A Basis for the Top Homology of a Generalized Partition Lattice
Julie Kerr
DOI: 10.1023/A:1018663030935
Abstract
For a fixed positive integer k, consider the collection of all affine hyperplanes in n-space given by xi - xj = m, where i, j 
[n], i 
j, and m {0, 1,..., k}. Let Ln,k be the set of all nonempty affine subspaces (including the empty space) which can be obtained by intersecting some subset of these affine hyperplanes. Now give Ln,k a lattice structure by ordering its elements by reverse inclusion. The symmetric group Gn acts naturally on Ln,k by permuting the coordinates of the space, and this action extends to an action on the top homology of Ln,k. It is easy to show by computing the character of this action that the top homology is isomorphic as an Gn-module to a direct sum of copies of the regular representation, CGn. In this paper, we construct an explicit basis for the top homology of Ln,k, where the basis elements are indexed by all labelled, rooted, (k + 1)-ary trees on n-vertices in which the root has no 0-child. This construction gives an explicit Gn-equivariant isomorphism between the top homology of Ln,k and a direct sum of copies of CGn.
[n], i j, and m {0, 1,..., k}. Let Ln,k be the set of all nonempty affine subspaces (including the empty space) which can be obtained by intersecting some subset of these affine hyperplanes. Now give Ln,k a lattice structure by ordering its elements by reverse inclusion. The symmetric group Gn acts naturally on Ln,k by permuting the coordinates of the space, and this action extends to an action on the top homology of Ln,k. It is easy to show by computing the character of this action that the top homology is isomorphic as an Gn-module to a direct sum of copies of the regular representation, CGn. In this paper, we construct an explicit basis for the top homology of Ln,k, where the basis elements are indexed by all labelled, rooted, (k + 1)-ary trees on n-vertices in which the root has no 0-child. This construction gives an explicit Gn-equivariant isomorphism between the top homology of Ln,k and a direct sum of copies of CGn.Pages: 47–60
Keywords: intersection lattice; partition lattice; homology; regular representation; rooted tree
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References
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2. A. Bj\ddot orner, “On the homology of geometric lattices,” Algebra Universalis 14 (1982), 107-128.
3. A. Garsia, “Combinatorics of the Free Lie Algebra and the symmetric group,” Analysis, et cetera, Academic Press, Boston, 1990, pp. 309-382.
4. R. Gill, Ph.D. Thesis, University of Michigan, in preparation.
5. P. Hanlon, “The fixed point partition lattices,” Pacific J. Math. 96 (1981), 319-341.
6. A. Joyal, “Foncteurs analytiques et esp`eces de structures,” Lecture Notes in Mathematics, Springer-Verlag, Berlin/Heidelberg/New York, 1986, Vol. 1234, pp. 126-160.
7. A.A. Klyachko, “Lie elements in the tensor algebra,” Siberian Math. J. 15(6) (1974), 1296-1304.
8. P. Orlik and H. Terao, Arrangements of Hyperplanes, Springer-Verlag, Berlin/ Heidelberg/New York, 1992.
9. G.-C. Rota, “On the foundations of combinatorial theory I. Theory of M\ddot obius functions,” Z. Wahrscheinlichkeitstheorie 2 (1964), 340-368.
10. R.P. Stanley, “Supersolvable lattices,” Algebra Universalis 2 (1972), 197-217.
11. R.P. Stanley, “Some aspects of groups acting on finite posets,” J. Combin. Theory Ser. A 32 (1982), 132-161.
12. R.P. Stanley, “Hyperplane arrangements, interval orders and trees,” preprint.
13. M. Wachs and J. Walker, “On geometric semilattices,” Order 2(4) (1986), 367-385.
14. H. Wilf, Generatingfunctionology, Academic Press, San Diego, 1990.