## Reference Request: Steinberg’s 1975 paper “On a paper of Pittie”(retrieved)

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I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference is a paper by Pittie( H.V. Pittie: Homogeneous vector bundles on homogeneous spaces, Topology II (1972) 199-203), but I could not find it online or in any books available in the library. Steinberg generalized Pittie's statement in his paper (Robert Steinberg, On a theorem by Pittie, Topology Vol. 14. pp. 173-177. Pergamon Press, 1975, Printed in Great Britain. Received 1 October 1974).

Since they already proved this in the past, I would like to see their papers before I finish my project, even if at some monetary cost. But I could not access either of them. Not knowing their work would not hinder my research, for I work in a much more elementary level than they did, but I think their work might be related to my eventual results and I should acknowledge in case they proved some formula I proved again on my own. So I want to ask where I can find them in paper or electronically. I can read parts of Steinberg's paper via google books, but I would like a pdf file or something (so I may check).

ADDED:

With advisor's help and the links provided by all the people below, I retrieved the two papers.

ADDED:

Received Steinberg's replying email. He notes "A correction should be made on p.175, line 6 ( which starts with "Consider now ") by putting the exponent "n sub a" on the item over which the product is being taken.The paper by Pittie appears in Topology, vol. 11, 1972, pp. 199-203, and, if I remember correctly, does not contain an explicit basis for the quotient. " This is important so I put in here.

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Changwei, both are available online. Pittie: sciencedirect.com/science/article/pii/… Steinberg: sciencedirect.com/science/article/pii/… Hopefully, your institution has a membership which will allow you to get them without paying full price... – BR Feb 1 2012 at 23:21
@BR: I accessed both websites via the link you provided, but when I press the button "view the full text", there is no output. Thanks for your help though. – Changwei Zhou Feb 1 2012 at 23:41
Changwei, I think you might have to either purchase the article or log in through an institutional account. I was able to download both through my institution (technically, I went through MathSciNet). – BR Feb 2 2012 at 3:25
@BR: This is disappointing, but at least I can try something now. Thank you. – Changwei Zhou Feb 2 2012 at 3:54

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Not to be too public about it, but I'm sure that if all else fails in retrieving these papers, various people on MO would be glad to download and email to you. – Steve D Feb 3 2012 at 20:46

## 2 Answers

To supplement Barry's citations, I'd point out that the journal Topology was at that time managed by a company which eventually gave up on it after editors resigned partly in protest against the high prices charged. While the online rights now belong to the ScienceDirect conglomerate, it's expensive to access. This can be frustrating because each paper discussed here is only 4+ pages long.

On the other hand, Steinberg's paper is reprinted in the moderately priced one volume Collected Papers (AMS 1997). Though Steinberg is long retired from UCLA, he maintains an email link there, and might be able to supply a reprint of his article. Pittie is an Indian mathematician who has taught at one of the colleges of City University of New York but has not published for many years; his entry in the combined membership list CML (www.ams.org) does give a current mailing address in New York City.

Some users of MO including myself do have access to both papers and might be able to answer precisely stated questions about them.

ADDED: I hadn't heard previously about the recent death of Harsh Pittie. I was somewhat acquainted with him when we were both at NYU-Courant decades ago and recall hearing some of his lectures on topology of Lie groups. His paper from that period was grounded in topology and K-theory, but Steinberg's follow-up (in his typical concise style) rounded out the discussion of representation rings in a more algebraic framework. Moreover, Steinberg exhibits an explicit basis for `$R(T)$` as a free `$R(G)$`-module in the crucial case where `$G$` is a semisimple simply connected compact Lie group and `$T$` any maximal torus. In particular, the rank here is the order of the Weyl group `$W$`. (He also observes that the same ideas work for algebraic groups over any algebraically closed field.)

Though I've never worked through the details of Steinberg's paper carefully, the underlying idea can be observed (in an oversimplified way) in the rank 1 case. Denoting the weight lattice (character group of `$T$` in additive notation) by `$X$`, the respective representation rings look like `$\mathbb{Z}[X]$` and `$\mathbb{Z}[X]^W$`. Then Steinberg's basis elements, one for each element `$w \in W$`, are defined by applying `$w^{-1}$` to a product of symbols (in my notation `$e^\lambda$`) with `$\lambda$` running over suitable fundamental weights. In rank 1, the basis just consists of `$e^0, e^{-\rho}$`.

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@Jim Humphreys: Thanks for the comment. I think I should contact them directly. My main question is whether their means to prove $R(T)$ is a free module over $R(G)$ related to the action of the Weyl group on fundamental weights(thus the weight lattice), for that is the approach I am going to take. I also wish to express the gratitude as I learnt a lot by reading your small book.  – Changwei Zhou Feb 2 2012 at 1:50
@Jim Humphreys: Unfortunately I found Pittie has already passed away (see jxxcarlson.wordpress.com/2012/01/25/…). But thank you for the information.  – Changwei Zhou Feb 2 2012 at 2:04

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If you're willing to pay, you can go to the Topology website and track the articles down. Here's a link that'll take you straight to the issue with the Pittie piece: http://www.sciencedirect.com/science/journal/00409383/11/2 -- you can find a link to the Steinberg issue there too. (Caveat: I don't know for a fact the articles are actually available; it's possible the site will say the order can't be filled. I didn't want to plunk down the coin to find out.)

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@Barry Cipra: Hi, I will try to access the Topology website after dinner. For the link you provided (as well the links provided from above comments), I just do not know how to get the full article as there is no output by pressing the button.  – Changwei Zhou Feb 1 2012 at 23:45
Unfortunately, the button you have to press is the one that says "Purchase."  Hopefully someone can arrange to get you reprints. – Barry Cipra Feb 2 2012 at 2:44
@Barry Cipra: I see, so essentially I need to log in to access both articles. Let me contact my school (Bard)'s librarian to see if there is something they can do in this situation.  – Changwei Zhou Feb 2 2012 at 11:11
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