## On the shape of Bruhat intervals.(English)Zbl 1226.05268

Summary: 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 $$f^{w,J}_i$$ 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 $0\leq i< j\leq\ell(w)- i\quad\text{implies}\quad f^{w,J}_i\leq f^{w,J}_j.$ Also, the case of equalities $$f^w_i= f^w_{\ell(w)-i}$$ for $$i= 1,\dots,k$$ 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 $$f^w_0, f^w_1,\dots, f^w_{\ell(w)}$$ cannot grow too rapidly. Further, in the finite case, for any given $$k\geq 1$$ and any $$w\in W$$ of sufficiently great length (with respect to $$k$$), we show $f^w_{\ell(w)-k}\geq f^w_{\ell(w)- k+1}\geq\cdots\geq f^w_{\ell(w)}.$ The proofs rely mostly on properties of the cohomology of Kac-Moody Schubert varieties, such as the following result: if $$\overline X_w$$ is a Schubert variety of dimension $$d=\ell(w)$$, and $$\lambda= c_1({\mathcal L})\in H^2(\overline X_w)$$ is the restriction to $$\overline X_w$$ of the Chem class of an ample line bundle, then $(\lambda^k)\cdot: H^{d-k}(\overline X_w)\to H^{d+k}(\overline X_w)$ is injective for all $$k\geq 0$$.

### MSC:

 05E99 Algebraic combinatorics 06A11 Algebraic aspects of posets 14F20 Étale and other Grothendieck topologies and (co)homologies 14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) 14M15 Grassmannians, Schubert varieties, flag manifolds 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05A05 Permutations, words, matrices
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### References:

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