{1w(1)}\cdots x_{n w(n)}$ in
\begin{align*}
  \sum \det X_{I_1} \cdots \det X_{I_k},
\end{align*}
where the sum runs over all ordered set partitions $(I_1,\ldots,I_k)$
of $[n]$ with $\Size{I_j} = \lambda_j$ for all $j$, is equal to
$(-1)^{\ell(w)} \pi_J(w)$, where $J$ is the subset of $S$ that corresponds
to the composition $\lambda$. This means that
\begin{align*}
  \sum (-1)^{n - \Size{\lambda}} (-1)^{\ell(w)} \pi_J(w) = 1,
\end{align*}
where the sum runs over all subsets $J$ of $S$. This is obviously
equivalent to Solomon's theorem.

\subsection*{Acknowledgement}
The second author wishes to acknowledge support from Science
Foundation Ireland.

redpajama_set_name: RedPajamaArXiv


text:
About ADFAS
Conservation & Philanthropy (PRF)
Cultural & Heritage Projects
Young Arts Projects
Society Events & Activities
Email: info@adfas.org.au
ADFAS Newcastle welcomes you.
ADFAS Newcastle is a thriving Society with over 185 members. While we are located in Newcastle we welcome members from Newcastle, Lake Macquarie, Port Stephens and the Hunter Valley (while noting that there is also ADFAS Pokolbin).
As well as hosting nine lectures a year and two special interest mornings, members and volunteers are also involved in Church Recording, School of Arts recording and raising funds to support Young Arts grants.
So far we have completed 11 School of Arts reports (East Gresford, Gresford, Minmi, Merewether, Plattsburg, Stroud, Tighes Hill, Wallsend 1, Wallsend 2, and Wickham) and one Church Recording (St John's Anglican Church, Cooks Hill).
In 2021 the Society received the prestigious Marsh Award for our community activities. Watch our presentation to the Marsh Committee.
VISIT THE ADFAS NEWCASTLE WEBSITE
View Presentation to the Marsh Committee
2023 Lecture Program
Young Arts
All lectures take place at the Hunter Theatre, Hunter School of the Performing Arts, Cameron St, Broadmeadow.
Parking is on Cameron Street and in the parking lots to the north and south of the theatre.
Lectures are on a Monday evening and begin at 6:30pm.
We offer nine lectures and two special interest mornings a year.
Click the links to find details of our 2023 lectures and special interest mornings.
Annual membership (for 9 lectures) is $170
Student membership is $85 (a student is under 25 and studying full-time at school, TAFE or university)
Click here for the ADFAS Newcastle membership form and return with payment.
If you have membership questions, please contact Jenny at membership@adfasnewcastle.org.au
Guests welcome:
Guests may attend up to three lectures per annum.
The cost is $25 per person per lecture.
For members of other ADFAS societies the cost is $5.
For more information visit ADFAS Newcastle website
For all enquiries please contact Jenny at membership@adfasnewcastle.org.au
Postal Address: PO Box 531, Newcastle, NSW 2300
Committee 2023
Chair: Kathy Heinrich
Click here for more information about the ADFAS Newcastle committee
NORA, STELLA AND SYBIL – WOMEN AT WAR
Presented by Gavin Fry
Time & Venue: 6:30pm. Hunter Theatre, Hunter School of the Performing Arts, Cameron St, Broadmeadow.
Just three women worked as Official war Artists during World War Two – Nora Heysen, Stella Bowen and Sybil Craig. They each made a significant body of work, bringing a fresh vision to the official expression of Australia at War.
Gavin Fry is a writer, artist and museum professional with fifty years experience working in curatorial and management positions in Australian museums, galleries and educational institutions. He is the author of twenty-five books on Australian art and history and a large number of catalogue and journal essays. In retirement Gavin has returned to his art training and exhibits as a painter in Newcastle and Melbourne. As well as working as a professional writer, he also designs and publishes books on behalf of other writers and artists. Gavin holds the degrees of Bachelor of Arts [Hons] and Master of Arts from Monash University and Master of Philosophy from Leicester University.
THE SCULPTURES OF THE PARTHENON OR HOW LORD ELGIN GAINED AND LOST HIS MARBLES
Presented by Alan Reed
In the two centuries since they were removed from the Parthenon by Lord Elgin, the meaning and significance of the 'Elgin marbles' has changed dramatically. From architectural decoration to disputed cultural objects this lecture looks at the response to them over their time in Britain, from the original controversy over their purchase to the current debate surrounding the restitution of the marbles to the new Acropolis Museum in Athens.
Alan holds a master's and first class honours degree in History of Art from Birkbeck College, London. He is a gallery guide at Tate Britain, Tate Modern, National Portrait Gallery and for Frieze Masters and regularly lectures at the NPG, Dulwich Picture Gallery, Plymouth City Art Gallery and other galleries in the UK. He also works as a London Blue Badge Guide and a City of London Guide.
Monday 1 May 2023
THE DARK HEART OF EMPIRE – PICTURING SLAVERY AND THE MOVEMENT FOR ABOLITION
Presented by Jos Hackforth-Jones
The triangular slave trade was seldom directly represented in art. However, many artists including Hogarth and Reynolds included slaves and slave owners in their portraits. This lecture will examine some of the complexities of this process. We will consider some of the abolitionist sympathies of artists such as Augustus Earle and Joshua Reynolds and the effect of the 'turn' against slavery and the abolition of the slave trade and the impact it had on the art of the period.
Professor Jos Hackforth-Jones has served as Director of the London Institute since 2008. Hackforth-Jones is a distinguished academic administrator, noted art historian, author, curator and lecturer. Since joining Sotheby's Institute she has worked closely with colleagues to refine the academic standing and raise the international profile of the London Institute. Hackforth-Jones has recently edited (with Iain Robertson) Art Business Today: 20 Key Topics (Lund Humphries and Sotheby's Institute of Art, 2016). This accompanies the launch of an exciting and innovative new MA curriculum in Art Business, Contemporary Art, Fine and Decorative Art and Design and Modern and Contemporary Asian Art. Prior to her post at the Institute, Hackforth-Jones served as President and Provost at Richmond The American International University in London.
GENDER AND THE BODY – KEPT BEHIND CURTAINS, STORY OF THE NUDE
Presented by Leslie Primo
The nude is still seen in our modern age, and indeed has been seen for quite some time as the pinnacle of creative artistic perfection but, throughout the course of art history the notion of the perfect body and consequently gender has been constantly reshaped and redefined. This lecture will look at the continuing fascination with representation of the body in sculpture and in painting across the ages.
Leslie Primo Holds a BA in Art History and an MA in Renaissance Studies from Birkbeck College, University of London. He was the Visiting Lecturer in Art History at the University of Reading in 2005 and 2007, and gives lectures and guided tours, plus special talks, at both the National Gallery and the National Portrait Gallery. He also lectures at the City Literary Institute and has presented a series of talks at the National Maritime Museum and the Courtauld Institute. The first lecture in this list is based on his book, called The Foreigners that Invented British Art: From Renaissance to Enlightenment, which will be published by Thames and Hudson in Spring 2023.
ART AND THE THIRD REICH
Presented by Leigh Capel
This lecture will look into Hitler's often overlooked artistic background; a frustrated artist in the traditional school, rejected due to the rise of Modern Art in the early 20th Century. It will also analyse some of the most important artworks and artists lost in Hitler's war on Modern Art, the Monuments Men and the search for the stolen art, and the aftermath and impact on Post Wat art, including the impact and influence of the Holocaust on past war and immigrant art in Australia.
I fell into art by accident. I studied filmmaking in Sydney and San Francisco, worked on the London Olympics Games with NBCUniversal, then moved home in 2013 without a job. As a favour to my mum, a small auction room gave me a start moving furniture and disposing of dead people's possessions. Within six months I was the fine art photographer for the business and within two years, the art researcher and valuer. Following stints at Menzies and Mossgreen, I finally rose to Sotheby's, working on Russell Crowe's 'Art of Divorce', 'Kirk Pengilly (INXS) Collection' and 'Important Art from the John Schaeffer Collection'. Disenchanted with the auction industry and working for other people's dreams, I quit my job to start my own art gallery in Petersham as an extension of mum's antique furniture business, Belle Epoque Fine Art and Antiques. I had just turned 30.
FIT FOR A KING – THE ULTIMATE SPIN DOCTOR
Presented by Jacqui Ansel
Had his older brother Henry not died in 1612, Charles I would never have been King. Had Van Dyck not arrived at the Stuart Court the stiff body language of Jacobean portraiture may never have been enlivened by the introduction of Renaissance values, giving rise to the splendour of Baroque. Van Dyck transformed the awkward and ungainly Charles; through his portraits the tiny King (and his even tinier wife) seemed to grow in stature. Van Dyck's male sitters exuded elegance, and with his depiction of sumptuous satins and shimmering silks he was hailed as 'the first that 'ere put 'ladies' dress into a careless romance'.
Jacqui Ansell is a Senior Lecturer at Christie's Education, tutoring online and face-to-face courses. She read History of Art and Theory at the University of Essex before going on to gain an MA in History of Dress from the Courtauld Institute under Professor Aileen Ribeiro. Formerly an Education Officer at the National Gallery, London, and a tutor and writer for the Open University, she lectures for the Art Society and has devised and delivered many short courses for the National Gallery and Wallace Collections. She lectures on aspects of art history and dress history 1450-1950 (with particular interest in portraiture). Specialist research and publications focus on European Courtly Cultures, The Grand Tour and Traditional Welsh Costume.
Presented by Chloe Sayer
The ancient goldsmiths of Peru and Colombia produced some of the most spectacular treasures of South America. Adept at casting and hammering, they transformed this sacred metal into elaborate pendants and breastplates, ear-ornaments, musical instruments, flasks, human figures, funerary masks, and lifelike representations of birds and animals. Colour tones went from pale yellow to deepest red.
Author, lecturer, photographer, and curator, Chloe specialises in the art and culture of Mexico. She has made an ethnographic collection featuring clothing and culture for the British Museums of Mexico, Belize, and the Royal Ontario Museum in Toronto. The Mexican Day of the Dead was also a source of inspiration as co-curator of the 'Exhibition of Mankind' for the British Museum.
PEDER SEVERIN KRØYER – A PAINTER OF LIGHT
Presented by Kathy McLauchlan
Looking back on his work towards the end of his life, Peder Severin Krøyer recalled his work at Skagen, the Danish artists' colony at the northernmost tip of Denmark and expressed his particular love for that time. Krøyer was referring to the 'blue hour' of northern Scandinavian summer nights, when sea and sky appear to merge into a single luminous whole. This lecture explores Krøyer's life and work in Skagen,and evaluates the paintings that made him one of Europe's most celebrated artists by the end of the 19th century.
Kathy McLauchlan is a lecturer specialising in 19th-century art history. She is currently a course Director at the Victoria & Albert Museum and also a freelance lecturer who teaches at several institutions, including the Arts Society and Art Pursuits. She is a graduate of Oxford University and the Courtauld Institute, with a Ph.D. in French 19th-Century painters in Rome. She has published catalogues and articles for the British Council and the Barbican Art Gallery.
A MEDITERRANEAN TOUR – NOTE JUST A LOAD OF OLD STONES
Presented by Dr Gillian Hovell
This extremely popular virtual tour of the ancient Mediterranean explores the distinctive civilizations. Learning how they fit together into a "big picture". Discover how to identify special art or architecture of each culture. Find out what to expect from each culture's sites and how to find the special personal details that thrill archaeologists. You'll never look at ancient sites in the same way again.
Author, historian, archaeologist and public speaker of considerable experience and range, I specialise in providing passion, colour and depth to the present by visiting the past – whether it be through books, lively illustrated talks, articles, archaeological tours and cruises, walks, community archaeology projects, curses and classes, media. My clients include the British Museum and York University.
Ex-BBC, I'm delighted to contribute via radio/audio, film, apps or printed media and I love meeting you in person at events too.
As Author, award-winning Writer, Tutor, Presenter and Public Speaker at events, in the media, on tours and cruises etc, I work as a freelance independent self-employed professional.
© All Rights Reserved 2022 Australian Decorative & Fine Arts Societies | Privacy Policy | Acceptable use policy | Cookies Policy | Terms and Conditions | Disclaimer


text:
MyMediabox Clients Among Most Influential Brands of 2013
Licensing.biz released their first Power List this month of the most influential brands of 2013. The list, which is voted for by a panel of 200 independent judges in the industry, is split into four categories – Character & Entertainment, Brands, Art & Design and Sports.
Another Successful Year of User Groups
Each year we set out on an educational road trip across Canada and the United States, and 2013 was our most engaging to date.
EZFacility's MemberMe Mobile Application Now Available on Android
EZFacility, Inc., a global leader in Cloud-based scheduling, management, membership, billing and financial solutions for health and fitness clubs and sports and recreation facilities, announced the launch of an Android version of its popular MemberMe mobile app.
Computrition Launching the Fusion Newsletter
Computrition will be launching a new quarterly publication called the Fusion Newsletter
Tackling Late Payment to Boost Business
For small businesses, cash flow is crucial so late payments can be a real minefield to negotiate.


text:
Reliving Anita Hill's Impact on Sex, Race and Gender
"It's about human beings trying to really struggle with issues of identity and morality and truth," Kerry Washington says. (Photo: HBO)
For many women, Kerry Washington's true gladiator role is serving as co-executive producer of Confirmation on HBO. The "Scandal" star portrays University of Oklahoma law professor Anita Hill, who testified about unwanted sexual advances from her former boss, Clarence Thomas, during his 1991 Senate confirmation hearings for the U.S. Supreme Court.
"It's about human beings trying to really struggle with issues of identity and morality and truth," Washington said during promotions for the film. "I love that we get to take this little moment in history and peel back the layers and allow people a peek behind the curtain so you can really understand more of what was truly happening,"
"It was a moment in our nation's history where we got to look at what power looks like in this country." She noted that those grilling Hill were "all white, all male, all of a certain age."
Confirmation marks the 25th anniversary of the Senate hearings and has introduced a new generation to Anita Hill and how she helped to spark more attention to sexual harassment as well as surge of women seeking political office.
"The takeaway for me is how truly courageous she is," Washington said.
Anita Hill sparked more attention to sexual harassment as well as surge of women seeking political office. (Photo: Brandeis University)
Hill, now 59, is a University Professor of Social Policy, Law, and Women's and Gender Studies at Brandeis University. She has also used her legal expertise as an attorney in civil rights and employment. Her books include Reimagining Equality: Stories of Gender, Race and Finding Home in 2011; Race, Gender and Power in America: The Legacy of the Hill-Thomas Hearings, co-edited with Emma Coleman Jordan in 1995; and her 1997 autobiography, Speaking Truth to Power.
Thomas has spent most of the last quarter-century without uttering a word during sessions at U.S. Supreme Court. Many Americans have considered the 67-year-old's silence irresponsible — even if they don't really want to hear what he has to say — because he is filling the seat of Thurgood Marshall, the only other African American to serve on the high court.
While Confirmation serves as an introduction for some, it's a reminder for legions of others caught up in he said-she said debates around the world. Melissa Harris-Perry said she was moved to tears by the scene in which Hill discovers boxes of letters from supporters.
"I was a college student just finding my feminist identity when Hill testified and was vilified," Harris-Perry, one of the letter writers, wrote in an Elle article. "Her testimony helped me contextualize my own sexual assault and helped ignite my political engagement."
However, Confirmation commits a sin of omission, Harris-Perry said in her article, "Where Are the Rest of the Black Feminists in HBO's Confirmation?"
Kimberlé Williams Crenshaw was a black feminist lawyer on Anita Hill's legal team. (Photo: Twitter)
"Confirmation forgets the contributions of Professor Kimberlé Williams Crenshaw, the black feminist lawyer on Hill's legal team," she wrote. Crenshaw, a law professor at UCLA and Columbia Law School, is co-founder and executive director of the African American Policy Forum, a think tank on structural inequality.
"The movie forgets Nov. 17, 1991, when 1,600 black women combined their limited resources to purchase a full page in the New York Times for $50,000 using the name 'African American Women in Defense of Ourselves.' These black feminists leveled policy-based opposition to Thomas' confirmation, situated Anita Hill's testimony in a history of black women's vulnerability to sexual violence, and articulated the need to analyze both race and gender in public discourse."
What Confirmation gets right is Hill's dignity. "Washington embodies Hill with poise and halting grace, mimicking the slow articulation of her complaints, and the humiliation the Senate panel put her through by forcing her to restate details of Thomas's behavior," Sophie Gilbert writes in The Atlantic.
"Far from being a scorned woman or a vengeful employee, she was a quiet law professor who simply wanted her concerns about a man appointed to the highest court in the land noted."
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text:
Q: understanding dependency injection in PHP class Author {
    private $firstName;
    private $lastName;

    public function __construct($firstName, $lastName) {
        $this->firstName = $firstName;
        $this->lastName = $lastName;
    }

    public function getFirstName() {
        return $this->firstName;
    }

    public function getLastName() {
        return $this->lastName;
    }
}

class Question {
    private $author;
    private $question;

    public function __construct($question, Author $author) {
        $this->author = $author;
        $this->question = $question;
    }

    public function getAuthor() {
        return $this->author;
    }

    public function getQuestion() {
        return $this->question;
    }
}

Class author is injected into the constructor of Question class am I correct? but how to call the Question class to get the author's name?
$question = new Question('What is PHP', 'Adam');
$question->getFirstname;

like this? I assume Question class inherited Author class so Question's instance can use the function of Author Class?

A: You could create a new method:
class Question
{
    // ...
    function getAuthorFirstname()
    {
        return $this->author->getFirstname();
    }
}

$question = new Question(.., new Author(..., ...));
echo $question->getAuthorFirstname();

Or, if you don't really care about Law of Demeter or feel that it doesn't apply:
$question = new Question(.., new Author(..., ...));
echo $question->getAuthor()->getFirstname();

In the end it all comes down to striking a balance between information hiding and pragmatism.

A: Simple
echo $question->getAuthor()->getFirstName();

Think of it this way if it helps
$author = $question->getAuthor();
echo $author->getFirstName();

Also note that you can't construct a Question with the string "Adam", you need to pass an instance of Author
$question = new Question('What is PHP', new Author('Adam', 'Lastname'));

redpajama_set_name: RedPajamaStackExchange


text:
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ADF News
Reaction mixed on dairy code2 months, 2 weeks ago
Reaction mixed on dairy code
David Inall, Australian Dairy Farmers CEO 5 Nov 2019, 1:49 p.m.
The ADF is urging industry players to stay engaged in the code consultation process.
Does the mandatory dairy code of conduct meet expectations?
It's fair to say that the Government's exposure draft of a mandatory dairy code of conduct has received a mixed reception.
The new code has been hotly anticipated since it was first announced by the government last year, and with all the noise around beefing up the provisions, expectations of the exposure draft were sky high.
At first glance, many provisions in the exposure draft align with the suggestions put to the Government by ADF, including that processors should be required to uniformly publish standard form contracts by June 1 each year. This is a definite win for farmers.
But several other clauses are less convincing and ADF is seeking legal advice.
Under the exposure draft, processors that are classified as small businesses are exempted from acting in good faith, a central pillar of the mandatory code.
The industry must also closely investigate a provision allowing a two-week cooling period during which either party can choose to terminate a milk supply agreement, despite processors having a full 30 days to to supply written confirmation of that contract.
By that logic, a farmer who strikes a verbal agreement but finds they are dissatisfied when they receive the written contract 30 days later cannot rely on the cooling period to escape the agreement without penalty.
Similarly, it is surely unreasonable to expect that farmers and processors have the same capacity to pay the same $63,000 penalty for breaching the code. An average size processor can make hundreds of millions of dollars in revenue, compared with the $136,000 generated by the average dairy farm over the same period.
The draft code, as it stands, has created a storm of controversy. I can attest to the complex and emotionally charged nature of this issue. ADF has been on the receiving end of a lot of criticism around codes over the last couple of years, despite taking the lead, through the ADIC, on developing the original industry code back in 2017.
ADF has been on the receiving end of a lot of criticism around codes... despite taking the lead, through the ADIC, on developing the original industry code back in 2017.
Much of this nuance is missing from the highly politicised environment in which the code is being viewed.
The spotlight is on dairy, with a senate inquiry now geared at investigating the profitability of dairy farmers during the twenty years since the industry was deregulated. The code will now be examined through this prism.
Politics is politics but at the end of the day, we need a code of conduct that levels the playing field for farmers.
A lot is resting on the code, but we must be clear - it will not solve pricing issues along the dairy supply chain.
The mandatory code has been established principally to ensure transparency in contracts between processors and farmers.
We hope that this will lead to more open and honest relationships between suppliers and their processor, but we must manage expectations in terms of what the code will achieve.
ADF will be actively engaging in the consultation period so we can ensure farmers get a code that looks after their interests.
Ultimately, we want a code that outlaws retrospective price step downs, ensures processors announce their opening prices before the start of the season (preferably on the same day), allows farmers to switch processors if they choose to make that decision, and above all else, establishes a fair and equitable appeals process for contractual disputes.
In the past, farmers have felt they didn't have a voice. That needs to change. We want farmers to feel empowered to appeal if they believe their contract terms are unfair. We are urging all industry players to stay engaged in the process so that we can have a robust mandatory code of practice.
David Inall is the CEO of Australian Dairy Farmers
Campaign promotes dairy benefits
Dairy Australia is directing consumers through Google search to a website where it can answer questions about dairy products.
Laos dairy buffalo challenge | Video
Buffalo are helping one of South East Asia's poorest countries improve its food production systems.
Surviving depression and alcoholism, what it took to break free and thrive
Dairy farmer Aaron Thomas has a lesson for anyone battling depression and alcoholism.
Fire-hit timber towns should become carbon sinks: expert
Timber towns ravaged by bushfires burning in south-eastern NSW and Victoria should become carbon sinks, a Canberra academic has argued.
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A giant cow once iconic on the Sunshine Coast has found a new home on the Darling Downs where it will play a role in a dairy museum.
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