text:
OUR MAN IN WASHINGTON
By John Lofton
Number 22, April, 1987
One of the biggest of the many big lies we are being told is the
big lie that the way America is today is the way we always were.
One of the things that is so insidious about this particular big
lie is that it sends a message of despair and hopelessness. And it is
a denial of our history. But what is even more egregious about this
big lie is that it also denies the Biblical truth that with God all
things are possible. Thus, even if we have always been the way we
are, which isn't true, so what? God can change this. And if we seek
and obey Him, He will change what we are, He has promised.
An example of this big lie I am talking about occurred recently
on the Phil Donahue television program.
In a discussion about AIDS,
when a viewer, a woman, called in to ask whatever happened to having
sex solely within marriage, Mr. Donahue grimaced as if this was the
craziest question he had ever heard. He asked his caller,
incredulously, "So, why don't we just behave?" When his female caller
says, well, yes, Mr. Donahue interrupts, saying: "But we never have."
He says the idea of having sex only within marriage "won't work."
At the end of this program, in one of the most sickening
performances I have ever seen on national TV (and I have seen many),
Mr. Donahue throws free packages of condoms into his audience as they
laugh, shriek in delight, and applaud. All of which causes him to say
that we should remember that AIDS is not funny. He says it is a
disease that threatens all of us (another big lie, at least at this
time). And he says AIDS is a human problem "and sexual preference has
nothing to do with this" (another big lie).
Now, in a way, I am glad Mr. Donahue said what he said. Because
what he said forced me to read a book I wanted to read anyway, a book
titled Sex in Middlesex: Popular Mores In A Massachusetts County,
1649-1699 (University of Massachusetts Press, Amherst, 1986) and
written by Roger Thompson, a University Reader in the School of
English and American Studies, University of East Anglia, Norwich,
England. And what this book demonstrates, beyond a shadow of a doubt,
is that it is, indeed a very big lie to say that Americans have never
sexually behaved themselves.
First, a little background. Middlesex County covered the area
north and west of Boston and originally included the towns of
Cambridge and Charlestown. Within Middlesex were varied communities:
a seaport, a college town, country villages with and without resident
magistrates, frontier settlements, so called "peaceable kingdoms" and
feud-wracked settlements. The population of Middlesex in 1647 was
about 2,990; by 1699, the population had grown to 12,017. The main
source for Prof. Thompson's study was Middlesex County court records,
buttressed by town, church and genealogical records.
Okay. So, what did Prof. Thompson discover? Did he find an
epidemic of sexually-transmitted diseases in Middlesex? Did he find
homosexuals on the march, vigorously defending their "right" to commit
sodomy? Was wife and child abuse rampant? Were unborn babies being
slaughtered on demand? Were drugs being abused in a massive scale?
Was it a common occurrence for unwed teenage girls to be getting
pregnant? In a word: no, Prof. Thompson did not discover this.
But why? Why weren't any of these things discovered? Well, as
Prof. Thompson explains it: "The churches if Christendom might
slaughter each other in their thousands, but they did agree that there
was only one proper place for sexuality and that was the marriage bed.
Newlyweds should enter it - 'a thing undefiled' - unspotted virgins.
The authority of Biblical, canon, and statute law was reinforced with
emphatic urgency by countless Puritan sermons, oral tracts, and
conduct books. All asserted that the primary agency for enforcing
chastity was the family, buttressed where necessary by the community
and its officials." Thus, comparatively speaking, "the ratios of
bastardy and pre-marital fornication in Middlesex were low."
And illicit sex was punished. In the 1680's a male adulterer was
offered the alternative of 30 stripes or a 20 pound fine. Promiscuous
women could expect similar treatment. During the 1650's punishment
for pre-marital fornication ranged from 40 shillings to 10 pounds and
from six stripes or lashes to 15. Sexual abuse against a woman was
almost always dealt with severely. Men could receive at least 20
stripes, sometimes 30 or 40. Often monetary alternatives were not
offered; when they were, they were often at the stunning amount of 15
or 20 pounds.
Prof. Thompson observes: "The sheer number of people in each town
who were officially responsible for law and order is impressive. Lay
officers included any resident magistrate or associate or commissioner
for small causes; the constable (some larger towns had two); the town
selectmen, who might number seven leading citizens; the grand juryman;
the watch, who patrolled the town after the 9 p.m. curfew; and the
tythingmen, responsible by the act of 1679 for the moral oversight of
10 families.
Among church members, the one or two ministers, the lay
ruling elders, deacons, occasional deaconesses and catechisers
frequently kept the whole community in oversight." All of these
officials were elected or nominated by their constituents.
Quite apart from law enforcement officials, however, all
citizens, especially church members had a duty to maintain a holy
watchfulness over their neighbors. Why? Because, Prof. Thompson
notes: "People understood that condoning breaches of the moral code
would invite God's wrath in the whole community; the person who turned
a blind eye to sin was as bad as the sinnner.... The people of
Middlesex therefore had compelling spiritual and material reasons for
vigilance."
And this system worked. Prof. Thompson says that, on the average
there were only two cases per decade of sexual abuse and harassment by
unmarried males in Middlesex. He says "the evidence in sexual
deviance, abuse, and harassment would seem to support the contention
that in these areas of potential sexual release the typical youth of
the county exercised exemplary self-control.... There were no
reported rapes in the legal definition of that term."
Prof. Thompson says that while the adolescent culture of that
time might have made waves, it did not produce floods, it did not
revolutionize values. He says: "The wide range of adolescents in the
Middlesex records do not come across as brainwashed, willbroken
robots.
They chose to conform as they entered the adult world." And,
he notes, everyone in Middlesex was required to live in a family.
Most would live in at least three families during their lifetime:
their family of birth or orientation; the family where they were put
out as servants to be trained; and the family of marriage or
procreation. Some young people served in more than one of these
families.
But it is the crucial role played by the Christian faith that
comes through over and over again in Prof. Thompson's excellent book.
And what he has to say deserves quotation at length. Rejecting the
idea that it was tyrannical fathers that made people behave
themselves, he writes: "If not patriarchal despotism, what then
produced such comparatively high moral standards? A major factor,
encountered time and again in this study, was popular piety.
The great majority of men and women encountered in the court records were,
quite literally, God-fearing. Their threatening and unpredictable
world was theocentric: God controlled their uncertain destinies. A
young man faced with death at sea makes a vow to God to mend his ways
if He will save him. Another, meeting resistance to his advances,
desists because "God smote him with a trembling." A threatened woman
tells her abuser that "God sees, if nobody (else) sees, for God sees
in the dark."
Confessors describe how "God withdrew His protection"
or "Our great and crying sin cries aloud in the ears of God for
vengeance."
A woman who successfully resisted attack explained that "'it
pleased God to keep in me a resolution not to yield." And the phrases
"as God is my witness" and "as in the presence of God" and "before God
and man" reinforced solemn assertions or denials; they were not empty
phrases. A suspected liar is told "God and his conscience did know
how far he was guilty."
Prof. Thompson points out that the devil and hellfire were just
as real as God to the people of Middlesex. He says: "Indictments in
the superior court conventionally spoke of the accused as "not having
the fear of God before his eyes and being instigated by the Devil."
Individuals referred to "this beast he tare (God's commands) in
pieces" or to "Satan's temptations in persuading me to deny it."
Those who swore by the devil were liable to prosecution. The
invisible world was real enough to such people.
Prof. Thompson says this spiritual world was real even to those
apparent subversives, seducers and harrassers who violated God's law.
On more than one occasion such transgressors spoke of their crimes
saying that what they had done "was no evil" or "it was a small sin
and might be repented of." Indeed, "there is little evidence in
Middlesex of a thriving counter-culture to Puritan values," he writes.
Prof. Thompson also reports that the ordinary men and women in
the pews were not mere passive listeners to a "speaking aristocracy."
Far from it. He says that there was no chasm between ministers and
their congregations. Instead, the rhetoric of clerics was intrinsic
to a collective mentality they shared with ordinary people.
Religiosity pervade everyday life in fact as well as prescription.
The servants, farmers and housewives.... were in part the makers of
their own faith.... The plebian of Middlesex saw themselves as
conscience-armed against the wiles of the devil."
The only riot in Middlesex was in favor of the status quo,
against a hated symbol of English folk culture, the maypole. Says
Prof. Thompson: "Without this social cohesion, the magistrates' task
would have been impossible. As it was, their burden was lightened by
the pervasiveness of informal community control. Middlesex was well-
stocked with moral monitors who did not miss much in the goldfish-bowl
existence of daily life.
The very use of the word 'uncivil' to
describe immoral behavior suggests it was regarded as an offense not
only against God but also against civic standards. A court
appearance, then, was often a last resort for the intractable."
And few people went to jail more than once. Prof. Thompson
writes: "Examples of recidivism were rare. Many convicted young women
went on to marry; many convicted youths later served their communities
in elected or appointed offices. Even such an unspeakable young rogue
as Jerathmeel Bowes eventually became a representative in the General
Where the object of justice was not simply punishment or
protecting society from criminal behavior, but also the reform of the
criminal in which society had a vested interest, breaches of the moral
law would be reduced, and the growth of criminal class deterred.
Prof. Thompson says certain crimes actually declined over time,
notably domestic violence, and after the 1680's marital infidelity.
He says the social "odiom," the "scandal and reproach," and "the lash
of the law" were feared and felt in new communities as much as the
old, and in the 1690's just as strong as in the 1650's. And he says
the statistics of criminality show that New Englanders in general, and
Middlesex people in particular, were markedly more law-abiding than
people in the old country from which they came.
In conclusion, noting that "the silent majority behaved
themselves and sustained the New England Way," Prof. Thompson writes
that "far from being a brutal, suspicious, intolerant, and bigoted
crew, most of the people of Middlesex were concerned, considerate, and
cooperative. Consensus and common sense helped to make it a law-
abiding place."
He says: "The remarkably high standards of popular mores in this
Massachusetts county in the second half of the 17th Century owed far
less to oppressive patriarchalism than we have been led to believe.
This conclusion coincides with the findings of other recent studies on
both sides of the Atlantic. Modern relationships and group dynamics
would be far less alien to our early modern ancestors than has been
suggested. The provocative and challenging theses of the 1970's
(which have disputed this - J.L.) simply do not stand up under
detailed testing and examination."
Amen! So put that in your pipe and smoke it, Mr. Donahue!
God works, Mr.Donahue! His righteousness does exalt a nation,
Mr. Donahue!
Indeed, there was a time when Americans behaved themselves, Mr.
Donahue! All things do work together for good to those who love God,
to those who are the called according to God's purposes, Mr.Donahue!
To God be the glory for all good things! And I pray that, once again,
God will make us a good, well-behaved people.
Vallecito, CA 95251
Electronic reprint courtesy of Genesis 1.28 (206) 361-0751
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text:
Nation Building
Arts & Sports
About Eritrea
Eritrea at a Glance
Haddas Ertra
Eritrea Alhaditha
Eritrea Haddas
Eritrea Profile
Men'esey Magazine
Shebab Magazine
Agizo Magazine
Men'esey Magazine (Arabic)
"Fight Against All Forms Of Gender Discrimination"
The 40th anniversary of the foundation of the National Union of Eritrean Women (NUEW) was commemorated with great enthusiasm under the theme "fighting against gender-based violence in the world of work".
The celebration, which was attended by ministers, diplomats, high government officials, members of the NUEW and other invited guests, was officially opened on 25 November at the NUEW's Headquarters in Asmara. In her opening speech at the ceremony, Ms. Zaid Mesfin, head of NUEW's finance and administration, stated that the forty years of honor being celebrated is the result of efforts the government made during the armed struggle as well as the result of the hardship and brawl that Eritrean female fighters went through to assure their rights and equality.
Ms. Zaid also noted that without the active participation of women, the Eritrean struggle for liberation and the national development programs would not have been successful. To this end, women were encouraged, mobilized and organized to participate and enjoy their equal access and opportunity. During the armed struggle women made up 30% of the freedom fighters. Overall, today more than 45% of the students in Eritrea are girls. Women's right to land use has been guaranteed by law.
She also highlighted that the implementation of policies on gender equality and women empowerment is not only left to women. To fight discrimination against women and girls, men and boys need to be involved. As part of measures to redress gender stereotypes in all spheres of life, massive awareness raising and sensitization programs have been conducted in government institutions, national civic associations and the communities at large. Administrations at national, regional and local levels are involved in organizing campaigns through drama, parades, cultural shows and other activities and have helped in challenging the existing attitudes and stereotypes.
This year's theme is "fighting against gender-based violence in the world of work. The 1994 Eritrean national charter explicitly states, 'In Eritrea there shall not be any position exclusively reserved for men that cannot be attained by women'. This shows that at policy level there is no segregation. Similarly, the labor proclamation has provided protection with a view to avoiding discrimination in giving access to employment and occupation on the grounds of gender. But for better and lasting results, the implementation of policies and proclamations needs to be continuously assessed and monitored by institutions and individuals. Ms. Zaid concluded her speech by calling on the government and the NUEW to strengthen their efforts in reducing female illiteracy rate in all ethnic groups and regions.
Ms.Shaheen Nilofer, UNICEF representative in Eritrea, said in her brief speech that she was very glad to be part of the commemoration and expressed her admiration for the NUEW for their strong organization as well as their effort to ensure women's equality and other rights. She also applauded the victory in reducing maternal mortality rate and the significant reduction of FGM.
Finally, members of the NUEW who graduated from different colleges shared their educational and work experiences. Ms. Sitel, a lawyer, Amira, a political scientist, Lwam, a journalist, and Millen, a marine engineer, all said that education is the base for a nation's development and an educated woman is the backbone of a civilized society. They called on the society to give all the support they can for girls' education.
As part of the celebrations to commemorate NUEW's 40th anniversary, there was a carnival that included a parade from Martyr's Avenue all the way to Bahti Meskerem Square. On the parade were young women in military outfit, women in colorful chiffons (traditional clothes) representing different regions of Asmara and women from different governmental and non-governmental organizations.
As is the tradition, the ceremony at Bahti Meskerem Square began with a moment of silence in honor of Eritrea's fallen heroines and heroes. In an opening speech, Ms. Alem Belay, head of Maekel region's NUEW, said that the NUEW, with its strong organization and enlightened vision, played a big role for independence. During the struggle for independence, armed with the slogan "we want peace but peace with justice", the Eritrean women's organization participated in international conferences, including one in Copenhagen in 1980 and Nairobi in 1985.
There were women in the ranks and files of many revolutionary movements in Asia and Africa, but they were not comparable to that of Eritrea both in number and potential. Ms Alem underlined the success made in reducing maternal mortality, which in 1990 was 1590 per 100,000 was reduced to 250 per 100,000 in 2019. She also highlighted that the ceremony is not only meant to glorify the achievements of the women's organization but also to reflect on what has not been done. She made a pledge that improved policies to allow full participation of women in policy making and in work places will be drafted and ratified in the future. The event was made colorful with music, poetry recitation and other performances.
In socioeconomic activities, women's participation in Eritrea has grown over the years. Women made up 55% of those who benefit from in microcredit programs. The role of women in politics has also risen. Almost a quarter of the ministerial posts in Eritrea are occupied by women. On average, women constitute 34% of village level leaders, 11% at subzones (kebabi) and 31% at regional assembly. Thirty seven percent of the community court judges and 14% of the regional court judges are women.
Major strides have been made in the fight against all harmful practices including Female Genital Mutilation (FGM) and underage marriage by raising people's awareness of the effects of the harmful practices and by drafting laws. A law was ratified to punish those who practice FGM. A study has shown that FGM on young girls aged 15 to 19 was reduced from 68% in 2002 to 33% in 2010.
The celebrations to commemorate the 40th anniversary of Eritrea's women's organization came to a close on Friday, 29th November, at an event held at the Expo hall. In their speeches on the occasion, Ms. Alem Belay, Major General Romodan Aweliyay, governor of the Central region, Mrs. Abraham Semere, chairperson of Maekel region's assembly, and Mrs. Samsom kifle, chairperson of the National Union of Eritrean Youth and Studetnts, paid tribute to the Eritrean women for their role in achieving Eritrea's independence as well as for safeguarding the nation's sovereignty. They also expressed their respect for all Eritrean women who are doing their part for the development of their nation in different sectors. At last they forwarded their best wishes to the NUEW on its future work.
Eritrea ratified the Convention on the Elimination of all Forms of Discrimination Against Women (CEDAW) in 1995, immediately after its independence 1991. The convention was adopted by the United Nations on December 18, 1979.
Glory to Eritrean women for achieving freedom and equality
Shamot Academy Award: a night the Old Guard passed the torch to the New Guard of Eritrea Entertainment Industry
Eritrean colleges strive to meet the demands of national appetite
Mental Illness and Disorders: A Snapshot
The International Year of Health and Care Workers
Book Review and Conversation with Amanuel Biedemariam
Ministry of Agriculture using Artificial Insemination to Improve Horse Breeds
EriPic
Fast tracking to slow fashion
An Agricultural Strategic Information System (SIS) in Eritrea
ሓበሬታ ካብ ሚኒስትሪ ጥዕና
Announcement from the Ministry of Health
UNHCR Eligibility Guidelines: Factual Findings or Recycled Defamation?
Local magazines
© 2021 - Eritrea Ministry Of Information. All Rights Reserved.

redpajama_set_name: RedPajamaCommonCrawl


Description: This tutorial will walk you through the process of taking generated random points of known invasive tamarisk stands and using them to map the species' distribution. This process involves inputting several environmental layers that the model will use to predict the tamarisk distribution. Only a single month will be run through the model at a time.


It is an honor and privilege to join Butterfield Canyon Elementary School.
After graduating from Radford University in Virginia with a degree in Elementary Education, I spent most of my adult life in El Paso, TX. There, I raised three amazing children and worked for the El Paso Independent School District for 25 years, teaching kindergarten through sixth grade, and finally technology.
About a year ago, I moved to Salt Lake to be closer to two of my children and three—soon to be four—grandchildren.
When I'm not teaching, I enjoy spending as much time as possible with them, traveling, photography and mountain biking.
I am looking forward to this opportunity and very excited to be a Bobcat!


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Do you like podcasts? Do you enjoy eavesdropping on chatty people on the bus or at the grocery store? Have you ever pondered, "Who is the best Golden Girl?"
Then you need Mohr Please!
Check us out on Apple Podcasts, Google Play, Stitcher, and more!


SpotCayman Makes Your Trip to Cayman Awesome!
We only get three off weekends during the NASCAR schedule, and we really wanted to make the most out of our family trip!
Cayman Islands - Came here on a cruise, and had a magical swim in these ridiculously clear waters.
Seven Mile Beach at The Caribbean Club Grand Cayman. On the day we were here it was very windy but still beautiful.
Look at this beautiful resort I found on IntervalWorld.com.
These rays are very safe, making this the number one tourist activity in the Cayman Islands.
Grand Cayman boasts it's first-ever boutique resort, directly on world-renowned Seven Mile Beach.


We are proud to Publish our 2nd Durga Puja Souvenir after a successful Durga Puja(Soft Copy attached). Hope you all like this Souvenir.
You may feel free to take a hard copy print out and share with well wishers and patrons.


text:
\section{Main Results}

\noindent
Consider the complex untwisted affine Lie algebra of type $A_{n-1}^{(1)}$ :
$$\mathfrak{g}=\widehat{\mathfrak{sl}}_n=
\sl_n({\C[t^{\pm 1}]})\ \oplus\ \mathbb{C} K\ \oplus\ \mathbb{C} d
,$$
where $\sl_n({\C[t^{\pm 1}]})$ denotes the traceless $n\times n$ matrices with entries in the Laurent polynomials ${\C[t^{\pm 1}]}=\mathbb{C}[t,t^{-1}]$,\ $K$ is a central element of $\mathfrak{g}$, and $d=t\frac{d}{dt}$ is a derivation (see \cite[Ch 7]{kac1}).  We have the Cartan decomposition $\mathfrak{g}=\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{n}_{-}$, where $\mathfrak{h}$ is the Cartan subalgebra
$$
\mathfrak{h}=\bigoplus_{1\leq i\leq n-1}
\!\!\! \mathbb{C}(E_{ii}\sh-E_{i+1,i+1})\
\oplus\ \mathbb{C} K \ \oplus\ \mathbb{C} d\,;
$$
and $\mathfrak{n}$ is the maximal nilpotent subalgebra
$$
\mathfrak{n}:=\mathop{\bigoplus_{1\leq i<j\leq n}}_{k\geq 0}\!\!\!\mathbb{C} t^kE_{ij}\ \oplus\mathop{\bigoplus_{1\leq i<j\leq n}}_{k\geq 1}\!\!\!\mathbb{C} t^k E_{ji}
\ \oplus\
\mathop{\bigoplus_{1\leq i\leq n-1}}_{k\geq 1}t^k
(E_{ii}-E_{i+1,i+1})\,.
$$
Here $E_{ij}\in\mathfrak{gl}_n(\mathbb{C})$ denotes a coordinate matrix.

Let $\Lambda_0,\Lambda_1,\ldots,\Lambda_{n-1}$ be the fundamental weights of $\mathfrak{g}$, and let $V(\Lambda_m)$ be the level 1 irreducible $\mathfrak{g}$-module with highest weight $\Lambda_m$. (Thus, $V(\Lambda_0)$ is the basic representation of $\mathfrak{g}$.)\  Let us recall the construction of $V(\Lambda_m)$ inside the fermionic Fock space $\mathcal{F}$ (cf.~\cite[Ch 14]{kac1}, \cite{kac2}).  Let $\mathbb{C}^\infty=\bigoplus_{i\in\mathbb{Z}} \mathbb{C} \epsilon_i$ be the $\mathbb{C}$-vector space with basis $\{\cdots,\epsilon_{-2},\epsilon_{-1},\epsilon_0,\epsilon_1,\epsilon_{2},\cdots\}$.

Let $\mathcal{T}$ denote the collection of all subsets $J\subset\mathbb{Z}$ which are comparable to the non-positive integers $\mathbb{Z}_{\leq 0}$, meaning that $J\sh\setminus\mathbb{Z}_{\leq 0}$ and $\mathbb{Z}_{\leq 0}\sh\setminus J$ are both finite:
$$
\mathcal{T}=\{\,J\subset\mathbb{Z}\ \ \ \text{s.t.}\ \ \
|J\sh\setminus\mathbb{Z}_{\leq 0}|,\,|\mathbb{Z}_{\leq 0}\sh\setminus J|<\infty\,\}\,.
$$
We write such a set as:
$$
J=\{\cdots<j_{-2}<j_{-1}<j_0\}\,.
$$
Define the Fock space as the semi-infinite wedge product of $\mathbb{C}^\infty$:
$$\mathcal{F}=\wedge^{\!\infty/2}\,\mathbb{C}^\infty:=\bigoplus_{J\in\mathcal{T}}\mathbb{C} \epsilon_J\,,$$
the $\mathbb{C}$-vector space with basis elements:
$$
\epsilon_J:=\,\cdots\sh\wedge\epsilon_{j_{-2}}\sh\wedge\epsilon_{j_{-1}}\sh\wedge\epsilon_{j_{0}}\,.
$$
Thus, the $J\in\mathcal{T}$ play the role of tableaux indexing the basis vectors of the Fock space.

For $i,j\in\mathbb{Z}$, let $E_{ij}'$ denote a coordinate
matrix acting on $\mathbb{C}^\infty$ by $E_{ij}'(\epsilon_j)=\epsilon_i$,\ \, $E_{ij}'(\epsilon_k)=0$ for $k\neq j$.
Then for $i<j$,\ $E_{ij}'$ acts on the Fock space in the expected way:
$$
E_{ij}'(\epsilon_J) = \left\{\begin{array}{cl}
\pm \epsilon_{J\setminus j\cup i}&\text{if } j\in J,\ i\not\in J\\[.5em]
0& \text{otherwise,}
\end{array}\right.$$
Here we denote:
$$
J\setminus j\cup i\,:=\,(\,J\sh\setminus\{j\}\,)\,\cup\,\{i\}\,,
$$
the operation which moves the element $j\in J$ to the vacant position $i\not\in J$;
and $\pm=(-1)^\ell$ with $\ell={|J\cap[i,j]|-1}$, the sign of the permutation needed to sort the wedge factors of $\epsilon_{J\setminus j\cup i}$ into increasing order.

We let:
$$
{\widehat{E}}_{pq}:=\sum_{k\in\mathbb{Z}} E_{p+nk,q+nk}'\,,
$$
which is a well-defined operator on $\mathcal{F}$.
Now, if $i<j$ or $k>0$, we let $t^k E_{ij}$ act on $\mathcal{F}$ by the operator
${\widehat{E}}_{pq}$, where $p=i-nk$, $q=j$:
$$
t^k E_{ij}={\widehat{E}}_{i-nk,\,j}:\mathcal{F}\to\mathcal{F}\,.
$$
This defines the action\begin{footnote}{
This action arises naturally
if we identify the free ${\C[t^{\pm 1}]}$-module
${\C[t^{\pm 1}]}^n=\bigoplus_{i=1}^n{\C[t^{\pm 1}]} \epsilon_i$ with the $\mathbb{C}$-vector space
$\mathbb{C}^\infty=\bigoplus_{j\in\mathbb{Z}}\mathbb{C}\epsilon_j$
via: $t^k \epsilon_i\leftrightarrow\epsilon_{i-nk}\,.$
This gives an embedding $\mathfrak{gl}_n({\C[t^{\pm 1}]})\subset\mathfrak{gl}(\mathbb{C}^\infty)$,
so that the natural action of the upper triangular part of $\mathfrak{gl}(\mathbb{C}^\infty)$ on the Fock space restricts to the specified action of $\mathfrak{n}\subset\mathfrak{gl}_n({\C[t^{\pm 1}]})$.
However, this gives only a projective representation of the entire $\mathfrak{gl}_n({\C[t^{\pm 1}]})$, which then lifts to a true representation of the central extension $\widehat{\mathfrak{gl}}_n$.}
\end{footnote}of $\mathfrak{n}$ on $\mathcal{F}$, and we can similarly define the action of $\mathfrak{n}_{-}$ and $\mathfrak{h}$.
Indeed, the Chevalley generators of $\mathfrak{n}_{-}$ are:
$F_i=E_{i+1,i}={\widehat{E}}_{i+1,i}$ for $i=1,\ldots,n\sh-1$ and $F_0=t^{-1}E_{1,n}={\widehat{E}}_{n+1,n}$.

Now let $L_m:=\mathbb{Z}_{\leq m}\in \mathcal{T}$.
It is well known that the $U(\mathfrak{g})$-span of
the highest-weight vector $\epsilon_{L_m}$ is an irreducible $\mathfrak{g}$-module:
$$
U(\mathfrak{g})\cdot \epsilon_{L_m} = U(\mathfrak{n}_{-})\cdot \epsilon_{L_m} \cong
V(\Lambda_m)\,,
$$
where we define $\Lambda_{m}:=\Lambda_{(m \mod n)}$.

Recall that we can realize the Weyl group $W$ of $\mathfrak{g}$ as a permutation group on $\mathbb{Z}$.  Indeed, we can write the simple reflection $s_i:\mathbb{Z}\to\mathbb{Z}$ as a product of commuting transpositions:
$$
s_i:=\prod_{k\in\mathbb{Z}} (i\sh+nk,\,i\sh+1\sh+nk)\ ,
$$
so that $s_i(i')=i'\sh+1$ whenever $i'\equiv i\mod n$.
Then $W=\langle s_0,\ldots,s_{n-1}\rangle$ is the corresponding Coxeter group.

The Weyl group $W$ acts on $\mathcal{T}$ via
$w(J):=\{w(j)\}_{j\in J}$.
Indeed, the extremal weight vectors of $V(\Lambda_m)\subset\mathcal{F}$ are just $\epsilon_J$ for $J=w(L_m)$.
Equivalently, a basis vector $\epsilon_J$ is an extremal weight vector whenever $J$ is {\it $n$-stable}:  that is, whenever $j-n\in J$ for all $j\in J$.
We define the {\it parabolic Bruhat order} between $K=\{\cdots\sh<k_{-1}\sh<k_0\}$ and $J=\{\cdots\sh<j_{-1}\sh<j_{0}\}$ as:
$$
K\stackrel{\rm B}{\leq} J \quad\Longleftrightarrow\quad
\left\{\begin{array}{cl}
k_i\leq j_i&\text{ for all } i\\
k_i=j_i&\text{ for all } i\sh\ll 0
\end{array}\right.
$$
This induces an order on the $n$-stable $J=w(L_m)$ which is consistent with the usual Bruhat order on $w\in W$.

The {\it Demazure modules} \cite{demazure} of $V(\Lambda)$ are the $\mathfrak{n}$-modules obtained by raising the extremal weights:
$$
V_w(\Lambda_m)\ \cong\ U(\mathfrak{n})\cdot \epsilon_{w(L_m)}\,.
$$
We can get the same modules also by lowering the highest weight:
$$
V_w(\Lambda_m)=\mathop{\rm Span}\nolimits_\mathbb{C}\{F_{i_1}^{k_1}\cdots F_{i_t}^{k_t}
\epsilon_{L_m}\
\mid k_1,\ldots,k_r\geq 0\}\,,
$$
where $w=s_{i_1}\cdots s_{i_t}$ is a reduced word.

Next we describe the sets $J\in\mathcal{T}$ which index basis vectors of $$V_w(\Lambda_m)\subset V(\Lambda_m)\subset\mathcal{F}\,.$$
Let us say that a set $J\in\mathcal{T}$ is
{\it n-bounded} if $j_i-j_{i-1}\leq n$ for all $i$.
Also, we define the {\it order} of a set $J$ by:
$\mathop{\rm ord}\nolimits(J):=|J\sh\setminus \mathbb{Z}_{\leq 0}|-|\mathbb{Z}_{\leq 0}\sh\setminus J|$ ; equivalently, $\mathop{\rm ord}\nolimits(J)=m$ means that
$j_i=m+i$ for all sufficiently large negative $i$.
Now let
$$\begin{array}{rcl}
\mathcal{C}(L_m)&:=&
\{J\in \mathcal{T}\ \mid\ \mathop{\rm ord}\nolimits(J)=m\,\text{ and $J$ is $n$-bounded}\}
\\[.3em]
&=&\left\{J=\{\cdots\sh<j_{-2}\sh<j_{-1}\sh<j_{0}\}\subset\mathbb{Z}\,\left|\begin{array}{c}
j_i=m+i\text{ for }i\ll 0\\
j_i-j_{i-1}\leq n\text{ for all }i
\end{array}\right.\right\}
\end{array}$$
(The reader should be aware of a frequently used alternative notation in terms of ``colored Young diagrams'' instead of subsets.\!\!
\footnote{In \cite{FLOTW} and related literature, the basis
 of $V(\Lambda_m)$ is indexed by the set of all
partitions $\lambda=(\lambda_1\sh\geq\lambda_2\sh\geq\cdots)$ with $\lambda_i\geq 0$, \ $\lambda_i=0$ for $i\gg0$, and
$\lambda_{i+1}\sh-\lambda_i\leq n\sh-1$.  Namely, a set $J=\{\cdots\sh< j_{-1}\sh<j_0\}$ of order $m$ corresponds to $\lambda$ with $\lambda_{i+1}=m-i-j_{-i}$.  It is useful to picture $\lambda$ as a Young diagram colored with a mod-$n$ checkerboard pattern: square $(i,j)$ has color $i-j\in\mathbb{Z}/n\mathbb{Z}$.
})

We can give $\mathcal{C}(L_m)$ a crystal graph structure by defining the crystal lowering operators ${f}_i$
for $i=0,\ldots,n\sh-1$, as recalled in Section 2 below.
If it is defined, the crystal operator $f_i$ on a set $J$ picks out a certain element $r\in J$ with $r\equiv i\mod n$, and replaces it with $r\sh+1\equiv i\sh+1\mod n$: that is,
$f_i(J)=J\setminus r\cup(r\sh+1)\,.$
We define the Demazure crystal as:
$$\mathcal{C}_w(L_m)=\{ {f}_{i_1}^{k_1}\cdots {f}_{i_t}^{k_t} L_m\mid k_1,\ldots,k_t\geq 0\},$$
where $w=s_{i_1}\cdots s_{i_t}$ is again a reduced word.

Our first theorem is a simpler description of the sets $J$ in this Demazure crystal, in analogy with the ``left key'' algorithm of Lascoux-Schutzenberger \cite{LS}.
If $J$ is $n$-bounded but not $n$-stable, define the following up-operation (which is different from the crystal operators):
$$\begin{array}{c}
\mathop{\rm up}\nolimits(J):=J\setminus p\cup q\quad \text{where:}\\[.5em]
p:=\max\{p'\ \mid\  p'\in J,\ \,p'\sh-n\not\in J\},\quad
\\[.5em]
q:=\min\{q'> p\ \mid\ q '\not\in J,\ \ q'\sh-n\in J,\ \
q'\not\equiv p \mod n\}\,.
\end{array}$$
To rephrase this in words, define a {\it seam} as a maximal arithmetic progression $S=\{\cdots<j\sh-2n<j\sh-n<j\}$ contained in $J$.  We call the vacant position $j\sh+n\not\in J$ the {\it tight end} of $S$\,; and if $S$ is finite, we call the minimal element $p\in S$ the {\it loose end} of $S$.  The up-operation moves $p\in J$ to $q\not\in J$, where $p$ is the maximal loose end in $J$, part of a seam $S=\{p,p\sh+n,\ldots\}$, and $q>p$ is the tight end of a different seam, the minimal such tight end.  See the examples below.

Iterating the up-operation ``pulls out" this seam, distributing all the elements of $S$ to the tight ends of different seams; and then the operation starts on another seam.  After each seam is pulled out, the number of loose ends decreases by one.  Once all the finite seams of $J$ are pulled out, the result is an $n$-stable set which we call the {\it roof} of $J$:
$$
\mathop{\rm roof}\nolimits(J):=\mathop{\rm up}\nolimits^\ell(J)=\mathop{\rm up}\nolimits(\cdots\mathop{\rm up}\nolimits(J)\cdots)=y(L_m)
\quad\text{for some}\ y\in W\,.
$$
{\bf Theorem 1}\quad {\it
Let $\mathcal{C}_w(L_m)$ be the Demazure crystal generated from the highest weight $L_m$ according to a reduced word for $w\in W$.  Then:
$$\begin{array}{rcl}
\mathcal{C}_w(L_m) &=&\{J\in\mathcal{C}(L_m) \ |\ \mathop{\rm roof}\nolimits(J)\leq w(L_m)\}
\end{array}$$
}
\\[-1.2em]
This gives a highly efficient algorithm for testing the membership of $J$ in $\mathcal{C}_w(L_m)$.  (Later in this section we give a corresponding algorithm for generating all $J\in\mathcal{C}_w(L_m)$.)

Next we give elementary bases of $V(\Lambda_m)$ and its dual which are compatible with the Demazure modules, in analogy to the construction of Raghavan-Sankaran \cite{raghavan} (generalized by Littelmann \cite{littelmann3}).
\\[1em]
{\bf Theorem 2}\quad {\it
(i) Given $J\in\mathcal{C}_w(L_m)$, suppose $\mathop{\rm up}\nolimits^i(J)=\mathop{\rm up}\nolimits^{i-1}(J)\setminus p_i\cup q_i$ for $i=1,\ldots,\ell$, and $\mathop{\rm roof}\nolimits(J)=\mathop{\rm up}\nolimits^\ell(J)=y(L_m)$.
Define $$ v_J:=
{\widehat{E}}_{p_1,q_1}\cdots{\widehat{E}}_{p_\ell,q_\ell}\epsilon_{y(L_m)}\,.$$
Then the irreducible highest-weight module $V(\Lambda_m)$,
a submodule of the Fock space $\mathcal{F}$, has basis $\{v_J\mid J\in\mathcal{C}(L_m)\}$; and the Demazure module $V_w(\Lambda_m)$ has basis $\{v_J\mid J\in\mathcal{C}_w(L_m)\}$.
\\[.5em]
(ii) The irreducible lowest-weight module $V(\Lambda_m)^*$,
a quotient of the dual Fock space $\mathcal{F}^*$, has basis
$\{\epsilon_J^*\mid J\in\mathcal{C}(L_m)\}$; and the dual Demazure module $V_w(\Lambda_m)^*$ has basis
$\{\epsilon_J^*\mid J\in\mathcal{C}_w(L_m)\}$.
Here $\epsilon_J^*$ denotes a dual basis vector of $\mathcal{F}^*$ restricted to $V(\Lambda_m)$ or to $V_w(\Lambda_m)$ respectively.
\\[.5em]
}
In geometric terms, the functions $\epsilon_J^*$ can be considered as Plucker coordinates on the affine Grassmannian embedded in the infinite projective space $\mathbb{P}(\mathcal{F})$.

The vectors $v_J$ possess a triangularity property
with respect to the standard basis of the Fock space.
Define {\it lexicographic order} on sets
$K,J$ as follows:
$$
K\stackrel{\rm lex}{<} J\quad\Longleftrightarrow\quad\left\{\begin{array}{cl}
k_N<j_N&\text{for some $N$}\\
k_i=j_i &\text{for all $i<N$\,.}
\end{array}\right.
$$
{\bf Proposition 3}\quad {\it
Let us write:
$$v_J=\sum_K a_K^J \epsilon_K\quad \text{with coefficients}\quad\ a_K^J\in\mathbb{C}\,.$$
\\[-1em]
(i) We have a non-zero coefficient
$a_K^J\neq 0$ only if $K\stackrel{\rm lex}{\leq} J$.\\
(ii) We have a non-zero leading coefficient $a_J^J\neq 0$  for every $J$, given explicitly as follows.
If $J$ is $n$-stable, then $a_J^J=1$.
If $J$ is $n$-bounded but not $n$-stable, suppose that $\mathop{\rm up}\nolimits^i(J)=\mathop{\rm up}\nolimits^{i-1}(J)\setminus p_i\cup q_i$, and $t$ is maximal such that
$p_1\equiv\cdots\equiv p_t\mod n\,.$
Define $\mu_d:=\#\{\,i\leq t\mid q_i-p_i=d\,\}\,$ and  ${\widetilde J}:=\mathop{\rm up}\nolimits^t(J)$. Then:
$$
a_J^J=\pm\left(\prod_{d\geq 1}\mu_d!\right) a_{\widetilde J}^{\widetilde J}\,.
$$
}
\\[0em]
In part (ii), note that the sequence $S=\{p_1<\ldots<p_t\}$ is actually the first seam of $J$ pulled out by the roof algorithm:\
$S=\{p\,,\,p\sh+n,\ldots, p\sh+n(t\sh-1)\}$.
Iterating part (ii), we get a combinatorial formula for
the leading coefficient $a_J^J$ of each $v_J$ depending only on the sequences $p_1,\ldots,p_\ell$ and $q_1,\ldots,q_\ell$ in the roof algorithm.
\\[1em]
{\bf Example}\quad Let $n=5$ and let:
$$
J:=\{\ldots,-4,-3,-2,-1, 0,3,4,7,10,12,14,17,18,23,27,32,33,35,37\}\,.
$$
Then $J\in\mathcal{C}(L_m)$ for $m=14$, since
$L_0\subset J$ and $|J\sh\setminus L_0|=14$, so that
$\mathop{\rm ord}\nolimits(J)=\mathop{\rm ord}\nolimits(L_0)\sh+14=14$.
We sort $J$ into its residue classes mod $n$ to show the seam structure. We mark the maximal loose end with boldface, and the tight end used by the up-operation with $\TT$.
$$\begin{array}{r@{\,}c@{\,}l}
J&=&\!\!
\mbox{\footnotesize$\left[\begin{array}{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdots&-4&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdots&-3&\cdot&7&12&17&\cdot&27&32&37\\
\cdots&-2&3&\cdot&\cdot&18&23&\cdot&33&\cdot\\
\cdots&-1&4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdots&\ \ 0&\cdot&10&\cdot&\cdot&\cdot&\cdot&{35}&\cdot&
\end{array}\!\!\!\right]$}
=
\mbox{\footnotesize$\left[\begin{array}{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&27&32&37\\
3&\cdot&\cdot&18&23&\cdot&33&\TT\\
4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\mathbf{35}&\cdot&
\end{array}\!\!\!\!\right]$}
\\[2.5em]
&\stackrel{\mathop{\rm up}\nolimits}{\to}&
\!\!\mbox{\footnotesize$\left[\begin{array}{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&27&32&37&\TT\\
3&\cdot&\cdot&18&23&\cdot&\mathbf{33}&38&\cdot\\
4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 2}}}{\longrightarrow}
\mbox{\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&\mathbf{27}&{32}&{37}&{42}&{47}\\
3&\cdot&\cdot&18&23&\TT&\cdot&\cdot&\cdot&\cdot\\
4&\cdot&14&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\\[2.5em]
&\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 5}}}{\longrightarrow}&
\!\!\mbox{\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&7&12&17&\cdot&\cdot&\cdot&\cdot&\cdot&\\
3&\cdot&\cdot&\mathbf{18}&23&28&33&38&43&48\\
4&\cdot&14&\TT&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&10&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\!\!\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 16}}}{\longrightarrow}\!\!\!\!
\mbox{
\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\!\!\mathbf{7}&\!\!12&17&22&27&32&37&42&47&52&57&62\\
3&\TT&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
4&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
\\[2.5em]
&\stackrel{\ \mathop{\rm up}\nolimits^{\mbox{\scriptsize 12}}}{\longrightarrow}&
\!\!\mbox{
\footnotesize$\left[\begin{array}
{@{\ }c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\ \,}c@{\,}}
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
3&8&13&18&23&28&33&38&43&48&53&58&63\\
4&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot\\
\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot&\cdot
\end{array}\!\!\!\right]$}
=\mathop{\rm roof}\nolimits(J)=y(L_{14})\,.
\end{array}$$
We thus have $(p_1,q_1)\sh=(35,38)$,\
$(p_2,q_2)\sh=(33,42)$,\
$(p_3,q_3)\sh=(38,47)$,\ldots, and:
$$\begin{array}{rcl}
v_J&=& {\widehat{E}}_{35,38}\,{\widehat{E}}_{33,42}\,{\widehat{E}}_{38,47}
\cdots \epsilon_{y(L_{14})}\\[.5em]
&=&{\widehat{E}}_{5,8}
{\widehat{E}}_{3,12}^{\,2}\,
{\widehat{E}}_{2,3}^{\,5}\,
{\widehat{E}}_{3,4}^{\,7}\,
{\widehat{E}}_{4,7}^{\,8}\,
{\widehat{E}}_{5,57}\,
{\widehat{E}}_{2,3}^{\,12}
\ \epsilon_{y(L_14)}\,.
\end{array}$$
Since $J$ has seven loose ends, we must apply Proposition 3(ii) seven times to compute:
$
a_J^J=\pm\ 1!\cdot 2!\cdot  5!\cdot  7!\cdot  8!\cdot
1!\cdot  1!\cdot  12!\,.
$
(Here we have only one factorial for each seam, though in general there will be several.)

To determine a reduced decomposition for the Weyl group element $y$, we start with the extremal weight $K=y(L_{14})$ and perform the simple reflection:
$K\mapsto s_{r}(K)$,
where
$$
r:=\min\,\{\,k\!\not\in\! K\ \mid\ k\sh+1\!\in\! K\,\}\,
$$
is the minimal ``hole'' of $K$, and $s_r:=s_{(r\mod n)}$.  This will always give
$K\stackrel{\mathrm{B}}{>}s_r(K)$, and iterating the operation produces a canonical reduced word for $y$.
Indeed, $$
y(L_{14})=s_{2}\,s_{1}\,s_{3}\,s_{2}\,s_{0}\,
(s_{4}s_{3}\,s_{2}\,s_{1}\,s_{0})^{11}\,s_{4}\, L_{14}
\ .$$
See also the Example in the next section.
\quad$\Box$
\\[1em]
{\bf Example}\quad Let $n=2$.  Then for fixed $m$, the Bruhat order on the sets $w(L_m)$ reduces to a linear order: for example,
$$L_0 = s_1(L_0)
\stackrel{\rm B}{<}
s_{0}(L_0)
\stackrel{\rm B}{<}
s_{1}s_{0}(L_0)
\stackrel{\rm B}{<}
s_{0}s_{1}s_{0}(L_0)
\stackrel{\rm B}{<}\cdots\,.$$
For $J$ an $n$-bounded set with order $m$, the roof operation reduces to:
$$\begin{array}{rcl}
\mathop{\rm roof}\nolimits(J)&=&\min\{\,w(L_m)\,\mid\, w\in W,\ J\stackrel{\rm B}{\leq} w(L_m)\,\}\\[.5em]
&=&L_a\cup\{a\sh+2,a\sh+4,\ldots,a\sh+2k\}\,,
\end{array}$$
where $a=\max\{a'\mid L_{a'}\subset J\}$ and $k=|J\setminus L_a|$.  That is, the Demazure crystal is simply $\mathcal{C}_w(L_m)=\{J\in\mathcal{C}(L_m)\,\mid\,J\stackrel{\rm B}{\leq} w(L_m)\}$.
This case is further considered in the context of completely integrable lattice models in \cite{FMO}.  \quad$\Box$
\\[1em]
{\bf Example}\quad Generalizing the previous case, let $n$ be arbitrary and suppose $w(L_m)$ is of the form:
$$
w(L_m) = L_a\cup
\{a\sh+n, a\sh+2n, \ldots, a\sh+kn\}
$$
for some $a$ and $k = m-a$.  Then for any $J\stackrel{\rm B}{\leq} w(L_m)$, we have $\mathop{\rm roof}\nolimits(J)\stackrel{\rm B}{\leq} w(L_m)$, so that again $\mathcal{C}_w(L_m)=\{J\in\mathcal{C}(L_m)\,\mid\,J\stackrel{\rm B}{\leq} w(L_m)\}$.  This case, which is considered in \cite{MW}, is exceptional: in general, it often happens that $J\stackrel{\rm B}{\leq} w(L_m)$, but $\mathop{\rm roof}\nolimits(J)\not\stackrel{\rm B}{\leq} w(L_m)$.
\quad$\Box$
\\[1em]
Next we consider the modifications which must be made to our theory to generalize it to positive characteristic.
Since the leading coefficients $a_J^J$ are not necessarily $\pm 1$, the vectors $v_J$ could become linearly dependent if we work over $\mathbb{Z}$ and then reduce modulo a prime. To define a characteristic-free basis $\{\,v^{\,\prime}_J\mid J\in\mathcal{C}_w(L_m)\,\}$, we start with $v^{\,\prime}_J:=\epsilon_J$ for $n$-stable $J$, then for a general $n$-bounded $J$ we define:
$$
v^{\,\prime}_J\ \ :=\ \ \frac
{{\widehat{E}}_{p_1,q_1}\cdots{\widehat{E}}_{p_t,q_t}}
{\prod_{d\in\mathbb{Z}}\mu_d!}\cdot v^{\,\prime}_{\widetilde J}
\ \ =\ \
\mathop{\prod_{d\in\mathbb{Z}}}_{d\not\equiv 0}\frac{{\widehat{E}}_{p,p+d}^{\,\mu_d}}{\mu_d!}\cdot v^{\,\prime}_{\widetilde J}\ ,
$$
where $t$ is maximal such that $p:=p_1\equiv\cdots\equiv p_t\mod n$.  The second equality follows because $d_i:=q_i-p_i\not\equiv 0\mod n$,  so all the operators ${\widehat{E}}_{p,p+d}$ commute with each other.  The basis $\{v^{\,\prime}_J\}$ clearly lies in the Kostant $\mathbb{Z}$-form  of the Demazure module $V_w(\Lambda_m)$, and it has leading coefficients $\pm1$, so it reduces to a basis over an arbitrary field.  (Cf. \cite[Ch. 26]{humphreys}.)

Theorem 1 also gives an alternative ``bottom-up'' algorithm to generate $\mathcal{C}_w(L_m)$, as opposed to the ``top-down'' definition in terms of crystal lowering operators.  We write:
$$\begin{array}{rcl}
\mathcal{C}_{=y}(L_m)&:=&\{J\in\mathcal{C}(L_m)\,\mid\,\mathop{\rm roof}\nolimits(J)=y\}\\[.5em]
&=&\{y(L_m)\}\,\cup\,\mathop{\rm up}\nolimits^{-1}(y(L_m))\,\cup\,
\mathop{\rm up}\nolimits^{-1}\mathop{\rm up}\nolimits^{-1}(y(L_m))\,\cup\, \cdots
\end{array}$$
where $\mathop{\rm up}\nolimits^{-1}({\widehat{J}})$ means the set of all $J$ such that $\mathop{\rm up}\nolimits(J)={\widehat{J}}$. To compute this for any given ${\widehat{J}}\in\mathcal{C}(L_m)$, we first find ${\widetilde{p}}<{\widehat{p}}$\,, the two maximal loose ends of ${\widehat{J}}$ (with one or both possibly $=-\infty$).  Next we choose any $q>{\widehat{p}}\sh-n$ such that $q\sh+n$ is the tight end of a seam $S\subset{\widehat{J}}$ of length $|S|\geq 2$,
and we let ${\widehat{q}}$ be the maximal tight end of ${\widehat{J}}$
less than $q$.  Finally, we define:
$$
P(q):=\left\{\, p\ \left|\begin{array}{c}
p\sh-n,p\not\in{\widehat{J}}\\[.3em]
\max({\widehat{p}},{\widehat{q}}\,)<p<q
\end{array}\right.\right\}
\,\cup\,
\left\{\, p={\widehat{p}}\sh-n\ \left|\begin{array}{c}
p\sh-n\not\in{\widehat{J}}\\[.3em]
\max({\widetilde{p}},{\widehat{q}}\,)<p
\end{array}\right.\right\}\,.
$$
Then we have:
$$
\mathop{\rm up}\nolimits^{-1}({\widehat{J}})=
\left\{J:={\widehat{J}}\setminus q\cup p\ \left|\
\begin{array}{c}
q\sh-n,q\in{\widehat{J}},\,\ q\sh+n\not\in{\widehat{J}}\\[.3em]
q>{\widehat{p}}\sh-n,
\,\ p\in P(q)
\end{array}\right.\right\}\,.
$$
Applying this to all $y\leq w$, we generate all $J\in\mathcal{C}_w(L_m)$.

We will prove Theorem 1 in Section 3 and Proposition 3 in Section 4.  Theorem 2 is a corollary of these, as follows.
By Theorem 1 and the definitions, we have:
$$
V':=\mathop{\rm Span}\nolimits_\mathbb{C}\{\,v_J\mid \mathop{\rm roof}\nolimits(J)\leq w\,\}
=\mathop{\rm Span}\nolimits_\mathbb{C}\{\,v_J\mid J\in\mathcal{C}_w(L_m)\,\}
\subset V_w(\Lambda_m)\,.
$$
Proposition 3 implies that the $v_J$ are linearly independent vectors in $\mathcal{F}$ (since they are triangular with respect to the standard  basis
$\{\epsilon_J\}$ ), so that
$\dim_\mathbb{C} V'= |\mathcal{C}_w(L_m)|\,;$
but it is well known from crystal graph theory  (Section 2 below) that $|\mathcal{C}_w(L_m)|=\dim_\mathbb{C} V_w(\Lambda_m)$, so that
$V'=V_w(\Lambda_m)$. This shows Theorem 2 for the Demazure module $V_w(\Lambda_m)$, and the claims for the irreducible module and the dual modules follow trivially.

We comment on related work which is closest to our point of view.  The pioneering paper \cite{DJKMO} by Date, Jimbo, Kuniba, Miwa, and Okado of the Kyoto school defined the tableaux $\mathcal{C}(L_m)$ for $V(\Lambda_m)$ (\,in fact for all $V(\ell\Lambda_m)$\,), and the crystal graph structure was first defined by Misra, Miwa, Jimbo, et al.~in \cite{MM}, \cite{jimbo}.  Certain Demazure crystals $\mathcal{C}_w(L_m)$ were considered by Kuniba, Misra, Miwa, Uchiyama and others in \cite{KMOTU},\cite{KMOU},\cite{FMO},\cite{MW}.
A useful survey of related work is \cite{FLOTW}, and \cite{kashiwara} is a fundamental reference.
\\[1em]
{\bf Notation}\quad  For a set $J\subset\mathbb{Z}$, we define:
$$
J^{\equiv i}:=\{j\in J\,\mid\, j\equiv i\mod n\}\,,
\qquad
J_{<r}:=\{j\in J\,\mid\, j<r\}\,.
$$
Similarly for\ $J_{>r}$ , for\ \
$J^{\equiv i}_{>r}:=J^{\equiv i} \cap J_{>r}$ ,
for\ \ ${}_{q\leq}J_{\leq r}:=J_{\geq q}\cap J_{\leq r}$ ,
etc.

\section{Crystal Operators}

In this section, we review the necessary facts about the crystal raising and lowering operators acting on $\mathcal{C}(L_m)$.\,\begin{footnote}{These operators are sometimes encoded in the {\it crystal graph} having vertices $J\in\mathcal{C}(L_m)$ and $i$-colored edges $J\stackrel{i}{\to}f_iJ$.}
\end{footnote}These operators were first defined in our case by the Kyoto school \cite{jimbo}, and they can also be derived from Littelmann's path model (as modified for semi-infinite paths in \cite{magyar}).   The crystal operators are basically different from the up-operation: indeed, by Theorem 1 the two are in some sense transversal to each other.

If it is defined, the lowering operator $f_i$ for $i=0,1,\ldots n\sh-1$ acting on a set $J\in\mathcal{T}$ picks out a certain element $r\in J$ with $r\equiv i\mod n$, and replaces it with $r\sh+1\equiv i\sh+1\mod n$.
(We say that $f_i(J)$ is ``lower'' than $J$ because it is farther from the highest-weight element $L_m$.)\ \
Similarly, the raising operator $e_i(J)$ picks out a certain element $r'\in J$ with $r'\equiv i\sh+1$ and replaces it with $r'\sh-1\equiv i$.  We have:\ \ $f_i(J)=J' \,\Longleftrightarrow\, J=e_i(J')$.
\\[1em]
{\bf Definition}\ \ {\it Given $J\in\mathcal{T}$.\\
(i) Let $$
R:=\{\,r\ \text{\ \ s.t.\ \ for all $k\geq r$}\,,\ \
|{}_{r\leq}J_{\leq k}^{\equiv i}\,|
> |{}_{r\leq}J_{\leq k}^{\equiv (i+1)}\,|\ \}\,.
$$
If $R$ is empty, then $f_i(J)$ is undefined.  Otherwise,
$$
f_i(J):=J\setminus r\cup(r\sh+1)\,,\quad\text{where}\quad
r:=\min(R)\,.
$$
(ii) Let
$$
R':=\{\,r'\text{\ \ s.t.\ \ for all $k\leq r'$,}\ \
|{}_{k\leq}J_{\leq r'}^{\equiv (i+1)}\,|
> |{}_{k\leq}J_{\leq r'}^{\equiv i}\,|\ \}\,.
$$
If $R'$ is empty, then $e_i(J)$ is undefined.  Otherwise,
$$
e_i(J):=J\setminus r'\cup (r'\sh-1)\,,\quad\text{where}\quad
r':=\max(R')\,.
$$
}
\\[-1em]
The importance of the crystal operators lies in the following
Refined Demazure Character Formula (cf. Jimbo, et al. \cite{jimbo}).  Define the {\it weight} of a tableau $J\in\mathcal{C}(L_m)$ by $\mathop{\rm wt}(L_m):=\Lambda_m$
and $\mathop{\rm wt}(f_i(J)):=\mathop{\rm wt}(J)-\alpha_i$.
\\[1em]
{\bf Proposition}\quad {\it The character of the Demazure module $V_w(\Lambda_m)$ is the weight generating function of the crystal graph $\mathcal{C}_w(L_m)$:  that is,
$$
\sum_{\mu}\ \dim_\mathbb{C}\! V_w(\Lambda_m)_\mu\ \,e^\mu
=\sum_{J\in\mathcal{C}_w(L_m)} e^{\mathop{\rm wt}(J)}\,.
$$
In particular,\ $\dim_\mathbb{C}\! V_w(\Lambda_m)=\#\mathcal{C}_w(L_m)$.
}
\\[1em]
Let us give a more pictorial way to understand these operators in the spirit of Lascoux-Schutzenberger \cite{LS}: we progressively remove elements of $J$ which are irrelevant to the action.
We call $j\in J^{\equiv i}$ the {\it $i$-elements} of $J$, and we write sets as usual in increasing order: $J=\{\cdots\sh<j_{-1}\sh<j_{0}\}$.  We start by removing all $j\in J$ except the $i$- and $(i\sh+1)$-elements.  We consider each remaining $i$-element which is immediately followed by an $(i\sh+1)$-element, and we remove these pairs.
Now we look again for remaining $i$-elements followed by $(i\sh+1)$-elements, and remove these pairs.
After finitely many iterations, we are left with
a finite subset
$$J'=\{j'_1<\cdots<j'_s<j''_1<\cdots<j''_t\}\subset J
\quad\text{with all}\ \ j'_k\equiv i\sh+1,\ \
j''_k\equiv i\,.
$$
Then we take $r=j''_1$, the smallest $i$-element, and $r':=j'_s$, the largest $(i\sh+1)$-element of $J'$, so that:
$$
f_i(J)=J\setminus j''_1\cup(j''_1\sh+1)\,,
\qquad
e_i(J)=J\setminus j'_s\cup(j'_s\sh-1)\,.
$$

\noindent {\bf Example}\quad
We exhibit the action of $e_2,f_2$ on the $J$ from our previous example.  This time, we write the elements of $J$ reduced modulo $n=5$:
since $J$ is $n$-bounded, this loses no information.
We have underlined the elements to be removed.
$$
\begin{array}{ccr@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}c@{\,}l}
J&=&\{\,\cdots,&\!-3,\,&\!-2,\,&\!-1,\,&\,0,\,&3,\,&4,\,&7,&10,&12,&14,&17,&18,&23,&27,&32,&33,&35,&37\,\}
\\
&=&\cdots\ &2&3&4&5&3&4&2&5&2&4&2&3&3&2&2&3&5&2
\\
&\Rightarrow&\cdots\ &\underline2&\underline3&&&3&&2&&2&&\underline{2}&\underline{3}&3&2&\underline2&\underline3&&2
\\
&\Rightarrow&&&&&&3&&2&&\underline2&&&&\underline3&2&&&&2
\\
J'&=&&&&&&\mathbf3&&\mathbf2&&&&&&&\mathbf2&&&&\mathbf2\\[.5em]
f_2(J)&=&\cdots\ &2&3&4&5&\mathbf3&4&\mathbf3&5&2&4&2&3&3&\mathbf2&2&3&5&\mathbf2 \\
f_2^2(J)&=& \cdots\ &2&3&4&5&\mathbf3&4&\mathbf3&5&2&4&2&3&3&\mathbf3&2&3&5&\mathbf2 \\
f_2^3(J)&=& \cdots\ &2&3&4&5&\mathbf3&4&\mathbf3&5&2&4&2&3&3&\mathbf3&2&3&5&\mathbf3 \\
f_2^4(J)&=&\text{undefined}\hspace{-2.8em}\\[.5em]
e_2(J)&=& \cdots\ &2&3&4&5&\mathbf2&4&\mathbf2&5&2&4&2&3&3&\mathbf2&2&3&5&\mathbf2 \\
e_2^2(J)&=&\text{undefined}\hspace{-2.8em}\\
\end{array}
$$
Note that the irrelevant elements removed from $J$ are the same as those from $e_i(J)$ and $f_i(J)$, so we can easily perform $e_i$ and $f_i$ repeatedly.

In the previous example we computed $\mathop{\rm roof}\nolimits(J)=y(L_m)$,
where:
$$
y=s_{2}\,s_{1}\,s_{3}\,s_{2}\,s_{0}\,
(s_{4}s_{3}\,s_{2}\,s_{1}\,s_{0})^{11}\,s_{4}\,.
$$
By Theorem 1, this means that $J\in\mathcal{C}_y(L_m)$:
$$
J=f_{2}^{\mbox{\tiny $\bullet$}}\,f_{1}^{\mbox{\tiny $\bullet$}}\,f_{3}^{\mbox{\tiny $\bullet$}}\,f_{2}^{\mbox{\tiny $\bullet$}}\,f_{0}^{\mbox{\tiny $\bullet$}}\,(f_{4}^{\mbox{\tiny $\bullet$}}\,f_{3}^{\mbox{\tiny $\bullet$}}\,f_{2}^{\mbox{\tiny $\bullet$}}\,f_{1}^{\mbox{\tiny $\bullet$}}\,f_{0}^{\mbox{\tiny $\bullet$}})^{11}\,f_{4}^{\mbox{\tiny $\bullet$}}\, L_{14}\ ,
$$
where each $f_i^{\mbox{\tiny $\bullet$}}$ represents some non-negative integer power of $f_i$.  We see from this the comparative rapidity of the roof algorithm in defining and generating Demazure crystals.\quad$\diamond$

\section{Proof of Theorem 1}

For a set $J\in\mathcal{C}(L_m)$, let $\mathcal{C}_y(L_m)$ be the unique minimal Demazure crystal containing $J$, and define the {\it ceiling of $J$} to be the extremal element of $\mathcal{C}_y(L_m)$:
$$
\ceil(J):=y(L_m)\,.
$$
Thus\
$\mathcal{C}_w(L_m) =\{J\in\mathcal{C}(L_m)\mid\ceil(J)\leq w(L_m)\}\,,$ and we can restate:
\\[1em]
{\bf Theorem 1}\quad {\it We have\ \ $\mathop{\rm roof}\nolimits(J)=\ceil(J)$\ \
for all $J\in\mathcal{C}(L_m)$.}
\\[1em]
For $J\neq L_m$, we define:
$$
a(J):=\max\{a\mid L_a\subset J\}
\quad\text{and}\quad
r(J):=\min J_{>a(J)}\,.
$$
That is, $a(J)<r(J)$ are the smallest consecutive elements of $J$ which are not consecutive integers.

We let $e^{\max}_i(J)$ denote the result of applying the highest possible power of the raising operator $e_i$ to $J$, and we let:
$$K:={e}_{r-1}^{\max}J\,,$$
where $r:=r(J)$. Observe that $r\sh-1\in K$
(and thus $K\neq J$), since in
$J_{<r}=L_{a(J)}$, the pairs of consecutive entries congruent to $r\sh-1$ and $r$ are irrelevant for the crystal operation.
\\[1em]
\noindent{\bf Ceiling Lemma}\\[.3em] {\it
(i) For all $J\in\mathcal{C}(L_m)$, we have:
$$r(\ceil(J))=r(J)
\quad\text{and}\quad
a(\ceil(J))=a(J)\,.$$
(ii) With $J\neq L_m$ and $K$ as above, we have:
$$
\ceil(J)>\ceil(K)=s_{r-1}\,\ceil(J)\,.
$$
}\\[0em]
{\bf Roof Lemma}\ \ {\it With $J\neq L_m$ and $K$ as above, we have:
$$
\mathop{\rm roof}\nolimits(J)>\mathop{\rm roof}\nolimits(K)=s_{r-1}\,\mathop{\rm roof}\nolimits(J)\,.
$$
}\\[0em]
Assuming these two Lemmas, we can immediately prove Theorem 1 by induction on the quantity:
$$
\mathop{\rm height}\nolimits(J):=\sum_{i\leq 0}\,(j_i-i-m)\,,
$$
a sum with finitely many non-zero terms for $J\in\mathcal{C}(L_m)$.
If $\mathop{\rm height}\nolimits(J)=0$, then $J=L_m$ and there is nothing to prove.  Otherwise, $\mathop{\rm height}\nolimits(K)<\mathop{\rm height}\nolimits(J)$, and we may assume $\mathop{\rm roof}\nolimits(K)=\ceil(K)$.  Then the
Roof and Ceiling Lemmas imply:
$$\qquad\qquad
\mathop{\rm roof}\nolimits(J)=s_{r-1}\mathop{\rm roof}\nolimits(K)=s_{r-1}\ceil(K)=\ceil(J)\,.
\qquad\Box
$$
\\[0em]
{\it Proof of Ceiling Lemma.}\ \
We first prove that $a(\ceil(J))\leq a(J)$.
Let $a:=a(J)$.
Let $\ceil(J)=s_{i_t}\cdots
s_{i_1}$, a reduced decomposition. Then for some
$c_t,\ldots,c_1\geq 0$, $J=f_{i_t}^{c_t}\cdots f_{i_1}^{c_1}L_m$.
The sequence $\{i_t,\ldots,i_1\}$ must contain a subsequence
$\{a\sh+1,\ldots,m\}$. Let $\{j_k,\ldots,j_1\}$ be the rightmost such subsequence: that is, ${j_1}$ is the rightmost occurrence of
$m$ in $\{i_t,\ldots,i_1\}$;  and for $k=1,\ldots,m\sh-a\sh-1$, after ${j_k}$ has been determined
let ${j_{k+1}}$ be the rightmost occurrence of $m\sh-k$ in
$\{i_t,\ldots,i_1\}$ to the left of ${j_k}$.
Let $f_{i}^{\max}T$ denote the result of
applying the lowering operator $f_{i}$ as many times as possible
to $T$; thus, for example, $\ceil(J)=f_{i_t}^{\max}\cdots
f_{i_1}^{\max}L_m$. Then
$$
a(f_{i_t}^{\max}\cdots f_{i_1}^{\max}L_m)\leq
a(f_{{j_k}}^{\max}\cdots f_{j_1}^{\max}L_m)=m\sh-k,
$$
for $k=1,\ldots,m\sh-a$.  Setting $k=m\sh-a$, we obtain the result.

We prove (i) and (ii) together by induction on $\mathop{\rm height}\nolimits(J)$. Let $r:=r(J)$.
If
$\mathop{\rm height}\nolimits(J)=0$ then $\ceil(J)=J=L_m$, so (i) is true, and (ii) is
vacuously true.  Assume $\mathop{\rm height}\nolimits(J)>0$.  Note that
$$\ceil(J)\geq\ceil(K)\geq s_{r-1}\,\ceil(J)\,,$$ so (ii) is
equivalent to $\ceil(J)\neq\ceil(K)$.

If $r(J)=a(J)+2$, then $a(K)=a(J)+1$.  Since
$\mathop{\rm height}\nolimits(K)<\mathop{\rm height}\nolimits(J)$, by induction, $a(\ceil(K))=a(K)$, thus
$a(\ceil(K))=a(K)>a(J)\geq a(\ceil(J))$, implying
$\ceil(K)\neq\ceil(J)$.  Therefore $\ceil(J)=s_{r-1}\ceil(K)$, and
clearly (i) follows.

If $r(J)>a(J)+2$, on the other hand, let $wL_m=\ceil(K)$. Since
$\mathop{\rm height}\nolimits(K)<\mathop{\rm height}\nolimits(J)$, by induction we have $a(wL_m)=a(K)=a$ and
$r(wL_m)=r(K)=r\sh-1$. Define
$$
v=s_{a+1}s_{a+2}\cdots s_{r-2}\,w\,.
$$
Note that $a(vL_m)=a\sh+1$.
Let $v=s_{i_t}\cdots s_{i_1}$ be a reduced decomposition.
Then $w=s_{r-2}\cdots s_{a+1}s_{i_t}\cdots s_{i_1}$,
also a reduced decomposition. Indeed, if we define $k$ by $r(wL_m)=w_k$
(where $wL_m=\{\cdots\sh>w_{-2}\sh>w_{-1}\sh>w_0\}$\,), then $$
(s_{a+(j+1)}\cdots
s_{a+1}s_{i_t}\cdots s_{i_1}L_m)_k =
1+(s_{a+j}\cdots
s_{a+1}s_{i_t}\cdots s_{i_1}L_m)_k
$$
for $j=1,\ldots,r\sh-a\sh-3$, and
$$
(s_{a+1}s_{i_t}\cdots s_{i_1}L_m)_k =
1+(s_{i_t}\cdots
s_{i_1}L_m)_k\,.
$$
In other words, with each successive
multiplication of $v=s_{i_t}\cdots s_{i_1}$ by $s_{a+j}$ for $j=1,\ldots, r\sh-a\sh-2$, the product increases.

Now suppose $\ceil(J)=wL_m$.
Then $J=f_{r-2}^{c_{r-2}}\cdots
f_{a+1}^{c_{a+1}}f_{i_t}^{d_t}\cdots f_{i_1}^{d_1}L_m$ for some
$c_{r-2},\ldots,c_{a+1},d_t,\ldots,d_1\geq 0$.  Thus,
$e_{a+1}^{c_{a+1}}\cdots e_{r-2}^{c_{r-2}}J\in C_v(L_m)$.  Thus\\
$a(e_{a+1}^{c_{a+1}}\cdots e_{r-2}^{c_{r-2}}J)\geq a(vL_m)=a\sh+1$, a
contradiction.  Therefore $\ceil(J)\neq wL_m$, so
$\ceil(J)=s_{r-1}wL_m$, from which (i) follows immediately.
\hfill$\Box$
\\[1em]
{\it Proof of Roof Lemma.} For $i\in\mathbb{Z}$, define
$\mathop{\rm roof}\nolimits_i(J)$ by
$$\mathop{\rm roof}\nolimits_i(J):=\mathop{\rm roof}\nolimits(L_i\cup J).$$ Several properties of
$\mathop{\rm roof}\nolimits_i(J)$ follow easily from the definition:
\begin{itemize}
\item[1.] $\mathop{\rm roof}\nolimits_i(J)=\mathop{\rm roof}\nolimits_{i+1}(J)$ \ if $i+1\in J$.
\item[2.] $\mathop{\rm roof}\nolimits_i(J)=\mathop{\rm roof}\nolimits(\mathop{\rm roof}\nolimits_{i+1}(J)\setminus \{i+1\})$ \ if $i+1\not\in J$.
\item[3.] $\mathop{\rm roof}\nolimits_i(J)=\mathop{\rm roof}\nolimits(J)$ if $J\supset L_i$ ; \ $\mathop{\rm roof}\nolimits_i(J)=L_i$ \ if $L_i\supset J$.
\end{itemize}
For $T\in\mathcal{C}(L_k)$, define
$$\lub(T)=\min_{\stackrel{\rm lex}{\geq}}\left\{J'\in \mathcal{C}(L_k)\Big|
\begin{array}{l}
J'\stackrel{\rm B}{\geq} T,\\
J'\hbox{ is n-stable}
\end{array}
\right\}.
$$
If $T$ has at most one seam, then $\mathop{\rm roof}\nolimits(T)=\lub(T)$. Since
$\mathop{\rm roof}\nolimits_{i+1}(J)\setminus (i\sh+1)$ has at most one seam, property
2 above can be modified:
\begin{itemize}
\item[4.] $\mathop{\rm roof}\nolimits_i(J)=\lub(\,\mathop{\rm roof}\nolimits_{i+1}(J)\setminus (i\sh+1)\,)$ \ if\,
$i\sh+1\not\in J$.
\end{itemize}

For $k\in\mathbb{Z}_{\geq 0}$, let $r[k]:=r+kn$. We will prove the following eight
statements $\bA_k$---$\bH_k$ together by decreasing
induction on $k$. Then we will show that the Roof Lemma is a
consequence of statement $\bC_0$ (i.e., $\bC_k$ for $k=0$).
\begin{itemize}
\item[$(\bA_k)$] Either $\mathop{\rm roof}\nolimits_{r[k]}(K)=\mathop{\rm roof}\nolimits_{r[k]}(J)$ or
$\mathop{\rm roof}\nolimits_{r[k]}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r[k]}(J)$.
\item[$(\bB_k)$] $\mathop{\rm roof}\nolimits_{r[k]}(K)\stackrel{\rm B}{\leq}\mathop{\rm roof}\nolimits_{r[k]}(J)$.\\[-.5em]
\item[$(\bC_k)$] If $r[k]\in J\setminus K$, $r[k]-1\in K\setminus J$,
then $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bD_k)$] If $r[k]\in J\cap K$, but $r[k]-1\not\in J$ or $K$,
then $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bE_k)$] If $r[k], r[k]-1\not\in J\cup K$, then
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bF_k)$] If $r[k]-1\in J\cap K$, but $r[k]\not\in J$ or $K$,
then $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\item[$(\bG_k)$] Either $\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$ or
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.
\item[$(\bH_k)$] $\mathop{\rm roof}\nolimits_{r[k]-2}(K)\stackrel{\rm B}{\leq}\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.\\[-.5em]
\end{itemize}
Our induction proof will establish
the following implications:
\[
(\bA_{k+1}\sh-\bF_{k+1})\ \Rightarrow\ (\bG_{k+1},\bH_{k+1})
\ \Rightarrow\ (\bA_{k},\bB_k)\ \Rightarrow\ (\bC_k,\bD_k,\bE_k)
\]
\[
(\bD_m, \bE_m, \bF_m\!:\,m\sh>k\,)\ \Rightarrow\ (\bF_k)
\]
For the starting point of induction, select $k$ large enough so
that $r[k-2]>j_0$. For such $k$, by property 3,
$\mathop{\rm roof}\nolimits_i(K)=\mathop{\rm roof}\nolimits_i(L)=L_i$ for $i=r[k\sh-2],r[k\sh-1],r[k]$. Thus
$\bA_k\sh-\bH_k$ are trivially true.
\\[1em]
$(\bA_{k+1}\sh-\bF_{k+1})\ \Rightarrow\ (\bG_{k+1}, \bH_{k+1})$ :
\\[1em]
Let us restate this as:
$(\bA_{k}\sh-\bF_{k})\Rightarrow(\bG_{k}, \bH_{k})$. If any of the
hypotheses of $\bC_k-\bF_k$ are satisfied, then $\bC_k-\bF_k$
imply $\bG_k$ and $\bH_k$.  There are two
possibilities omitted from the hypotheses of
$\bC_k-\bF_k$:
\begin{itemize}
\item[(i)] $r[k]-1\in J\setminus K$, $r[k]\in K\setminus J$, and
\item[(ii)] $r[k]-1, r[k]\in J\cap K$.
\end{itemize}
However, (i) cannot occur, since $K$ is obtained from $J$ by
applying the raising operator $e_{r-1}$ several times. If (ii)
occurs, then by property 1, $\mathop{\rm roof}\nolimits_{r[k-2]}(J)=\mathop{\rm roof}\nolimits_{r[k]}(J)$,
and $\mathop{\rm roof}\nolimits_{r[k-2]}(K)=\mathop{\rm roof}\nolimits_{r[k]}(K)$.  Thus, in this case as
well, $\bG_k$ and $\bH_k$ follow immediately from $\bA_k$ and
$\bB_k$.
\\[1em]
$(\bG_{k+1}, \bH_{k+1})\Rightarrow (\bA_k,\bB_k)$ :
\\[1em]
 Define $t$ by
$\mathop{\rm roof}\nolimits_{r[k]}(J)=\mathop{\rm up}\nolimits^t(\mathop{\rm roof}\nolimits_{r[k+1]-2}(J))$. Then necessarily
$\mathop{\rm roof}\nolimits_{r[k]}(K)=\mathop{\rm up}\nolimits^t(\mathop{\rm roof}\nolimits_{r[k+1]-2}(K))$. Letting
$T=\mathop{\rm roof}\nolimits_{r[k+1]-2}(J)$, $U=\mathop{\rm roof}\nolimits_{r[k+1]-2}(K)$, we show that
\begin{eqnarray}\label{eqn_TU}
&&\hbox{Either }\ \mathop{\rm up}\nolimits^i(U)=\mathop{\rm up}\nolimits^i(T)\ \hbox{ or
}\ \mathop{\rm up}\nolimits^i(U)=s_{r-1}\mathop{\rm up}\nolimits^i(T),\ \hbox{ and}\\
&&\mathop{\rm up}\nolimits^i(U)\stackrel{\rm B}{\leq}\mathop{\rm up}\nolimits^i(T)
\end{eqnarray}
 $0\leq i\leq t$, by induction on $i$; the
result is then obtained by setting $i=t$.
\\[1em]
We have $\mathop{\rm up}\nolimits^{0}(T):=T$, $\mathop{\rm up}\nolimits^{0}(U):=U$; thus (1) and (2) hold
for $i=0$. Let $0<i\leq t$, and assume that (1) and (2) hold for
$i-1$. Then either
\\[.5em]
(i) $\mathop{\rm up}\nolimits^{i-1}(U)=\mathop{\rm up}\nolimits^{i-1}(T)$, in which case
$$\mathop{\rm up}\nolimits^i(U)=\mathop{\rm up}\nolimits(\mathop{\rm up}\nolimits^{i-1}(U))=\mathop{\rm up}\nolimits(\mathop{\rm up}\nolimits^{i-1}(T))=\mathop{\rm up}\nolimits^{i}(T),\hbox{ or }$$
(ii) $\mathop{\rm up}\nolimits^{i-1}(U)=s_{r-1}\mathop{\rm up}\nolimits^{i-1}(T)$ and $\mathop{\rm up}\nolimits^{i-1}(U)\stackrel{\rm B}{\leq}
\mathop{\rm up}\nolimits^{i-1}(T)$. In this case, define $p, q$ by
$$\mathop{\rm up}\nolimits^i(T)=\mathop{\rm up}\nolimits^{i-1}
(T)\setminus p\cup q.$$ Then it is easy to see that
$\mathop{\rm up}\nolimits^i(U)=\mathop{\rm up}\nolimits^{i-1}(U)\setminus p\cup q'$, where
\[
q'=
\begin{cases}
q, &\hbox{ if }q\not\equiv r-1, r\mod n\\
q+1, &\hbox{ if }q\equiv r-1\mod n\\
q-1, &\hbox{ if }q\equiv r\mod n\\
\end{cases}.
\]
Thus $\mathop{\rm up}\nolimits^i(U)=s_{r-1}\mathop{\rm up}\nolimits^i(T)$ and $\mathop{\rm up}\nolimits^{i}(U)\stackrel{\rm B}{\leq} \mathop{\rm up}\nolimits^{i}(T)$.
This proves (1) and (2).
\\[1em]
$(\bD_m, \bE_m, \bF_m: m>k) \Rightarrow(\bF_k)$ :
\\[1em]
Let $m>k$ be the minimum integer such that not both $r[m]-1$ and
$r[m]$ are in $K$. Then, by the definition of the raising operator
$e_{r-1}$, it is not possible that $r[m]\in J\setminus K$,
$r[m]-1\in K\setminus J$. Thus either the hypotheses of $\bD_m$,
$\bE_m$, or $\bF_m$ must hold. Thus
$\mathop{\rm roof}\nolimits_{r[m]-2}(J)=\mathop{\rm roof}\nolimits_{r[m]-2}(K)$. Since also
$$
_{r[k]-2\leq}J_{\leq r[m]-2}\ =\ _{r[k]-2\leq }K_{\leq r[m]-2}\,,$$
we have
$\mathop{\rm roof}\nolimits_{x}(J)=\mathop{\rm roof}\nolimits_{x}(K)$, for $r[k]-2\leq x\leq r[m]-2$.
\\[1em]
$(\bA_k,\bB_k)\Rightarrow(\bD_k)$ :
\\[1em]
By Property 1, $\mathop{\rm roof}\nolimits_{r[k]-1}(J)=\mathop{\rm roof}\nolimits_{r[k]}(J)$ and
$\mathop{\rm roof}\nolimits_{r[k]-1}(K)=\mathop{\rm roof}\nolimits_{r[k]}(K)$. Now $\bA_k$, $\bB_k$ imply that
if $J'\in\mathcal{C}(L_m)$ is $n$-stable and $J'\stackrel{\rm B}{\geq}
\mathop{\rm roof}\nolimits_{r[k]-1}(K)\setminus \{r[k]-1\}$, then $J'\stackrel{\rm B}{\geq}
\mathop{\rm roof}\nolimits_{r[k]-1}(J)\setminus \{r[k]-1\}$ (the same statement with $J$
and $K$ switched holds obviously). The result follows from
Property 4.
\\[1em]
$(\bA_k,\bB_k)\Rightarrow(\bC_k)$ :
\\[1em]
Let
$J'=\mathop{\max}_{\stackrel{\rm B}{\geq}}\{\mathop{\rm roof}\nolimits_{r[k]}(J),s_{r-1}\mathop{\rm roof}\nolimits_{r[k]}(J)\}$. By
Property 1, $\mathop{\rm roof}\nolimits_{r[k]-1}(J)=\mathop{\rm roof}\nolimits_{r[k]}(J)$. If $J'\in\mathcal{T}$ is
$n$-stable and $J'\stackrel{\rm B}{\geq} \mathop{\rm roof}\nolimits_{r[k]-1}(J)\setminus \{r[k]-1\}$, then
$J'\stackrel{\rm B}{\geq} \hat{J}\setminus \{r[k]-1\}$; conversely, if $J'\in\mathcal{T}$ is
$n$-stable and $J'\stackrel{\rm B}{\geq} \hat{J}\setminus \{r[k]-1\}$, then $J'\stackrel{\rm B}{\geq}
\mathop{\rm roof}\nolimits_{r[k]-1}(J)\setminus \{r[k]-1\}$. Thus
$\mathop{\rm roof}\nolimits_{r[k]-2}(\hat{J})=\mathop{\rm roof}\nolimits_{r[k]-2}(J)$.

We claim that $K\stackrel{\rm B}{\leq} s_{r-1}K$.   Indeed, let $m>k$ be the
minimum integer greater than $k$ such that not both $r[m]-1,r[m]\in
K$. Then by the definition of the raising operator $e_{r-1}$, it
is not possible that $r[m]\in K$. Thus $\mathop{\rm roof}\nolimits_{r[m]-2}(K)\stackrel{\rm B}{\leq}
s_{r-1}\mathop{\rm roof}\nolimits_{r[m]-2}(K)$.  Define $t'$ by
$\mathop{\rm roof}\nolimits_{r[k]}(K)=\mathop{\rm up}\nolimits^{t'}(\mathop{\rm roof}\nolimits_{r[m]-2}(K))$.  Then it is easy to
see that $\mathop{\rm up}\nolimits^{i}(\mathop{\rm roof}\nolimits_{r[m]-2}(K))\stackrel{\rm B}{\leq}
s_{r-1}\mathop{\rm up}\nolimits^{i}(\mathop{\rm roof}\nolimits_{r[m]-2}(K))$, $1\leq i\leq t'$. This proves
the claim.

Define $t$ by $\mathop{\rm roof}\nolimits_{r[k]-2}(\hat{J})=\mathop{\rm up}\nolimits^t(\hat{J})$.  Then by
Property 1,
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits_{r[k]-1}(K)=\mathop{\rm up}\nolimits^t(\mathop{\rm roof}\nolimits_{r[k]}(K))$. For
$1\leq i\leq t$, let
$$\mathop{\rm up}\nolimits^i(\hat{J})=\mathop{\rm up}\nolimits^{i-1}
(\hat{J})\setminus p\cup q.$$ Then
$\mathop{\rm up}\nolimits^i(\mathop{\rm roof}\nolimits_{r[k]}(K))=\mathop{\rm up}\nolimits^{i-1}(\mathop{\rm roof}\nolimits_{r[k]}(K))\setminus p\cup
q'$, where
\[
q'=
\begin{cases}
q, &\hbox{ if }q\not\equiv r \mod n\\
q-1, &\hbox{ if }q\equiv r\mod n\\
\end{cases}.
\] The result follows from this.
\\[1em]
$(\bA_k,\bB_k)\Rightarrow(\bE_k)$ :
\\[1em]
We have that
$\mathop{\rm roof}\nolimits_{r[k]-2}(J)=\mathop{\rm roof}\nolimits(\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\})$,
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\mathop{\rm roof}\nolimits(\mathop{\rm roof}\nolimits_{r[k]}(K)\setminus\{r[k]-1,r[k]\})$. Let
$L=\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\}$.  If $L$ has only one seam, then
the result is obvious. Thus assume
that $L$ has two seams:
\begin{eqnarray*}
S_1&=&\{r[k]=p_1<\cdots<p_t\}\\
S_2&=&\{r[k]-1=p_{t+1}<\cdots<p_{t+s}\}
\end{eqnarray*}
where $p_{i+1}=p_i+n$, $1\leq i\leq t+s-1$, $i\neq t$. Then
$\mathop{\rm up}\nolimits^t(L)$ has exactly one seam, namely $S_2$ with possibly some
additional elements added to its tight end. Thus
$\mathop{\rm roof}\nolimits_{r[k]-2}(J)=\mathop{\rm roof}\nolimits(\mathop{\rm up}\nolimits^t(L))=\lub(\mathop{\rm up}\nolimits^t(L))$. We claim that
$\lub(\mathop{\rm up}\nolimits^t(L))=\lub(L)$. The claim implies
$\mathop{\rm roof}\nolimits_{r[k]-2}(J)=\lub(L)=\lub(\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\})$;
replacing $J$ with $K$, in precisely the same manner we show that
$\mathop{\rm roof}\nolimits_{r[k]-2}(K)=\lub(\mathop{\rm roof}\nolimits_{r[k]}(K)\setminus\{r[k]-1,r[k]\})$. But
it is clear that $\lub(\mathop{\rm roof}\nolimits_{r[k]}(J)\setminus\{r[k]-1,r[k]\})=
\lub(\mathop{\rm roof}\nolimits_{r[k]}(K)\setminus\{r[k]-1,r[k]\})$.  Thus the result
follows from the claim.

To prove the claim, note that since $\mathop{\rm up}\nolimits^t(L)\stackrel{\rm B}{\geq} L$,
$\lub(\mathop{\rm up}\nolimits^t(L))\stackrel{\rm lex}{\geq}\lub(L)$. It suffices to show that
$\lub(L)\stackrel{\rm B}{\geq} \mathop{\rm up}\nolimits^t(L)$, since this implies
$\lub(L)\stackrel{\rm lex}{\geq}\lub(\mathop{\rm up}\nolimits^t(L))$. We show something slightly
stronger: if $M\stackrel{\rm B}{\geq} L$ is $n$-stable, then $M\stackrel{\rm B}{\geq} \mathop{\rm up}\nolimits^t(L)$. We
can express $M=L\setminus\{p_1,\ldots,p_{t+s}\}\cup
\{p_1,\ldots,p_{t+s}\}$, where $x_i> p_i$, $i=1,\ldots,t+s$.
Likewise, $\mathop{\rm up}\nolimits^t(L)=L\setminus\{p_1,\ldots,p_{t}\}\cup
\{q_1,\ldots,q_{t}\}$, where $\mathop{\rm up}\nolimits^i(L)=\mathop{\rm up}\nolimits^{i-1}(L)\setminus
p_i\cup q_i$. To show $M\stackrel{\rm B}{\geq} \mathop{\rm up}\nolimits^t(L)$, it suffices to show that
$q_i\leq x_i$, $i=1,\ldots,t$. This is clear from the definition
of the $\mathop{\rm up}\nolimits$ operation. Indeed, let $i_{\max}$ be the largest $i$
for which $q_i-p_i> n$. There are no tight ends in $\mathop{\rm up}\nolimits^t(L)$
between $r[k]$ and $q_{i_{\max}}$; thus $q_i\leq x_i$, $1\leq i\leq
i_{\max}$. If $i_{\max}<t$, then for $i_{\max}<i\leq t$,
$$q_i=\min\{q\not\in L\mid q-p_i\leq n-1,\,
q-n\in\mathop{\rm up}\nolimits^{i-1}(L),\, q\not\equiv r\mod n,\}.$$ Inductively, this
implies that $q_i\leq x_i$.
\\[2em]
This completes the proof of $\bA_k-\bH_k$. Noting that $r[0]=r$, we see that
$\bC_0$ implies $\mathop{\rm roof}\nolimits_{r-2}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{r-2}(J)$.  By
property 3, $\mathop{\rm roof}\nolimits(J)=\mathop{\rm roof}\nolimits_{a(J)}(J)$ and
$\mathop{\rm roof}\nolimits(K)=\mathop{\rm roof}\nolimits_{a(J)}(K)$. Using identical arguments as in the
proof of $(\bG_{k+1}, \bH_{k+1})\Rightarrow (\bA_k,\bB_k)$, we see
that $\mathop{\rm roof}\nolimits_{a(J)}(K)=s_{r-1}\mathop{\rm roof}\nolimits_{a(J)}(J)$, which completes the
proof of the Roof Lemma. \hfill$\Box$

\section{Proof of Proposition 3}

\subsection{Proof of Proposition 3(i)}
The result states that:
$$
v_J=\sum_K a_K^J\, \epsilon_K\,,
$$
where the sum runs over $K\stackrel{\rm lex}{\leq} J$.
We use induction on $\ell$, where $\mathop{\rm roof}\nolimits(J)=\mathop{\rm up}\nolimits^\ell(J)$.  If $\ell=0$, then $v_J=\epsilon_J$ and there is nothing to prove.

Now let $\ell>0$. We inductively apply the Proposition to
$
{\widehat{J}} :=\mathop{\rm up}\nolimits(J)=J\setminus p\cup q\,,
$\ \ so that:
$$
v_{\widehat{J}} = \sum_{\widehat{K}} a_{\widehat{K}}^{\widehat{J}} \,\epsilon_{\widehat{K}}\,,
$$
where the sum runs over ${\widehat{K}}\stackrel{\rm lex}{\leq}
{\widehat{J}}$.  Thus:
$$
v_J={\widehat{E}}_{pq}\, v_{\widehat{J}}
=\sum_{\widehat{K}} \sum_{h\in\mathbb{Z}}
a_{\widehat{K}}^{\widehat{J}}\, E_{p+nh,q+nh}\,\epsilon_{{\widehat{K}}}\,.
$$
It suffices to show the following:
\\[1em]
{\bf Lemma}\quad {\it
Let ${\widehat{J}}:=\mathop{\rm up}\nolimits(J)=J\setminus p\cup q$. Consider any ${\widehat{K}}\stackrel{\rm lex}{\leq}{\widehat{J}}$ and any $h\in\mathbb{Z}$ such that:
$$
p':=p\sh+nh\not\in{\widehat{K}}\quad\text{and}\quad
q':=q\sh+nh\in{\widehat{K}}\,.
$$
Then we have:
$$
K:={\widehat{K}}\setminus q'\cup p'\ \stackrel{\rm lex}{\leq}\ J={\widehat{J}}\setminus q\cup p\,.
$$
}
\\
We prove the Lemma using several facts which follow easily from the definitions.  Let $K,J\in\mathcal{C}(L_m)$.
For $J=\{\cdots\sh<j_{-1}\sh<j_0\}$ with $\mathop{\rm ord}\nolimits(J)=m$, recall that:\ \
$
\mathop{\rm height}\nolimits(J):=\sum_{i\leq 0}\,(j_i-i-m)\,.
$
\begin{enumerate}

\item  If $a_K^J\neq 0$, then $\mathop{\rm height}\nolimits(K)=\mathop{\rm height}\nolimits(J)$.

\item  If $a_K^J\neq 0$,\
then $|K^{\equiv i}_{>N}| = |J^{\equiv i}_{>N}|$ for all $N\ll0$.

\item  If ${\widehat{J}}=\mathop{\rm up}\nolimits(J)=J\setminus p\cup q$,\
then ${\widehat{J}}^{\equiv q}\subset {\widehat{J}}_{\leq q}$.

\item
If $J_{>p}$ contains no loose ends of $J$ (that is, $j\sh-n\in J$ for all $j\in J_{>p}$),
\ \ and\ \
$|K^{\equiv i}_{\geq p}|=
|J^{\equiv i}_{\geq p}|$ for some $i$, then
$K^{\equiv i}_{\geq p} \stackrel{\rm B}{\geq}
J^{\equiv i}_{\geq p}$.
\end{enumerate}

Proceeding with the proof of the Lemma, suppose first that ${\widehat{K}}={\widehat{J}}$.  By Fact 3 we must have $q'\leq q$, so $p'\leq p$ and clearly
$K={\widehat{J}}\setminus q'\cup p'\,\stackrel{\rm lex}{\leq}\,
{\widehat{J}}\setminus q\cup p =J$.

Now let ${\widehat{K}}\stackrel{\rm lex}{<}{\widehat{J}}$, and let ${\widehat{k}}$ be the {\it split point}, the value such that:
$$
{\widehat{k}}\in{\widehat{K}},\ \ {\widehat{k}}\not\in {\widehat{J}},
\quad\text{and}\quad
K_{<{\widehat{k}}}=J_{<{\widehat{k}}}.
$$

Case (a):  ${\widehat{k}}<p$.  Then: $$
{\widehat{k}}\in{\widehat{K}},\ \ {\widehat{k}}\not\in J
\quad\text{and}\quad {\widehat{K}}_{<{\widehat{k}}}={\widehat{J}}_{<{\widehat{k}}}=J_{<{\widehat{k}}}\,,
$$
so ${\widehat{K}}\stackrel{\rm lex}{\leq} J$.  But clearly $K\stackrel{\rm lex}{\leq} {\widehat{K}}$, so $K\stackrel{\rm lex}{\leq} J$ as desired.

Case (b):  $p\leq{\widehat{k}}$. If $p'\leq p$, then clearly
$K\stackrel{\rm lex}{\leq} J$ as desired.
On the other hand, suppose $p<p'$.
Then $K_{<p}={\widehat{K}}_{<p}={\widehat{J}}_{<p}=J_{<p}$ by the definition of ${\widehat{k}}$.
Furthermore, for all $i$ and some $N<p$ we have
$|K^{\equiv i}_{>N}|=|J^{\equiv i}_{>N}|$ by Fact 2,
and thus
$|K^{\equiv i}_{\geq p}|=|J^{\equiv i}_{\geq p}|$. By definition $J_{>p}$ contains no loose ends, so Fact 4 implies that $K^{\equiv i}_{\geq p}\stackrel{\rm B}{\geq} J^{\equiv i}_{\geq p}$ for all $i$.  We also have $K_{<p}=J_{<p}$, so $K\stackrel{\rm B}{\geq} J$.  If $K\stackrel{\rm B}{>}J$, then clearly $\mathop{\rm height}\nolimits(K)>\mathop{\rm height}\nolimits(J)$, contradicting Fact 1.  We conclude that
$K=J$, and we are done.

This proves the Lemma, and hence Proposition 3(i).

\subsection{Proof of Proposition 3(ii)}
To derive the formula relating the leading coefficients $a_J^J$ and $a_{\widetilde J}^{\widetilde J}$, note first that
$$\begin{array}{rcl}
v_J&=&({\widehat{E}}_{p_1q_1}\cdots{\widehat{E}}_{p_tq_t})v_{\widetilde J}\\[.5em]
&=&\displaystyle\sum_K\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}\!\!\!\!
a_K^{\widetilde J}\, (E_{p_1+nh_1,q_1+nh_1}\cdots E_{p_t+nh_t,q_t+nh_t})\,\epsilon_K\\[1.5em]
&=&\displaystyle\sum_K\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}\!\!
\pm\,a_K^{\widetilde J}\ \epsilon_{K{\uparrow}(h_1,\ldots,h_t)}\ ,
\end{array}$$
where we use notation:
$$
K{\uparrow}(h_1,\ldots,h_t):=K\setminus\{q_1\sh+nh_1
,\ldots ,q_t\sh+nh_t\}\cup\{p_1\sh+nh_1,\cdots
p_t\sh+nh_t\}
$$
provided $q_i\sh+nh_i\in K$ and $p_i\sh+nh_i\not\in K$ for all $i\leq t$ ; otherwise $K{\uparrow}(h_1,\ldots,h_t)$ is undefined, and $\epsilon_{K{\uparrow}(h_1,\ldots,h_t)}:=0$.
\\[1em]
{\bf Lemma} {\it (i) If $J=K{\uparrow}(h_1,\ldots,h_t)$ for some $K$ with $a_K^{\widetilde J}\neq 0$, then $K={\widetilde J}$.
\\
(ii) If $J={\widetilde J}{\uparrow}(h_1,\ldots,h_t)$, then there is a unique permutation $\sigma$ of $\{1,2,\ldots,r\}$ such that
$$
p_i\sh+nh_i=p_{\sigma(i)}\,,\qquad q_i\sh+nh_i=q_{\sigma(i)}\,.
$$
We obtain in this way every permutation $\sigma$ satisfying
$q_i-p_i=q_{\sigma(i)}-p_{\sigma(i)}$ for all $i\leq t$.
}
\\[1em]
The Proposition follows easily from (i) and (ii) of the Lemma, since:
$$\begin{array}{rcl}
v_J&=&\displaystyle\sum_K\!\!\!\!
\mathop{\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}}
_{K{\uparrow}(h_1,\ldots,h_t)=J}\!\!\!\!
a_K^{\widetilde J}\, (E_{p_1+nh_1,q_1+nh_1}\cdots E_{p_t+nh_t,q_t+nh_t})\,\epsilon_K
\\[3em]
&\stackrel{(i)}=&\left(a_{\widetilde J}^{\widetilde J}\!\!\!\!
\displaystyle
\mathop{\sum_{h_1,\ldots,h_t\in\mathbb{Z}\!\!\!\!\!}}
_{{\widetilde J}{\uparrow}(h_1,\ldots,h_t)=J}\!\!\!\!
(E_{p_1+nh_1,q_1+nh_1}\cdots E_{p_t+nh_t,q_t+nh_t})
\,\epsilon_{{\widetilde J}}\right)+
\text{lower}
\\[3em]
&\stackrel{(ii)}=&\left(a_{\widetilde J}^{\widetilde J}\,
\displaystyle\sum_{\sigma}
(E_{p_{\sigma(1)},q_{\sigma(1)}}\cdots E_{p_{\sigma(r)},q_{\sigma(r)}})
\,\epsilon_{{\widetilde J}}\right)+
\text{lower}\\[2em]
&\stackrel{(*)}=&\left(a_{\widetilde J}^{\widetilde J}\,
\displaystyle
\mathop{\sum_{\sigma}}
(E_{p_1q_1}\cdots E_{p_tq_t})
\,\epsilon_{{\widetilde J}}\right)+
\text{lower}\\[2em]
&=&\pm\, a_{\widetilde J}^{\widetilde J}\cdot\#\{\sigma\}\cdot\epsilon_J\ +\ \text{lower}\,,
\end{array}$$
where $\sigma$ runs over the set of all permutations of $\{1,\ldots,t\}$ such that $q_i-p_i=q_{\sigma(i)}-p_{\sigma(i)}$ for $i\leq t$ : clearly $\#\{\sigma\}=\prod_{d\geq 1} \mu_d!$ .
Equation $(*)$ holds because the operators $E_{p_iq_i}$ all commute for $i\leq t$.  It remains to prove the Lemma.
\\[1em]
{\it Proof of Lemma (i).}  Suppose
$J=K{\uparrow}(h_1,\ldots,h_t)={\widetilde J}{\uparrow}(0,\ldots,0)$ and $a_K^{{\widetilde J}}\neq 0$. Let
$$
p':=\min\{p_1\sh+nh_1,\ldots,p_t\sh+nh_t\}\,,\qquad
p:=p_1=\min\{p_1,\ldots,p_t\}\,.
$$
We clearly have $K_{<\min(p,p')}={\widetilde J}_{<\min(p,p')}$.
If $p'<p$, then $p'\not\in K$, $p'\in{\widetilde J}$, and $K_{<p'}={\widetilde J}_{<p'}$, so ${\widetilde J}\stackrel{\rm lex}{<} K$, which contradicts Proposition 3(i).

Thus $p\leq p'$, and $K_{<p}={\widetilde J}_{<p}$.  Furthermore, by Fact 2 in the proof of Prop. 3(i), for any $i\leq t$ we have $|K^{\equiv i}_{>N}|=|{\widetilde J}^{\equiv i}_{>N}|$ for some $N<p$.  Hence for any $i$,\ \ $|K^{\equiv i}_{\geq p}|=|{\widetilde J}^{\equiv i}_{\geq p}|$ .  Since ${\widetilde J}_{>p}$ clearly has no loose ends, Fact 4 implies  $K^{\equiv i}_{\geq p}\stackrel{\rm B}{\geq}{\widetilde J}^{\equiv i}_{\geq p}$ for any $i$, and we also know $K_{<p}={\widetilde J}_{<p}$.  We conclude that $K\stackrel{\rm B}{\geq}{\widetilde J}$, and a fortiori $K\stackrel{\rm lex}{\geq}{\widetilde J}$.  Since $K\stackrel{\rm lex}{\leq}{\widetilde J}$ by Proposition 3(i), we must have $K={\widetilde J}$.
\\[1em]
{\it Proof of Lemma (ii).}  Suppose
$J={\widetilde J}{\uparrow}(0,\ldots,0)={\widetilde J}{\uparrow}(h_1,\ldots,h_t)$.  Define:
$$
p'_i:=p_i+nh_i\,,\qquad q'_i:=q_i+nh_i\,,\qquad
d_i:=q_i-p_i:=q'_i-p'_i$$
Then we have
$\{p_1,\ldots,p_t\}=\{p'_1,\ldots,p'_t\}$
and $\{q_1,\ldots,q_t\}=\{q'_1,\ldots,q'_t\}$,
so there exist permutations $\alpha,\beta$ of $\{1,\ldots,r\}$ such that:
$$
p_i = p'_{\alpha(i)}\,,\qquad
q_i = q'_{\beta(i)}\,.
$$
We will use the following facts:
\begin{enumerate}
\item We have $p_i=p+n(i-1)$ for $i=1,\ldots,r$,\, and also\ $q_1<q_2<\cdots<q_t$ .

This follows from the seam-pulling action of the up-operation.
\item If $i<j$ and $q_i\equiv q_j\mod n$, then
$d_i\geq d_j$ .

Indeed, for a fixed $k$, the set of all
$q_i\equiv k\mod n$ forms an arithmetic progression
$\{q,q\sh+n,\ldots\}$, whereas the corresponding set of $\{p_i\mid q_i\equiv k\}$ is a subset of the arithmetic progression $\{p_1, p_2,\ldots\}=\{p,p\sh+n,\ldots\}$.  Hence, if $i<j$ and $q_i\equiv q_j\mod n$, then $p_j-p_i\geq q_j-q_i$, and so $d_i=q_i-p_i\geq q_j-p_j=d_j\,.$

\item  $\alpha(i)=\beta(i)
\quad\Longleftrightarrow\quad
d_i=d_{\alpha(i)}
\quad\Longrightarrow\quad
q_i\equiv q_{\alpha(i)}\mod n\quad
$

If $\alpha(i)=\beta(i)$, then
$
d_i=q_i-p_i=q'_{\beta(i)}-p'_{\alpha(i)}
=q'_{\alpha(i)}-p'_{\alpha(i)}=d_{\alpha(i)}
\,,$
and also $q_i=q'_{\beta(i)}\equiv q_{\beta(i)}=q_{\alpha(i)}$.

\end{enumerate}
\smallskip

\noindent
Assume $\alpha\neq\beta$, and let $j$ be the smallest value such that $\alpha(j)\neq \beta(j)\,.$\
Then $j$ is minimal with $\beta^{-1}\alpha(j)\neq j$, and necessarily:
$$
\beta^{-1}\alpha(j)>j\quad\text{with}\quad
q_{\beta^{-1}\alpha(j)}=q'_{\alpha(j)}\equiv q_{\alpha(j)}\,.
$$

Consider the sequence:\
$
j,\, \alpha(j),\, \alpha^2(j),\, \alpha^3(j),\ldots\,.
$\ \
If $\alpha(j),\alpha^2(j),\cdots,\alpha^{c}(j)\sh<j$, then by the definition of $j$
and Fact 3 we have:
$$\begin{array}{c}
d_j\neq d_{\alpha(j)}=d_{\alpha\alpha(j)}=d_{\alpha\alpha\alpha(j)}=\cdots=d_{\alpha^{c+1}(j)}\\[.7em]
q_{\alpha(j)}\equiv q_{\alpha\alpha(j)}\equiv
q_{\alpha\alpha\alpha(j)}\equiv\cdots\equiv q_{\alpha^{c+1}(j)}\,.
\end{array}$$
But we eventually have $\alpha^{c+1}(j)=j$, so to avoid the contradiction $d_j\neq d_j$, there must exist some $k:=\alpha^{c+1}(j)$ such that:
$$
k>j\quad\text{with}\quad
d_k=d_{\alpha(j)}
\quad\text{and}\quad
q_k\equiv q_{\alpha(j)}\,.
$$

Case (a): $j<\beta^{-1}\alpha(j) \leq k$.
Then by Fact 2, we have $d_{\beta^{-1}\alpha(j)}\geq d_k=d_{\alpha(j)}$.  But:
$$\begin{array}{rcl}
d_{\beta^{-1}\alpha(j)}&=&
q_{\beta^{-1}\alpha(j)}-p_{\beta^{-1}\alpha(j)}\\
&<&
q_{\beta^{-1}\alpha(j)}-p_{j}\\
&<&
q'_{\alpha(j)}-p'_{\alpha(j)}=d_{\alpha(j)}
\,,\end{array}$$
so this case is impossible.

Case (b): $j<k<\beta^{-1}\alpha(j)$.
Then by Fact 1, we have $p_j<p_k<q_k<q_{\beta^{-1}\alpha(j)}$.
But:
$$\begin{array}{rcl}
q_{\beta^{-1}\alpha(j)}
&=&q'_{\alpha(j)}\\
&=&p'_{\alpha(j)}+d_{\alpha(j)}\\
&=&p_j+d_{k}\\
&<&p_k+d_k\ =\ q_k\,.
\end{array}$$
Thus, this case is impossible also.

The above contradictions show that $\alpha=\beta$.  Hence we have
$$
p+nh_i=p'_i=p_{\sigma(i)}\,,\qquad
q+nh_i=q'_i=q_{\sigma(i)}\,,
$$
where $\sigma=\alpha^{-1}=\beta^{-1}$, which is the first part of Lemma (ii).

To see the second part of Lemma (ii), suppose $p_i,q_i$ given and let $\sigma$ satisfy
$d_i=d_{\sigma(i)}$.  Then define $h_i:=(p_{\sigma(i)}-p_i)/n$, so that
$p'_i:=p_i+nh_i=p_{\sigma(i)}$ and:
$$
q'_i:=q_i+nh_i=p'_i+d_i=p_{\sigma(i)}+d_{\sigma(i)}=q_{\sigma(i)}\,.
$$
Thus $\{p_1,\ldots,p_t\}=\{p'_1,\ldots,p'_t\}$
and $\{q_1,\ldots,q_t\}=\{q'_1,\ldots,q'_t\}$,
so $J={\widetilde J}{\uparrow}(h_1,\ldots,h_t)$, as desired.

This proves the Lemma, and hence Proposition 3(ii).

redpajama_set_name: RedPajamaArXiv


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Communication at its most profound level is the giving of self in love. Consequently, good communications has always been at the heart of the Society of the Divine Word.
The founder of the Society of the Divine Word, St Arnold Janssen began his life's work of awakening the missionary responsibility of the Church by publishing a popular newsletter, called The Little Messenger of the Sacred Heart.
Carrying on this tradition, the Australian Province of the Divine Word Missionaries produces a quarterly magazine, Society Matters, which brings to colourful life the stories of our missionaries and the people they serve from all around the world.
It also explores various missionary themes and offers a prayerful reflection for readers, supporters and benefactors.
Within the Province itself, a monthly newsletter keeps confreres informed of each other's activities and various other occasional publications are distributed containing matters of missiological or theological interest.
St Arnold once said: "The pastor of souls must use every means which the signs of times dictate". For us, this means not only our publications, but good use of our website, the distribution of a monthly e-newsletter and a social media presence.


text:
INTEGRATED ADVANTAGE
CERTIFIED FIELD TECHNICIAN PROGRAM
Christopher Ryan
Home Depot Rolls Out Nationwide Sensormatic Anti-Theft Systems and Slashes Shrinkage
How EES provided heavy construction installation logistics to install 1,800 Sensormatic Anti-Theft Systems at 450 Home Depot locations
Home Depot had been experiencing
higher than acceptable shrinkage
at the store level nationwide. They
made the decision to invest in a
top of the line anti-theft security
product from the Sensormatic Co.
based in southern Florida. The roll out was
implemented into all existing stores
nationwide, which presented enormous
logistical challenges in distribution,
scheduling, installation, and materials
placement. EES successfully executed the
installation under Home Depot's "no
interruption to business" directive.
Launched in 1989, Vermont-based Electrical Equipment Solutions, LLC (formerly Integrated Solutions, Inc.) provides global, manufacturer to end-user, customized installation logistics services for electrified products and equipment. Through an extensive network of licensed electrical contractors, construction service providers, warehousing and distribution facilities, and transportation partners, EES is positioned to meet the needs of our clients' most demanding installation logistics requirements.
Retrofitting Anti-Theft Systems Into Existing Stores:
Home Depot stores serve a customer base of both homeowners and contractors. The stores typically have operating hours from 6:00 AM to 10:00 PM, seven days a week. Each store location has multiple entrances and exits for walking traffic, as well as one large overhead door for contractors, which allows for forklift activity. All entrances and exits were scheduled to receive the new anti-theft systems.
The Sensormatic anti-theft system selected by Home Depot required that a receiving wire antenna, which runs between the two detection stations on either side of the doors, be placed under the concrete flooring. At each location, large sections of concrete flooring directly in front of each door were removed by saw cutting or jackhammering. After removing the concrete debris in EES containers, the antenna wire was laid and a specialized polymer fast-dry concrete was mixed and poured. Our teams would also install a large steel plate in the contractor door to allow the use of the forklifts in and out of the building. EES ran new electrical circuits to the Sensormatic equipment, in preparation for the arrival of Sensormatic's local technician, who would then connect, activate, and test the system.
EES implemented the entire scope of installation during each store's off hours, from 10:00 PM to 5:30 AM, so as not to interrupt important early morning contractor business. EES was responsible for having the work areas 100% clean and functioning by 5:30 AM. One location per night was completed per EES crew, with up to twelve crews operating simultaneously up and down our assigned region, the Eastern Seaboard.
"Partnering with EES to provide the heavy construction services they do so well, freed our technicians to stay customer-focused at the store level."
Fixed Costs, Multiple Crews, and Flexible Scheduling:
By entering into a regional, multiple-location contract to develop processes and manage on-site installations, EES was able to provide Sensormatic with a pre-set menu of services and associated costs based solely on the type of door (walk-in or forklift), regardless of geographical location.
EES' access to an extensive network of electricians, tradesmen, and labor pools allowed work to commence simultaneously at multiple locations, which drastically accelerated the completion schedules for this large-scale roll out.
When scheduling delays did occur because of weather, material shortages, etc., our independent crews just moved onto the next location, with no lost time.
"The flexibility of EES' crews and their commitment to our clients, no disturbance to business directive, made our roll out a success."
Federal Express Grows Into Global Supply Chain Management and Rebrands as FedEx
PHILIP MORRIS USA Outsources its Entire POS Logistics Department
DIEBOLD Gains Competitive Advantage
© 2023 by Electrical Equipment Solutions. Created By Our Partners At SIX


text:
Babyshambles [Peter Doherty]
Babyshambles is the first single for Pete Doherty's post-Libertines band of the same name. It was released in May 2004 on High Society Records. The single was limited to 2,000 copies of the CD and 1,000 copies of the 7" vinyl and both can now be found changing hands for more than £100 on internet auction sites such as eBay. It peaked at number #32 on the UK Singles Chart.
#1 8 Babyshambles 02:23 (Doherty)
#2 4 At the Flophouse 02:36 (Alison, Doherty)
#3 14 What Katie Did 03:06 (Doherty)
LabelHigh Society
Oh What a Lovely Tour (Babyshambles)
The Libertines (The Libertines)
Nothing Comes to Nothing (Babyshambles)
Duke of Clarence (The Libertines)
O2 London (The Libertines)
Horrorshow in London Town (The Libertines)
Broken Love Song (Peter Doherty)
Time for Heroes (The Libertines)
Un Bilo Titled (Babyshambles)
She is Far (The Libertines)
Can't Stand Me Now (The Libertines)
Kolly Kibber (Peter Doherty)