The homology representations of the 𝑘-equal partition lattice

Sundaram, Sheila; Wachs, Michelle

doi:10.1090/S0002-9947-97-01806-0

The homology representations of the $k$-equal partition lattice
HTML articles powered by AMS MathViewer

by Sheila Sundaram and Michelle Wachs PDF

Trans. Amer. Math. Soc. 349 (1997), 935-954 Request permission

Abstract:

We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set $\{1,2,\ldots ,n\}$ obtained by restricting block sizes to the set $\{1,k,k+1,\ldots \}$. A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice.

References

V. I. Arnol′d, The cohomology ring of the group of dyed braids, Mat. Zametki 5 (1969), 227–231 (Russian). MR 242196
Kenneth Bacławski, Whitney numbers of geometric lattices, Advances in Math. 16 (1975), 125–138. MR 387086, DOI 10.1016/0001-8708(75)90145-0
Hélène Barcelo, On the action of the symmetric group on the free Lie algebra and the partition lattice, J. Combin. Theory Ser. A 55 (1990), no. 1, 93–129. MR 1070018, DOI 10.1016/0097-3165(90)90050-7
Anders Björner, Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159–183. MR 570784, DOI 10.1090/S0002-9947-1980-0570784-2
Anders Björner, On the homology of geometric lattices, Algebra Universalis 14 (1982), no. 1, 107–128. MR 634422, DOI 10.1007/BF02483913
Björner, A., Lovász, L., Yao, A.: Linear decision trees: volume estimates and topological bounds. In: Proc. 24th ACM Symp. on Theory of Computing, New York: ACM Press 1992, pp. 170–177.
Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. MR 1333388, DOI 10.1090/S0002-9947-96-01534-6
Anders Björner and Volkmar Welker, The homology of “$k$-equal” manifolds and related partition lattices, Adv. Math. 110 (1995), no. 2, 277–313. MR 1317619, DOI 10.1006/aima.1995.1012
Egbert Brieskorn, Sur les groupes de tresses [d’après V. I. Arnol′d], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lecture Notes in Math., Vol. 317, Springer, Berlin, 1973, pp. 21–44 (French). MR 0422674
Phil Hanlon, The fixed-point partition lattices, Pacific J. Math. 96 (1981), no. 2, 319–341. MR 637975, DOI 10.2140/pjm.1981.96.319
Phil Hanlon, The generalized Dowling lattices, Trans. Amer. Math. Soc. 325 (1991), no. 1, 1–37. MR 1014249, DOI 10.1090/S0002-9947-1991-1014249-X
André Joyal, Foncteurs analytiques et espèces de structures, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) Lecture Notes in Math., vol. 1234, Springer, Berlin, 1986, pp. 126–159 (French). MR 927763, DOI 10.1007/BFb0072514
G. I. Lehrer and Louis Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra 104 (1986), no. 2, 410–424. MR 866785, DOI 10.1016/0021-8693(86)90225-5
I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979. MR 553598
Peter Orlik and Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent. Math. 56 (1980), no. 2, 167–189. MR 558866, DOI 10.1007/BF01392549
Christophe Reutenauer, Dimensions and characters of the derived series of the free Lie algebra, Mots, Lang. Raison. Calc., Hermès, Paris, 1990, pp. 171–184. MR 1252663
Louis Solomon, A decomposition of the group algebra of a finite Coxeter group, J. Algebra 9 (1968), 220–239. MR 232868, DOI 10.1016/0021-8693(68)90022-7
Richard P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), no. 2, 132–161. MR 654618, DOI 10.1016/0097-3165(82)90017-6
Sheila Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math. 104 (1994), no. 2, 225–296. MR 1273390, DOI 10.1006/aima.1994.1030
Sheila Sundaram, Applications of the Hopf trace formula to computing homology representations, Jerusalem combinatorics ’93, Contemp. Math., vol. 178, Amer. Math. Soc., Providence, RI, 1994, pp. 277–309. MR 1310588, DOI 10.1090/conm/178/01904
Sundaram, S. and Welker, V.: Groups actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc., to appear.
Wachs, M. L.: A basis for the homology of the $d$-divisible partition lattice, Advances in Math.117(1996), 294–318.
Wachs, M. L.: On the (co)homology of the partition lattice and the free Lie algebra, Proceedings of the 1994 Taormina Conference in honour of A. Garsia, Discr. Math., to appear.