On the shape of Bruhat intervals

2005/07/31 by Anders Björner, Bjorner, Anders, Torsten Ekedahl +1 · 4 citations
Mathematics · #05E99 #06A11 #14M15 #20F55 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.math/0508022

openalex publication_date 2005/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (W,S) be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and J a subset of S. Let W^J denote the set of minimal coset representatives modulo the parabolic subgroup W_J. For w in W^J, let f^w,J_i denote the number of elements of length i below w in Bruhat order on W^J (notation simplified to f^w_i in the case when J=S). We show that f^w,J_i is less than or equal to f^w,J_j when i < j and j is less than or equal to the length of w minus i. Furthermore, we express when an initial and final interval of the f's is symmetric around the middle in terms of Kazhdan-Lusztig polynomials. It is also shown that if W is finite then the sequence of f's cannot grow too rapidly. Som result mirroring our first result are obtaind, again in the finite case. The proofs rely for the most part on properties of the cohomology of Kac-Moody Schubert varieties.

Cited by

• Billey-Postnikov posets, rationally smooth Schubert varieties, and Poincaré duality
• Intersection cohomology without spaces
• A weak Lehmer code for type F₄
• An anticanonical perspective on G/P Schubert varieties

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