Kevin Dilks ; T. Kyle Petersen ; John R. Stembridge - Affine descents and the Steinberg torus

dmtcs:3631 - Discrete Mathematics & Theoretical Computer Science, January 1, 2008, DMTCS Proceedings vol. AJ, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008) - https://doi.org/10.46298/dmtcs.3631
Affine descents and the Steinberg torus

Authors: Kevin Dilks 1; T. Kyle Petersen 1; John R. Stembridge 1

  • 1 Department of Mathematics - University of Michigan

Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori.


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: Coxeter complex,$\gamma$-nonnegativity,descent statistic,Eulerian polynomial,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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Source : ScholeXplorer IsRelatedTo ARXIV math/0102105
Source : ScholeXplorer IsRelatedTo DOI 10.1006/jabr.2001.8814
Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.math/0102105
  • math/0102105
  • 10.1006/jabr.2001.8814
  • 10.48550/arxiv.math/0102105
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