Deligne categories and reduced Kronecker coefficients

Abstract

The Kronecker coefficients are the structural constants for the tensor categories of representations of the symmetric groups; namely, given three partitions $\lambda ,\mu ,\tau$ of $n$, the multiplicity of $\lambda$ in $\mu \otimes \tau$ is called the Kronecker coefficient $g_{\mu ,\tau }^{\lambda }$. 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 the Deligne categories $\underline{Rep}(S_t)$. This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of the categories $\underline{Rep}(S_t)$.

14 pages. arXiv admin note: substantial text overlap with arXiv:1403.5509

References in corpus (2)

• Schur-Weyl duality for Deligne categories
• The partition algebra and the Kronecker coefficients

Cited by in corpus (2)

• A new approach to the representation theory of the partition category
• Generators for the representation rings of certain wreath products