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Báo cáo toán học: " Combinatorial Laplacian of the Matching Complex"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Combinatorial Laplacian of the Matching Complex. | Combinatorial Laplacian of the Matching Complex Xun Dong School of Mathematics University of Minnesota Minneapolis MN 55455 xdong@math.umn.edu Michelle L. Wachs Department of Mathematics University of Miami Coral Gables FL 33124 wachs@math.miami.edu Submitted September 28 2000 Accepted April 2 2002. MR Subject Classifications Primary 05E10 05E25 Secondary 05E05 20C30 55U10. Abstract A striking result of Bouc gives the decomposition of the representation of the symmetric group on the homology of the matching complex into irreducibles that are self-conjugate. We show how the combinatorial Laplacian can be used to give an elegant proof of this result. We also show that the spectrum of the Laplacian is integral. 1 Introduction The matching complex of a graph G is the abstract simplicial complex whose vertex set is the set of edges of G and whose faces are sets of edges of G with no two edges meeting at a vertex. The matching complex of the complete graph known simply as the matching complex and the matching complex of the complete bipartite graph known as the chessboard complex have arisen in a number of contexts in the literature see eg. 6 16 2 19 3 4 8 12 1 9 15 13 17 18 . Closely related complexes have been considered in 7 and 14 . Let Mn denote the matching complex of the complete graph on node set 1 . ng. The symmetric group n acts on the matching complex Mn by permuting the graph Research supported in part by a University of Minnesota McKnight Land Grant Professorship held by Victor Reiner. Current address Department of Mathematics Caltech Pasadena CA 91125. Research supported in part by NSF grant DMS 9701407. THE ELECTRONIC JOURNAL OF COMBINATORICS 9 2002 R17 1 nodes. This induces a representation on the reduced simplicial homology Hr Mn k where throughout this paper k is a held of characteristic 0. The Betti numbers for the matching complex and the decomposition of the representation into irreducibles were computed by Bouc 3 and later independently by .