On blocks of Delignes category Rep(S_t)

Recently P. Deligne introduced the tensor category Rep(S_t) (for t not necessarily an integer) which in a certain precise sense interpolates the categories Rep(S_d) of representations of the symmetric groups S_d. In this paper we describe the blocks of Deligne’s category Rep(S_t).

💡 Research Summary

The paper investigates the block decomposition of Deligne’s interpolating tensor category Rep(Sₜ), where the parameter t may be any element of a characteristic‑zero field, not necessarily an integer. After recalling Deligne’s construction, the authors note that Rep(Sₜ) is additive but not abelian in general, and it becomes semisimple (hence abelian) when t is not a non‑negative integer. The main goal is to understand the additive category in the non‑semisimple regime by splitting it into blocks, a standard technique in representation theory.

The authors begin by constructing many non‑trivial endomorphisms of the identity functor of Rep(Sₜ). They do this by encoding morphisms between tensor powers of the natural representation of the symmetric group S_d via set partitions. For a fixed integer d, a partition π of the set {1,…,n,1′,…,m′} determines a linear map f(π): V^{⊗n}d → V^{⊗m}d. The composition of such maps is governed by a simple combinatorial rule: if µ ∈ P{m,l} and π ∈ P{n,m}, then f(µ)∘f(π)=d^{δ(µ,π)} f(µ·π), where δ(µ,π) counts the connected components of the concatenated diagram that lie entirely in the middle layer, and µ·π is the partition obtained by deleting those components. This rule shows that the structure constants are polynomials in d.

Replacing the integer d by an arbitrary scalar t, the authors define a “t‑completion” of the partition algebra: the category Rep₀(Sₜ) has objects