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

Björner, Anders; Ekedahl, Torsten

Ann. Math. (2) 170, No. 2, 799-817 (2009).

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

Keywords:

crystallographic Coxeter group; Weyl group; Bruhat order; Schubert variety; \(\ell \)-adic cohomology; intersection cohomology; Kazhdan-Lusztig polynomial

References:

[1]	A. Björner and F. Brenti, Combinatorics of Coxeter Groups, New York: Springer-Verlag, 2005. · Zbl 1110.05001
[2]	A. A. Beuilinson, J. Bernstein, and P. Deligne, ”Faisceaux pervers,” in Analyse et Topologie sur les Espaces Singuliers, I, Paris: Soc. Math. France, 1982, pp. 5-171. · Zbl 0536.14011
[3]	J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, New York: Springer-Verlag, 1994. · Zbl 0808.14038
[4]	T. Braden and R. MacPherson, ”From moment graphs to intersection cohomology,” Math. Ann., vol. 321, iss. 3, pp. 533-551, 2001. · Zbl 1077.14522
[5]	M. Brion, ”Poincaré duality and equivariant (co)homology,” Michigan Math. J., vol. 48, pp. 77-92, 2000. · Zbl 1077.14523
[6]	J. B. Carrell, ”The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties,” in Algebraic Groups and their Generalizations: Classical Methods, Haboush, W. J. and Parshall, B. J., Eds., Providence, RI: Amer. Math. Soc., 1994, pp. 53-61. · Zbl 0818.14020
[7]	J. B. Carrell, ”On the smooth points of a Schubert variety,” in Representations of Groups, Allison, B. N. and Cliff, G. H., Eds., Providence, RI: Amer. Math. Soc., 1995, pp. 15-33. · Zbl 0854.14022
[8]	P. Deligne, ”La conjecture de Weil, II,” Inst. Hautes Études Sci. Publ. Math., vol. 52, pp. 137-252, 1980. · Zbl 0456.14014
[9]	M. Dyer, ”On the “Bruhat graph” of a Coxeter system,” Compositio Math., vol. 78, iss. 2, pp. 185-191, 1991. · Zbl 0784.20019
[10]	M. Goresky and R. MacPherson, ”Intersection homology, II,” Invent. Math., vol. 72, iss. 1, pp. 77-129, 1983. · Zbl 0529.55007
[11]	M. Goresky, ”Kazhdan-Lusztig polynomials for classical groups,” Northeastern University, technical report , 1981.
[12]	T. Hausel and B. Sturmfels, ”Toric hyperKähler varieties,” Doc. Math., vol. 7, pp. 495-534, 2002. · Zbl 1029.53054
[13]	J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge: Cambridge Univ. Press, 1990. · Zbl 0725.20028
[14]	A. Hultman, ”Bruhat intervals of length 4 in Weyl groups,” J. Combin. Theory Ser. A, vol. 102, iss. 1, pp. 163-178, 2003. · Zbl 1060.20036
[15]	D. Kazhdan and G. Lusztig, ”Representations of Coxeter groups and Hecke algebras,” Invent. Math., vol. 53, iss. 2, pp. 165-184, 1979. · Zbl 0499.20035
[16]	V. G. Kac, Infinite-Dimensional Lie Algebras: An Introduction, Boston: Birkhäuser, 1983. · Zbl 0537.17001
[17]	S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Boston: Birkhäuser, 2002. · Zbl 1026.17030
[18]	Théorie des Topos et Cohomologie Étale des Schémas, III, Artin, M., Grothendieck, A., Verdier, J. -L., Deligne, P., and Saint-Donat, B., Eds., New York: Springer-Verlag, 1973. · Zbl 0245.00002
[19]	M. Saito, ”Introduction to mixed Hodge modules, Actes du Colloque de Théorie de Hodge, (Luminy, 1987),” Astérisque, pp. 179-180, 1989. · Zbl 0753.32004
[20]	P. Slodowy, ”On the geometry of Schubert varieties attached to Kac-Moody Lie algebras,” in Proc. of the 1984 Vancouver Conference in Algebraic Geometry, Carrell, J., Geramita, A. V., and Russell, P., Eds., Providence, RI: Amer. Math. Soc., 1986, pp. 405-442. · Zbl 0591.14038
[21]	T. A. Springer, ”A purity result for fixed point varieties in flag manifolds,” J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 31, iss. 2, pp. 271-282, 1984. · Zbl 0581.20048
[22]	R. P. Stanley, ”Hilbert functions of graded algebras,” Advances in Math., vol. 28, iss. 1, pp. 57-83, 1978. · Zbl 0384.13012
[23]	R. P. Stanley, ”Weyl groups, the hard Lefschetz theorem, and the Sperner property,” SIAM J. on Algebraic Discrete Methods, vol. 1, iss. 2, pp. 168-184, 1980. · Zbl 0502.05004
[24]	D. Stanton, ”Unimodality and Young’s lattice,” J. Combin. Theory Ser. A, vol. 54, iss. 1, pp. 41-53, 1990. · Zbl 0736.05009
[25]	E. Swartz, ”\(g\)-elements, finite buildings and higher Cohen-Macaulay connectivity,” J. Combin. Theory Ser. A, vol. 113, iss. 7, pp. 1305-1320, 2006. · Zbl 1102.13025
[26]	A. Weber, ”Pure homology of algebraic varieties,” Topology, vol. 43, iss. 3, pp. 635-644, 2004. · Zbl 1072.14023