Annals of Combinatorics 2 (1998) 1-6

Orbifold Euler Characteristics and the Number of Commuting m-tuples in the Symmetric Groups

Jim Bryan and Jason Fulman

Department of Mathematics,University of California, Berkeley, CA 94720, USA

Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA

Received November 4, 1997

AMS Subject Classification: 55N20, 20B30

Abstract. Generating functions for the number of commuting m-tuples in the symmetric groups are obtained. We define a natural sequence of "orbifold Euler characteristics" for a finite group G acting on a manifold X. Our definition generalizes the ordinary Euler characteristic of X/G and the string-theoretic orbifold Euler characteristic. Our formulae for commuting m-tuples underlie formulae that generalize the results of Macdonald and Hirzebruch-Höfer concerning the ordinary and string-theoretic Euler characteristics of symmetric products.

Keywords: Euler characteristic, orbifold, cohomology, symmetric groups


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