Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties
Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 863-878.

In the computation of the intersection cohomology of Shimura varieties, or of the L 2 cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood (see for example the appendix of [8]). We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group. Apart from some introductory remarks about the origin of the identities, our paper is entirely combinatorial and does not require any knowledge of Shimura varieties or of representation theory.

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DOI: 10.5802/alco.66
Classification: 14G35,  05E45,  14F43,  05A18,  06A07,  06A11,  52B22
Keywords: Averaged discrete series characters, permutahedron, intersection cohomology, ordered set partitions, shellability
Ehrenborg, Richard 1; Morel, Sophie 2; Readdy, Margaret 1

1 University of Kentucky Department of Mathematics Lexington, KY 40506, USA
2 Princeton University, Department of Mathematics, Princeton, NJ 08540, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ehrenborg, Richard; Morel, Sophie; Readdy, Margaret. Some combinatorial identities appearing in the calculation of the cohomology of Siegel modular varieties. Algebraic Combinatorics, Volume 2 (2019) no. 5, pp. 863-878. doi : 10.5802/alco.66. https://alco.centre-mersenne.org/articles/10.5802/alco.66/

[1] Björner, Anders Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc., Volume 260 (1980) no. 1, pp. 159-183 | DOI | MR | Zbl

[2] Björner, Anders Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math., Volume 52 (1984) no. 3, pp. 173-212 | DOI | MR | Zbl

[3] Björner, Anders; Wachs, Michelle L. Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc., Volume 348 (1996) no. 4, pp. 1299-1327 | DOI | MR | Zbl

[4] Ehrenborg, Richard; Hedmark, Dustin Filters in the partition lattice, J. Algebraic Combin., Volume 47 (2018) no. 3, pp. 403-439 | DOI | MR | Zbl

[5] Ehrenborg, Richard; Jung, JiYoon The topology of restricted partition posets, J. Algebraic Combin., Volume 37 (2013) no. 4, pp. 643-666 | DOI | MR | Zbl

[6] Forman, Robin A user’s guide to discrete Morse theory, Sém. Lothar. Combin., Volume 48 (2002), p. Art. B48c, 35 | MR | Zbl

[7] Herb, Rebecca A. Characters of averaged discrete series on semisimple real Lie groups, Pacific J. Math., Volume 80 (1979) no. 1, pp. 169-177 | DOI | MR | Zbl

[8] Morel, Sophie Cohomologie d’intersection des variétés modulaires de Siegel, suite, Compos. Math., Volume 147 (2011) no. 6, pp. 1671-1740 | DOI | MR | Zbl

[9] Stanley, Richard P. Combinatorics and commutative algebra, Progress in Mathematics, 41, Birkhäuser Boston, Inc., Boston, MA, 1996, x+164 pages | MR | Zbl

[10] Stanley, Richard P. Enumerative combinatorics. Volume 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 2012, xiv+626 pages | MR | Zbl

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