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Let $(W, S)$ be a Coxeter system. Soergel defined a category of bimodules $B$ over a polynomial ring whose split Grothendieck group is isomorphic to the Hecke algebra $H$ of $W$. Conjecturally, the image of certain indecomposable (projective?) bimodules in $B$ is the well-known Kazhdan-Lusztig basis of $H$. Assuming the conjecture, Soergel showed that the coefficients of the Kazhdan-Lusztig polynomials of $W$ are given by the dimensions of certain Hom-spaces in $B$. It follows that these coefficients are non-negative, which was already known by work of Kazhdan-Lusztig in the Weyl group case by linking these coefficients to intersection cohomology of the corresponding Schubert varieties.

Soergel proved this conjecture in 1992 for $W$ a Weyl group, and Härterich proved it in 1999 for $W$ an affine Weyl group. Unfortunately, I can't access the first paper, and the second paper is in German, so I don't know anything about either of these proofs.

Question: Do these proofs depend on the relationship of the coefficients of the K-L polynomials to intersection cohomology, or are they independent of the corresponding machinery?

(The reason I ask is that I am potentially interested in relating known combinatorial proofs of positivity to Soergel's work, and I want to get an idea of how much machinery I would need to learn to do this.)

Edit: Soergel's 1992 paper is here, if only I had the appropriate journal access. If anybody does and would like to send me this paper, that would be excellent - my contact information is at a link on my profile.

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It seems that you can find Soergel's paper at an open access site called DigiZeitschriften. A link to the journal containing this paper is digizeitschriften.de/main/dms/toc/?IDDOC=502899. Issue number 429 has the relevant paper. –  Mike Skirvin Jun 24 '10 at 21:06
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Thanks! Unfortunately, the paper only brings me more bad news; the proof depends on results from another paper of Soergel's (jstor.org/stable/1990960) which is in German... –  Qiaochu Yuan Jun 24 '10 at 21:57
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@Mike: This is a good site for online viewing, though U.S. users might want to read the copyright notice there before photocopying. (That's like illegal downloads of music, videos.) On his Webpage, Wolfgang suggests that reprints might be sent by his secretary: home.mathematik.uni-freiburg.de/soergel/ (Both reprints and secretaries are however vanishing breeds.) @Qiaochu: In any language you'll find Soergel's papers challenging to read, if that's any consolation. But they are often important papers, as are those of his adviser Jantzen in Hamburg. And some papers are intertwined. –  Jim Humphreys Jun 24 '10 at 22:12
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Qiaochu: 1 As someone who had to learn mathematical German in order to understand Jantzen's book on primitive ideals (which is still the best source for Harish-Chandra bimodules), I feel for you. At the same time, it's well worth it! 2 Can you, please, give a reference for "known combinatorial proofs of positivity"? I am not aware of any that work for general $y,w$ even in the case of the $A$ series. –  Victor Protsak Jun 25 '10 at 5:00
    
Neither am I! The best result I can track down is dmtcs.org/dmtcs-ojs/index.php/proceedings/article/download/… . –  Qiaochu Yuan Jun 25 '10 at 15:13

2 Answers 2

up vote 8 down vote accepted

My understanding is that Soergel's approach applies just to finite Weyl groups and not directly to other finite Coxeter groups (or more generally), since what he can actually prove depends on some of the geometric machinery used to prove the Kazhdan-Lusztig Conjecture. The same must be true of the 1999 thesis work of his student Martin Harterich involving affine Weyl groups, which doesn't seem to have been formally published. In those situations the coefficients of KL polynomials were seen to be nonnegative in the early steps taken by Kazhdan and Lusztig toward understanding their conjecture via Schubert varieties: they occur as dimensions of certain cohomology groups.

Later on, Soergel made his program more explicit for proving the nonnegativity for arbitrary Coxeter groups using his more algebraic/categorical setting of bimodules: MR2329762 (2009c:20009) 20C08 (20F55) Soergel,Wolfgang (D-FRBG), Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ¨uber Polynomringen. (German. English, German summaries) [Kazhdan-Lusztig polynomials and indecomposable bimodules over polynomial rings] J. Inst. Math. Jussieu 6 (2007), no. 3, 501–525. This is in a French journal but written in German; the helpful review by Ulrich Goertz is however in English if you have access to MathSciNet. (In any case, J. Reine Angew. Math. has become super-expensive for libraries, so print or online access gets tricky.)

A helpful follow-up paper (in English) by Soergel's later student Peter Fiebig (now at Erlangen) should also be consulted, though it is still unclear to me how far one can get with Soergel's conjectural approach in this spirit: MR2395170 (2009g:20087) 20F55 (20C08) Fiebig, Peter (D-FRBG), The combinatorics of Coxeter categories. Trans. Amer. Math. Soc. 360 (2008), no. 8, 4211–4233. (Fiebig's papers are on arXiv, by the way.)

I'll have to take another look at this literature, but in any case the nonnegativity of coefficients of KL polynomials for arbitrary Coxeter groups (predicted in 1979 by Kazhdan and Lusztig) remains an intriguing question. The general setting is far from the kind of representation theory or geometry one encounters in Lie theory, but a purely combinatorial approach seems at the moment unlikely to succeed.

ADDED: Special cases where Kazhdan-Lusztig polynomials have been computed are discussed in section 7.12 of my 1990/1992 book on reflection groups and Coxeter groups. In particular, noncrystallographic finite Coxeter groups all yield nonnegative coefficients. For dihedral groups, the polynomials are all 1, while for type $H_3$ the computer tables found by Mark Goresky are still on his Webpage at IAS. The 1987 paper in J. Algebra by Dean Alvis which I cited involved his unpublished computer results on the polynomials for $H_4$, for which his current Webpage gives details: http://mypage.iusb.edu/~dalvis/h4data/index.html

These polynomials were later recovered by Fokko du Cloux using his computer system Coxeter: see his last published paper MR2255133 (2007e:20010) 20C08 (20F55) du Cloux, Fokko (F-LYON-ICJ), Positivity results for the Hecke algebras of noncrystallographic finite Coxeter groups. J. Algebra 303 (2006), no. 2, 731–741.

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Perhaps I can supplement Jim's answer a little.

In the paper "Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln uber Polynomringen" Soergel shows that there are certain graded indecomposable bimodules over a polynomial ring (now known as Soergel bimodules) which categorify the Hecke algebra. (Note that these are not projective!!)

That is, the indecomposable objects are classified (up to shifts and isomorphism) by the Weyl group, and one has an isomorphism between the Hecke algebra and the split Grothendieck group of the category of Soergel bimodules.

As a consequence, one obtains a basis for the Hecke algebra which is positive in the standard basis (as follows from the construction of the isomorphism of the Grothendieck group with the Hecke algebra) and has positive structure constants (because it is a categorification).

Soergel conjectures that this basis is in fact the Kazhdan-Lusztig basis, which would imply positivity in general.

Up until now there are only two cases when one can verify Soergel's conjecture:

  • when one has some sort of geometry (in which case one can show that the indecomposable Soergel bimodules are the equivariant intersection cohomology of Schubert varieties, and then use old results of Kazhdan and Lusztig). This shows Soergel's conjecture for Coxeter groups associated to Kac-Moody groups (in particular finite and affine Weyl groups).
  • when the combinatorics is very simple (i.e. for dihedral groups (Soergel) or universal Coxeter groups (Fiebig, Libedinsky)).

Hence, up until now there are no examples where Soergel's conjecture has yielded positivity when it was not known by other means. Also note that in the vast majority of cases, the proof using Soergel bimodules is strictly more complicated than the geometry proof, as one needs an extra step to get from geometry to Soergel bimodules.

Soergel's conjecture would however have more far reaching consequences than a proof that Kazhdan-Lusztig polynomials have positive coefficients. For example it provides a natural "geometry" for arbitrary Coxeter groups. For example, generalising some sort of Soergel bimodules to complex reflection groups would yield a natural setting for the study of "spetses" (unipotent characters associated to complex reflection groups).

One should also note that Dyer has developed a very similar conjectural world associating commutative algebra categories to Coxeter groups. He instead considers modules over the dual nil Hecke ring (which is the analogue of the cohomology of a flag variety), and has many nice results and conjectures. (Much of his work considers more general orders than the Bruhat order, and so will probably come in handy soon ...!)

While I am at it I should mention Peter Fiebig's theory of Braden-MacPherson sheaves on moment graphs. This is (in a sense made precise in one of Peter's papers) a local version of Soergel bimodules, and hence many questions become more natural on the moment graph.

Finally, one should mention the recent work of Elias-Khovanov and Libedinsky, which give generators and relations for the monoidal category of Soergel bimodules for certain Coxeter groups. (Elias-Khovanov in type A and Libedinsky in right-angled type.) These are very interesting results, but it is unclear to what extent they can be used to attack Soergel's conjecture.

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