The combinatorics of twisted involutions in Coxeter groups

Hultman, Axel

doi:10.1090/S0002-9947-07-04070-6

The combinatorics of twisted involutions in Coxeter groups
HTML articles powered by AMS MathViewer

by Axel Hultman PDF

Trans. Amer. Math. Soc. 359 (2007), 2787-2798 Request permission

Abstract:

The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein$^*$ over $\mathbb {Z}_2$. We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the $h$-polynomial of the complex coincides with the polynomial counting twisted involutions by descents. In particular, this gives a type-independent proof that the latter is symmetric.

References

A. Björner, Posets, regular CW complexes and Bruhat order, European J. Combin. 5 (1984), no. 1, 7–16. MR 746039, DOI 10.1016/S0195-6698(84)80012-8
Anders Björner, Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math. 52 (1984), no. 3, 173–212. MR 744856, DOI 10.1016/0001-8708(84)90021-5
Anders Björner, The Möbius function of factor order, Theoret. Comput. Sci. 117 (1993), no. 1-2, 91–98. Conference on Formal Power Series and Algebraic Combinatorics (Bordeaux, 1991). MR 1235170, DOI 10.1016/0304-3975(93)90305-D
Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler, Oriented matroids, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999. MR 1744046, DOI 10.1017/CBO9780511586507
Anders Björner and Michelle Wachs, Bruhat order of Coxeter groups and shellability, Adv. in Math. 43 (1982), no. 1, 87–100. MR 644668, DOI 10.1016/0001-8708(82)90029-9
Francesco Brenti, $q$-Eulerian polynomials arising from Coxeter groups, European J. Combin. 15 (1994), no. 5, 417–441. MR 1292954, DOI 10.1006/eujc.1994.1046
Francesco Brenti, Kazhdan-Lusztig polynomials: history problems, and combinatorial invariance, Sém. Lothar. Combin. 49 (2002/04), Art. B49b, 30. MR 2006565
Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123, DOI 10.1007/978-1-4612-1019-1
R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 318337
Vinay V. Deodhar, Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function, Invent. Math. 39 (1977), no. 2, 187–198. MR 435249, DOI 10.1007/BF01390109
Fokko du Cloux, An abstract model for Bruhat intervals, European J. Combin. 21 (2000), no. 2, 197–222. MR 1742435, DOI 10.1006/eujc.1999.0343
W. M. B. Dukes, Permutation statistics on involutions, European J. Combin. 28 (2007), no. 1, 186–198. MR 2261870, DOI 10.1016/j.ejc.2005.07.014
M. J. Dyer, Hecke algebras and reflections in Coxeter groups, Ph.D. thesis, University of Sydney, 1987.
A. M. Garsia and D. Stanton, Group actions of Stanley-Reisner rings and invariants of permutation groups, Adv. in Math. 51 (1984), no. 2, 107–201. MR 736732, DOI 10.1016/0001-8708(84)90005-7
Axel Hultman, Fixed points of involutive automorphisms of the Bruhat order, Adv. Math. 195 (2005), no. 1, 283–296. MR 2145798, DOI 10.1016/j.aim.2004.08.011
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Federico Incitti, The Bruhat order on the involutions of the symmetric group, J. Algebraic Combin. 20 (2004), no. 3, 243–261. MR 2106960, DOI 10.1023/B:JACO.0000048514.62391.f4
Federico Incitti, The Bruhat order on the involutions of the hyperoctahedral group, European J. Combin. 24 (2003), no. 7, 825–848. MR 2009396, DOI 10.1016/S0195-6698(03)00103-3
F. Incitti, Bruhat order on the involutions of classical Weyl groups, Ph.D. thesis, Università di Roma “La Sapienza”, 2003.
M. Marietti, Combinatorial properties of zircons, Institut Mittag-Leffler, http://www.mittag-leffler.se /preprints/0405s/info.php?id=34.
Nathan Reading, The cd-index of Bruhat intervals, Electron. J. Combin. 11 (2004), no. 1, Research Paper 74, 25. MR 2097340
R. W. Richardson and T. A. Springer, The Bruhat order on symmetric varieties, Geom. Dedicata 35 (1990), no. 1-3, 389–436. MR 1066573, DOI 10.1007/BF00147354
R. W. Richardson and T. A. Springer, Complements to: “The Bruhat order on symmetric varieties” [Geom. Dedicata 35 (1990), no. 1-3, 389–436; MR1066573 (92e:20032)], Geom. Dedicata 49 (1994), no. 2, 231–238. MR 1266276, DOI 10.1007/BF01610623
T. A. Springer, Some results on algebraic groups with involutions, Algebraic groups and related topics (Kyoto/Nagoya, 1983) Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 525–543. MR 803346, DOI 10.2969/aspm/00610525
Richard P. Stanley, $f$-vectors and $h$-vectors of simplicial posets, J. Pure Appl. Algebra 71 (1991), no. 2-3, 319–331. MR 1117642, DOI 10.1016/0022-4049(91)90155-U
V. Strehl, Symmetric Eulerian distributions for involutions, Sémin. Lothar. Combin. 1, Strasbourg 1980. Publications de l’I.R.M.A. 140/S-02 , Strasbourg, 1981.