Deligne categories and reduced Kronecker coefficients

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Abstract

The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups, namely, given three partitions λ, μ, τ of n, the multiplicity of λ in μ⊗ τ is called the Kronecker coefficient gμ,τλ. When the first part of each of the partitions is taken to be very large (the remaining parts being fixed), the values of the appropriate Kronecker coefficients stabilize; the stable value is called the reduced (or stable) Kronecker coefficient. These coefficients also generalize the Littlewood–Richardson coefficients and have been studied quite extensively. In this paper, we show that reduced Kronecker coefficients appear naturally as structure constants of Deligne categories Rep̲(St). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories Rep̲(St) and derive new combinatorial identities.

Original languageEnglish
Pages (from-to)345-362
Number of pages18
JournalJournal of Algebraic Combinatorics
Volume44
Issue number2
DOIs
StatePublished - 1 Sep 2016
Externally publishedYes

Keywords

  • Deligne categories
  • Kronecker coefficients
  • Representations of symmetric groups

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