Character orthogonality for the partition algebra and fixed points of permutations

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Abstract

We give a closed formula for the trace of the partition algebra Pn (x) acting on an irreducible representation whose basis is indexed by the set partitions of {1, ..., n}. This trace counts the set partitions that are fixed under the action of the symmetric group Sn. We use this trace to determine an analog of the "second orthogonality of characters" formula for Pn(x) and to compute the trace of the biregular representation of Pn(x). We use the Schur-Weyl duality between Pn(r) and the symmetric group Sr to study fixed points of random permutations in Sr. In particular, we compute the joint mixed moments of the random variables Tr(σj) for σ ∈ Sr. © 2003 Elsevier Inc. All rights reserved.

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Farina, J., & Halverson, T. (2003). Character orthogonality for the partition algebra and fixed points of permutations. Advances in Applied Mathematics, 31(1), 113–131. https://doi.org/10.1016/S0196-8858(02)00555-9

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