Computing Kazhdan--Lusztig cells for unequal parameters

Following Lusztig, we consider a Coxeter group $W$ together with a weight function $L$. This gives rise to the pre-order relation $\leq_{L}$ and the corresponding partition of $W$ into left cells. We introduce an equivalence relation on weight functions such that, in particular, $\leq_{L}$ is constant on equivalent classes. We shall work this out explicitly for $W$ of type $F_4$ and check that several of Lusztig's conjectures concerning left cells with unequal parameters hold in this case, even for those parameters which do not admit a geometric interpretation. The proofs involve some explicit computations using {\sf CHEVIE}

Conjectures about certain parabolic Kazhdan--Lusztig polynomials

Irreducibility results for parabolic induction of representations of the general linear group over a local non-archimedean field can be formulated in terms of Kazhdan--Lusztig polynomials of type $A$. Spurred by these results and some computer calculations, we conjecture that certain alternating sums of Kazhdan--Lusztig polynomials known as parabolic Kazhdan--Lusztig polynomials satisfy properties analogous to those of the ordinary ones.Comment: final versio

Infrared Computations of Defect Schur Indices

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.Comment: 63 pages + appendices, 15 figures. v2 published version, references added, representations of SO(8) Kac-Moody discusse

Leading Coefficients of Kazhdan--Lusztig Polynomials in Type DD

Kazhdan--Lusztig polynomials arise in the context of Hecke algebras associated to Coxeter groups. The computation of these polynomials is very difficult for examples of even moderate rank. In type $A$ it is known that the leading coefficient, $\mu(x,w)$ of a Kazhdan--Lusztig polynomial $P_{x,w}$ is either 0 or 1 when $x$ is fully commutative and $w$ is arbitrary. In type $D$ Coxeter groups there are certain "bad" elements that make $\mu$-value computation difficult. The Robinson--Schensted correspondence between the symmetric group and pairs of standard Young tableaux gives rise to a way to compute cells of Coxeter groups of type $A$. A lesser known correspondence exists for signed permutations and pairs of so-called domino tableaux, which allows us to compute cells in Coxeter groups of types $B$ and $D$. I will use this correspondence in type $D$ to compute $\mu$-values involving bad elements. I will conclude by showing that $\mu(x,w)$ is 0 or 1 when $x$ is fully commutative in type $D$.Comment: Author's Ph.D. Thesis (2013) directed by R.M. Green at the University of Colorado Boulder. 68 page

Computing Kazhdan-Lusztig polynomials for arbitrary Coxeter groups

this paper, we shall present an algorithm which, for any Coxeter group W , constructs directly from the Coxeter matrix (m s;t ) and a word a = (s 1 ; : : : ; s p ) in the generators, the interval [e; y] in the Bruhat ordering, where y = s 1 : : : s p is the element of W represented by a, together with the (partially de ned) left and right actions of all the generators on [e; y], without any prior implementation of the group operations. This provides us with exactly the data we need to compute P x;z for all x z y using the recursion formula of Kazhdan and Lusztig. The algorithm is based on the analysis of the structure of Bruhat intervals and some other Bruhat-like posets in [9]; the essential ingredient in its correctness proof is a remarkable theorem due to Matthew Dyer in his thesis [10], which we shall recall in section

Geometric and Topological Combinatorics

The 2007 Oberwolfach meeting “Geometric and Topological Combinatorics” presented a great variety of investigations where topological and algebraic methods are brought into play to solve combinatorial and geometric problems, but also where geometric and combinatorial ideas are applied to topological questions