On the shape of Bruhat intervals

Abstract

Let $(W,S)$ be a crystallographic Coxeter group (this includes all finite and affine Weyl groups), and let $J\subseteq S$. Let $W^J$ denote the set of minimal coset representatives modulo the parabolic subgroup $W_J$. For $w\in W^J$, let $f^{w\smash{,J}}_{i}$ denote the number of elements of length $i$ below $w$ in Bruhat order on $W^J$ (with notation simplified to $f^{w}_{i}$ in the case when $W^J=W$). We show that $$ 0\le i\lt j\le \ell (w)-i \quad\hbox{implies}\quad f^{w\smash{,J}}_{i} \le f^{w\smash{,J}}_{j}.$$ Also, the case of equalities $\smash{f^{w}_{i} = f^{w}_{\ell(w)-i}}$ for $i=1, \ldots,k$ is characterized in terms of vanishing of coefficients in the Kazhdan-Lusztig polynomial $P_{e,w}(q)$.

We show that if $W$ is finite then the number sequence $\smash{f^{w}_{0}, f^{w}_{1}, \ldots, f^{w}_{\ell (w)}}$ cannot grow too rapidly. Further, in the finite case, for any given $k\ge 1$ and any $w\in W$ of sufficiently great length (with respect to $k$), we show $$ f^{w}_{\ell(w)-k} \ge f^{w}_{\ell(w)-k+1} \ge\cdots \ge f^{w}_{\ell (w)}.$$

The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if $\mskip3mu\overline{\mskip-3mu X} _w$ is a Schubert variety of dimension $d=\ell (w)$, and $\lambda=c_1 (\mathscr L)\in H^2 (\mskip3mu\overline{\mskip-3mu X} _w)$ is the restriction to $\mskip3mu\overline{\mskip-3mu X} _w$ of the Chern class of an ample line bundle, then \[(\lambda^k)\,\cdot \,{} : H^{d-k}(\mskip3mu\overline{\mskip-3mu X} _w) \rightarrow H^{d+k}(\mskip3mu\overline{\mskip-3mu X} _w) \] is injective for all $k\ge 0$.

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      ZBLNUMBER = {1102.13025},
      MRREVIEWER = {Dmitrii V. Pasechnik},
      DOI = {10.1016/j.jcta.2005.11.007},
      }
  • [weber::pure] Go to document A. Weber, "Pure homology of algebraic varieties," Topology, vol. 43, iss. 3, pp. 635-644, 2004.
    @article{weber::pure, MRKEY = {2041634},
      AUTHOR = {Weber, Andrzej},
      TITLE = {Pure homology of algebraic varieties},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {43},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {635--644},
      ISSN = {0040-9383},
      CODEN = {TPLGAF},
      MRCLASS = {14F43 (14C30 55N33)},
      MRNUMBER = {2004m:14036},
      ZBLNUMBER = {1072.14023},
      MRREVIEWER = {Joost van Hamel},
      DOI = {10.1016/j.top.2003.09.001},
      }

Authors

Anders Björner

Department of Mathematics
Royal Institute of Technology
100 44 Stockholm
Sweden

Torsten Ekedahl

Department of Mathematics
Stockholm University
106 91 Stockholm
Sweden