Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table

2010/07/11 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT

paper · pdf · doi:10.48550/arxiv.1007.1791

15 pages, to appear in Journal of Algebraic Combinatorics

arxiv created 2010/07/11 · openalex publication_date 2010/07/11 · arxiv updated 2010/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let R be the regular representation of a finite abelian group G and let C_n denote the cyclic group of order n. For G=C_n, we compute the Poincare series of all C_n-isotypic components in S^⋅ R⊗ \wedge^⋅ R (the symmetric tensor exterior algebra of R). From this we derive a general reciprocity and some number-theoretic identities. This generalises results of Fredman and Elashvili-Jibladze. Then we consider the Cayley table, M_G, of G and some generalisations of it. In particular, we prove that the number of formally different terms in the permanent of M_G equals (Sⁿ R)^G, where n is the order of G.

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