Math 280 spring 2005

Below are some suggested texts and papers, brief synopses, and a possible order in which to read/present them.

KL polys have applications to the representation theory of semisimple algebraic groups, Verma modules, algebraic geometry and topology of Schubert varieties, canonical bases, immanant inequalities, etc.

KL polys occur as
• change of basis coeffs (more or less) from the standard basis to the KL basis in a Hecke algebra (see Humphreys)
• giving the dimensions of local  intersection cohomology for Schubert varieties.   (this interpretation proves their positivity and integrality, but NO combinatorial interpretation is known!!) (see Geometry papers below; proved by KL in "Schubert varieties and Poincare duality.")
• the multiplicity of standard modules in indecomposable tilting modules (Soergel) are given by parabolic, affine KL polys eval at q=1. [ n_{\lambda + \rho, \mu + \rho}(q=1) ]
• the change of basis coeffs for the canonical basis in terms of the standard basis for the Fock space (basic representation) [d_{\lambda, \mu}(q). note Goodman-Wenzl ; Varagnolo-Vasserot show d_{\lambda, \mu}(q) = n_{\lambda + \rho, \mu + \rho}(q),]
• the multiplicity of simple modules in Specht modules for Hecke algebras at roots of unity over a field of characteristic 0are given by parabolic, affine KL polys eval at q=1. [ie decomposition numbers d_{\lambda, \mu}= d_{\lambda, \mu}(q=1)]
• the multiplicity of Verma modules (proved by Beilinson-Bernstein and Brylinski-Kashiwara) when eval at q=1
• the coefficient (+/-) of a Weyl module character (ie Schur function) in the expression of the character of an irreducible highest weight module (ch L(\lambda) ) of a quantum group at a root of unity , when evaluated at q=1. [this is Lusztig's conjecture, proved by KL, Kashiwara-Tanisaki]
• the multiplicities of standards in the Jantzen filtration on Vermas [Beilinson-Bernstein] and hence the multiplicity of irreducibles in  the Jantzen filtration of standard modules [by a result of Suzuki by functoriality].  Note, as a consequence, when we evaluate at q=1 we get overall multiplicity (not w/ the grading of the filtration).
• there is more about geometry, perverse sheaves, Lagrangian subvarieties, Springer resolution ...
• ...
Warning: the phrase "Lusztig conjecture" appears all over this area.  He has many conjectures, and many of them are now theorems.

 Basics (defns) Humphreys Reflection groups and coxeter groups   Shields Library QA171 .H833 1990 Regular Loan Ch 7, pg 145; introductory lectures and basic definitions will largely follow this text. If you are rusty on the idea of coxeter groups and length, read here (Ch 1, Ch 5), and also try HW1. Andersen,  Jantzen,  Soergel. Representations of quantum groups at Ap-TH root of unity and of semisimple groups in characteristic p, independence of p Shields Library QA176 .A53 1994 Regular Loan Andersen The irreducible characters for semi-simple algebraic groups and for quantum groups, Proc. ICM94 (Vol II, 1995), 731--743. see section 4 Brenti Kazhdan-Lusztig polynomials: history problems, and combinatorial invariance.  Sém. Lothar. Combin.  49  (2002/04), Art. B49b, 30 pp. (electronic). Good intro, defn's (a more combinatorical POV than Humphreys) Ariki Lectures on cyclotomic Hecke algebras; http://arxiv.org/abs/math.QA/9908005 a nice overview Parabolic KL polys Frenkel,  Khovanov,  Kirillov Kazhdan-Lusztig polynomials and canonical basis http://xxx.lanl.gov/abs/q-alg/9709042 A good place for basic defns of parabolic KL polys, and also very hand-on feel for quantum groups and canonical bases in a certain repr.     In this paper they show that the Kazhdan-Lusztig polynomials (and, more generally, parabolic KL polynomials) for the group $S_n$ coincide with the coefficients of the canonical basis in $n$th tensor power of the fundamental representation of the quantum group $U_q sl_k$. Leclerc,  Thibon Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials; http://arxiv.org/ps/math.QA/9809122 good exposition!  They show that the Littlewood-Richardson coefficients are values at 1 of certain parabolic Kazhdan-Lusztig polynomials for affine symmetric groups. Deodhar On some geometric aspects of Bruhat orderings II. The parabolic analogue of Kazhdan-Lusztig polynomials, Journal of Algebra 111 (1987), 483-506. this might be where parabolic KLs are first defined. computation and combinatorics Brenti Kazhdan-Lusztig and $R$-polynomials, Young's lattice, and Dyck partitions.  Pacific J. Math.  207  (2002),  no. 2, 257--286. Brenti many other papers Brenti has many papers on combinatorial properties of KL polys and using them for computation.  Combinatorics of posets come in. as to q-analogues of fibonacci numbers,  and things about increasing subsequences.   If interested, let me know and i will expand on this list. Lascoux; Schützenberger Polynômes de Kazhdan & Lusztig pour les grassmanniennes. (French) [Kazhdan-Lusztig polynomials for Grassmannians] Young tableaux and Schur functors in algebra and geometry (Toru\'n, 1980), pp. 249--266, Astérisque, 87--88, Soc. Math. France, Paris, 1981. does anyone read french? this is if you are interested in the geometry, and what KL polys have to do with intersection cohomology of schubert cells. Relating to canonical bases/ Fock space Leclerc Decomposition numbers and canonical bases; http://arxiv.org/abs/math.QA/9902006 They obtain some simple relations between decomposition numbers of quantized Schur algebras at an n-th root of unity (over a field of characteristic 0). These relations imply that every decomposition number for such an algebra occurs as a decomposition number for some Hecke algebra of type A. We prove similar relations between coefficients of the canonical basis of the q-deformed Fock space previously introduced in a joint work with Thibon. [nice summary of history] (LLT) Lascoux, Leclerc,  Thibon Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), 205-263. They give the algorithm for computing the global crystal basis of the basic representation of a quantum affine algebra of type An(1).  They conjecture [since proved by Ariki, Grojnowski]  that at q=1 , they get decomposition numbers for  Hecke algebras at  n th root of 1.  Very concrete description of Fock space and action.  [you might find cleaner exposition in later papers] Goodman Fock space and KL polys some very nice lecture notes. Goodman and Wenzl A path algorithm for affine Kazhdan-Lusztig polynomials, Math. Z. 237 (2001), no. 2, 235--249. Preprint version at math.RT/0011245 They develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A (see paper below), which in turn is a faster version of an algorithm due to Lascouz, Leclerc and Thibon.  (there is some relation to Littelmann paths) Goodman and Wenzl Crystal bases of quantum affine algebras and affine Kazhdan-Lusztig polynomials. Internat. Math. Res. Notices 1999, no. 5, 251--275. Preprint version at math.QA/9807014 This is type A.  They  show that the coefficients of the lower global crystal base for the Fock representation of quantum affine sl_n coincide with certain affine Kazhdan-Lusztig polynomials. These are q-analogues of decomposition numbers for Specht modules of the Hecke algebra of type A_n; the coefficients of the affine Kazhdan-Lusztig polynomials are q-analogues of decomposition numbers for tilting modules for quantum sl_k. Their algorithm allows fast computation of these decomposition numbers, improving on the algorithm of Lascoux, Leclerc, and Thibon for the lower global crystal base . Students interested in algorithms, computation, complexity, might like to read this.  Also you get a very hands-on feel for the Fock space. Goodman and Wenzl Iwahori-Hecke algebras of type $A$ at roots of unity. J. Algebra 215 (1999), no. 2, 694--734. Preprint version at q-alg/9610033 Mark Goresky TABLES OF KAZHDAN-LUSZTIG POLYNOMIALS once we've gone through the basic def'ns, you may want to look here to see examples Geometry Varagnolo  and Vasserot On the decomposition matrices of the quantized Schur algebra http://www.arxiv.org/abs/math.QA/9803023 Kashiwara, Saito Geometric Construction of Crystal Bases;  Research Institute for Mathematical Sciences, Kyoto University, preprint RIMS-1081 The result is very geometric, but the intro is a good read. They realize a crystal  as a set of Lagrangian subvarieties of the cotangent bundle of the quiver variety. Kashiwara,  Tanisaki Parabolic Kazhdan-Lusztig polynomials and Schubert varieties math.QA/9809122 They give a description of the intersection cohomology groups of the Schubert varieties in partial flag manifolds over symmetrizable Kac-Moody Lie algebras in terms of parabolic Kazhdan-Lusztig polynomials introduced by Deodhar. Billey,  Warrington Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations; http://arxiv.org/abs/math.CO/0005052 This is a mixture of geometry and combinatorics.  It uses the Schubert point of view.  If you are interested in this, Billey has several other papers along these lines.  They give a combinatorial formula for the Kazhdan-Lusztig polynomials $P_{x,w}$ in the symmetric group when $w$ is a 321-hexagon-avoiding permutation. Immanants Haiman Hecke algebra characters and immanant conjectures J. Amer. Math. Soc. 6 (1993) no. 3, 569-595. PostScript, PDF Two conjectures on characters of the Hecke algebra of type An, evaluated on Kazhdan-Lusztig basis elements. Theorem: immanants of Jacobi-Trudi matrices are positive combinations of Schur functions. If Conjecture 1 holds, then "monomial" immanants are also Schur positive. Rhoades , Skandera Kazhdan-Lusztig immanants this and the paper above are a very different sort of way to study the KL basis and symmetric functions Cells Kazhdan and Lusztig Representations of Coxeter groups and Hecke algebras; Invent. Math. 53 (1979), no. 2, 165 -- 184 This is where KL polys are defined, as is the relation on W by which right, left, 2-sided cells are defined. Graham ; Lehrer Cellular algebras, Invent. Math. 123 (1996), 1–34. more details on cells and the resulting representations LUSZTIG Left cells in Weyl groups, Lie Group Representations, I (R. L. R. Herb and J. Rosenberg, eds.), Lecture Notes in Math., vol. 1024, Springer-Verlag, 1983, pp. 99-111. or Cells in affine Weyl groups, Advanced Studies in Pure Math. 6, Algebraic groups and related topics, Kinokuniya and North-Holland, 1985, 255-287. or Lectures on affine Hecke algebras with unequal parameters. Available at arXiv:math.RT/0108172. More about cells. In type A: When you look at the w in a certain left cell that defines a Specht module of shape lambda, all those w give a pair P,Q under RSK both of that shape lambda, and all share the same P. (or is it Q? one for left cells the other for right)