Bases for certain cohomology representations of the symmetric group
Anthony Henderson
University of Sydney School of Mathematics and Statistics NSW 2006 Sydney Australia NSW 2006 Sydney Australia
DOI: 10.1007/s10801-006-0018-3
Abstract
We give a combinatorial description (including explicit differential-form bases) for the cohomology groups of the space of n distinct nonzero complex numbers, with coefficients in rank-one local systems which are of finite monodromy around the coordinate hyperplanes and trivial monodromy around all other hyperplanes. In the case where the local system is equivariant for the symmetric group, we write the cohomology groups as direct sums of inductions of one-dimensional characters of subgroups. This relies on an equivariant description of the Orlik-Solomon algebras of full monomial reflection groups (wreath products of the symmetric group with a cyclic group). The combinatorial models involved are certain representations of these wreath products which possess bases indexed by labelled trees.
Pages: 361–390
Keywords: keywords hyperplane complement; cohomology; representation; symmetric group
Full Text: PDF
References
1. H. Barcelo, “On the action of the symmetric group on the free Lie algebra and the partition lattice,” J. Combin. Theory Ser. A, 55 (1990), 93-129.
2. D.C. Cohen, “Triples of arrangements and local systems,” Proc. Amer. Math. Soc. 130(10) (2002), 3025-3031.
3. J.M. Douglass, “On the cohomology of an arrangement of type Bl ,” J. Algebra 147 (1992), 265-282.
4. P. Hanlon, “The characters of the wreath product group acting on the homology groups of the Dowling lattices,” J. Algebra 91 (1984), 430-463.
5. A. Henderson, “Representations of wreath products on cohomology of De Concini-Procesi compactifications,” Int. Math. Res. Not. 20 (2004), 983-1021.
6. A. Henderson, “The symmetric group representation on cohomology of the regular elements of a maximal torus of the special linear group,” math.RT/0312006.
7. Y. Kawahara, “Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements,” Proc. Amer. Math. Soc. 133(7) (2005), 1907-1915.
8. G.I. Lehrer, “On the Poincaré series associated with Coxeter group actions on complements of hyperplanes,” J. London Math. Soc. 36(2) (1987), 275-294.
9. G.I. Lehrer, “On hyperoctahedral hyperplane complements,” in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Vol. 47 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1987, pp. 219-234.
10. G.I. Lehrer, “Poincaré polynomials for unitary reflection groups,” Invent. Math. 120 (1995), 411-425.
11. G.I. Lehrer and L. Solomon, “On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes,” J. Algebra. 104 (1986), 410-424.
12. A. Libgober and S. Yuzvinsky, “Cohomology of local systems,” in Arrangements-Tokyo 1998, Vol. 27 of Adv. Stud. Pure Math., Kinokuniya, Tokyo, 2000, pp. 169-184. Springer J Algebr Comb (2006) 24:361-390
13. P. Orlik and H. Terao, “ Arrangements of Hyperplanes. Springer-Verlag, 1992.
14. V. Schechtman, H. Terao, and A. Varchenko, “Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors,” J. Pure Appl. Algebra 100(1-3) (1995), 93- 102.
15. S. Yuzvinsky, “Orlik-Solomon algebras in algebra and topology,” Uspekhi Mat. Nauk 56(2) (2001), (338), pp. 87-166, translation in Russian Math. Surveys 56(2) (2001), 293-364.
2. D.C. Cohen, “Triples of arrangements and local systems,” Proc. Amer. Math. Soc. 130(10) (2002), 3025-3031.
3. J.M. Douglass, “On the cohomology of an arrangement of type Bl ,” J. Algebra 147 (1992), 265-282.
4. P. Hanlon, “The characters of the wreath product group acting on the homology groups of the Dowling lattices,” J. Algebra 91 (1984), 430-463.
5. A. Henderson, “Representations of wreath products on cohomology of De Concini-Procesi compactifications,” Int. Math. Res. Not. 20 (2004), 983-1021.
6. A. Henderson, “The symmetric group representation on cohomology of the regular elements of a maximal torus of the special linear group,” math.RT/0312006.
7. Y. Kawahara, “Vanishing and bases for cohomology of partially trivial local systems on hyperplane arrangements,” Proc. Amer. Math. Soc. 133(7) (2005), 1907-1915.
8. G.I. Lehrer, “On the Poincaré series associated with Coxeter group actions on complements of hyperplanes,” J. London Math. Soc. 36(2) (1987), 275-294.
9. G.I. Lehrer, “On hyperoctahedral hyperplane complements,” in The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Vol. 47 of Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI, 1987, pp. 219-234.
10. G.I. Lehrer, “Poincaré polynomials for unitary reflection groups,” Invent. Math. 120 (1995), 411-425.
11. G.I. Lehrer and L. Solomon, “On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes,” J. Algebra. 104 (1986), 410-424.
12. A. Libgober and S. Yuzvinsky, “Cohomology of local systems,” in Arrangements-Tokyo 1998, Vol. 27 of Adv. Stud. Pure Math., Kinokuniya, Tokyo, 2000, pp. 169-184. Springer J Algebr Comb (2006) 24:361-390
13. P. Orlik and H. Terao, “ Arrangements of Hyperplanes. Springer-Verlag, 1992.
14. V. Schechtman, H. Terao, and A. Varchenko, “Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors,” J. Pure Appl. Algebra 100(1-3) (1995), 93- 102.
15. S. Yuzvinsky, “Orlik-Solomon algebras in algebra and topology,” Uspekhi Mat. Nauk 56(2) (2001), (338), pp. 87-166, translation in Russian Math. Surveys 56(2) (2001), 293-364.