Abstract
Equivariant Euler characteristics are important numerical homotopy invariants for objects with group actions. They have deep connections with many other areas like modular representation theory and chromatic homotopy theory. They are also computable, especially for combinatorial objects like partition posets, buildings associated with finite groups of Lie types, etc. In this article, we make new contributions to concrete computations by determining the equivariant Euler characteristics for all subgroup complexes of symmetric groups \(\varSigma _n\) when n is prime, twice a prime, or a power of two and several variants. There are two basic approaches to calculating equivariant Euler characteristics. One is based on a recursion formula and generating functions, and another on analyzing the fixed points of abelian subgroups. In this article, we adopt the second approach since the fixed points of abelian subgroups are simple in this case.
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Acknowledgements
I would like to thank my Ph.D. advisor Jesper Møller for his support, encouragement, inspiration, and computer calculation help during this project. I would also like to thank the anonymous referee who read this article carefully and gave me many insightful comments and suggestions, especially a wonderful group-theoretic trick which simplifies the argument in Sect. 3 greatly.
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Duan, Z. Equivariant Euler Characteristics of Subgroup Complexes of Symmetric Groups. Ann. Comb. 27, 67–85 (2023). https://doi.org/10.1007/s00026-022-00630-2
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DOI: https://doi.org/10.1007/s00026-022-00630-2