Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The generalized Dowling lattices

Author: Phil Hanlon
Journal: Trans. Amer. Math. Soc. 325 (1991), 1-37
MSC: Primary 06B05; Secondary 06B99, 20C15
MathSciNet review: 1014249
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study a new class of lattices called the generalized Dowling lattices. These lattices are parametrized by a positive integer $ n$, a finite group $ G$, and a meet sublattice $ K$ of the lattice of subgroups of $ G$. For an appropriate choice of $ K$ the generalized Dowling lattice $ {D_n}(G,K)$ agrees with the ordinary Dowling lattice $ {D_n}(G)$. For a different choice of $ K$, the generalized Dowling lattices are the lattice of intersections of a set of subspaces in complex space. The set of subspaces, defined in terms of a representation of $ G$, generalizes the thick diagonal in $ {\mathbb{C}^n}$.

We compute the Möbius function and characteristic polynomial of the lattice $ {D_n}(G,K)$ along with the homology of $ {D_n}(G,K)$ in terms of the homology of $ K$. We go on to compute the character of $ G$ wr $ {S_n}$ acting on the homology of $ {D_n}(G,K)$. This computation provides a nontrivial generalization of a result due to Stanley concerning the character of $ {S_n}$ acting on the top homology of the partition lattice.

References [Enhancements On Off] (What's this?)

  • [1] F. Bergeron, N. Bergeron, and A. M. Garsia, Idempotents for the free Lie algebra and $ q$-enumeration, preprint.
  • [2] A. Bjorner and J. Walker, A homotopy complementation formula for partially ordered sets, European J. Combin. 4 (1983), 11-19. MR 694463 (84f:06003)
  • [3] D. Burghelea and M. Vigué-Poirrier, Cyclic homology of commutative algebras, Proc. Meeting on Algebraic Homotopy (Louvain 1986), Lecture Notes in Math., vol. 1318, Springer, 1988. MR 952571 (89k:18027)
  • [4] H. H. Crapo, The Möbius function of a lattice, J. Combin. Theory 1 (1966), 126-131. MR 0193018 (33:1240)
  • [5] T. A. Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14 (1973), 61-86. MR 0307951 (46:7066)
  • [6] W. Feit, Characters of finite groups, Benjamin, 1967. MR 0219636 (36:2715)
  • [7] M. Gerstenhaber and S. Schack, A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebras 48 (1987), 229-247. MR 917209 (88k:13011)
  • [8] M. Goresky and R. Macpherson, Stratified Morse theory, Springer-Verlag, 1988. MR 932724 (90d:57039)
  • [9] P. Hanlon, The fixed-point partition lattices, Pacific J. Math. 96 (1981), 319-341. MR 637975 (83d:06010)
  • [10] -, The action of $ {S_n}$ on the components of the Hodge decomposition of Hochschild homology, Michigan Math. J. 37 (1990), 105-124. MR 1042517 (91g:20013)
  • [11] G. James and A. Kerber, The representation theory of the symmetric group, Encyclopedia of Math., Vol. 16, Addison-Wesley, New York, 1981. MR 644144 (83k:20003)
  • [12] J. L. Loday, Opérations sur l'homologie cyclique des algèbres commutatives, Invent. Math. 96 (1989), 205-230. MR 981743 (89m:18017)
  • [13] D. Passman, Permutation groups, Yale Univ. Press, 1967.
  • [14] C. Reutenauer, Theorem on Poincaré-Birkhoff-Witt, logarithm, and representations of the symmetric group whose orders are the Stirling numbers, preprint.
  • [15] G. C. Rota, On the foundations of combinatorial theory I: theory of Mobius functions, Z. Wahrsch. Verw. Gebiete 2 (1964), 340-368. MR 0174487 (30:4688)
  • [16] R. P. Stanley, Some aspects of groups acting on finite posets, J. Combin. Theory Ser. A 32 (1982), 132-161. MR 654618 (83d:06002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 06B05, 06B99, 20C15

Retrieve articles in all journals with MSC: 06B05, 06B99, 20C15

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society