Discrete Mathematics & Theoretical Computer Science |

- 1 Department of Mathematics [Univ California Davis]

The Kazhdan-Lusztig polynomials for finite Weyl groups arise in representation theory as well as the geometry of Schubert varieties. It was proved very soon after their introduction that they have nonnegative integer coefficients, but no simple all positive interpretation for them is known in general. Deodhar has given a framework, which generally involves recursion, to express the Kazhdan-Lusztig polynomials in a very attractive form. We use a new kind of pattern-avoidance that can be defined for general Coxeter groups to characterize when Deodhar's algorithm yields a non-recursive combinatorial formula for Kazhdan-Lusztig polynomials $P_{x,w}(q)$ of finite Weyl groups. This generalizes results of Billey-Warrington which identified the $321$-hexagon-avoiding permutations, and Fan-Green which identified the fully-tight Coxeter groups. We also show that the leading coefficient known as $\mu (x,w)$ for these Kazhdan―Lusztig polynomials is always either $0$ or $1$. Finally, we generalize the simple combinatorial formula for the Kazhdan―Lusztig polynomials of the $321$-hexagon-avoiding permutations to the case when $w$ is hexagon avoiding and maximally clustered.

Source: HAL:hal-01185181v1

Volume: DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)

Section: Proceedings

Published on: January 1, 2008

Imported on: May 10, 2017

Keywords: Kazhdan―Lusztig polynomial,pattern avoidance,$321$-hexagon,freely-braided,maximally-clustered,$2$-sided weak Bruhat order,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

Funding:

- Source : OpenAIRE Graph
*PECASE: Combinatorial Structures in Algebra and Geometry*; Funder: National Science Foundation; Code: 9983797*EMSW21-VIGRE: Focus on Mathematics*; Funder: National Science Foundation; Code: 0636297

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