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 λ, μ, τ of n, the multiplicity of λ in μ ⊗ τ is called the Kronecker coefficient g λ μ,τ . 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 Rep(S t ). This allows us to interpret various properties of the reduced Kronecker coefficients as categorical properties of Deligne categories Rep(S t ) and derive new combinatorial identities.

Kronecker Comultiplication of Stable Characters and Restriction From $S_{mn}$ to $S_m \times S_n$

“…Stability properties follow by taking t 1 , t 2 to be sufficiently large integers and applying specialisation functors to obtain representations of S t1 × S t2 . The argument is similar to stable Kronecker coefficients in Subsection 5.4 of [EA16]. The author is grateful to Pavel Etingof for this remark.…”

Generators for the representation rings of certain wreath products

“…Understanding these non-semisimple Deligne categories and their split Grothendieck rings in better detail is of independent interest as they seem to lie somewhere between classical representation theory and stable representation theory. Recently Inna Entova-Aizenbud [4] used the nonsemisimple Deligne categories for symmetric groups to find new identities involving the reduced and non-reduced Kronecker coefficients.…”

Global symmetry and conformal bootstrap in the two-dimensional $Q$-state Potts model

“…For the logarithmic case, conformal blocks with primary fields as external fields have been completely determined in [7]. [18,19]. Thus, the resulting Young diagrams are independent of Q.…”