A special class of prime ideals for infinite symmetric group algebras

TL;DR

This paper explores a special class of prime ideals in the infinite symmetric group algebra, revealing their semiring structure and connections to representation theory in various characteristics.

Contribution

It introduces a new class of prime ideals, analyzes their algebraic structure, and links them to spherical representations and tensor categories.

Findings

The set of these prime ideals forms a semiring.

In complex numbers, they relate to spherical representations.

In positive characteristic, they connect with symmetric tensor categories.

Abstract

We identify an interesting special class of prime ideals in the finitary infinite symmetric group algebra. We show that the set of such ideals carries a semiring structure. Over the complex numbers, we establish a connection with spherical representations of (the Gelfand pair corresponding to) the infinite symmetric group. In positive characteristic, we investigate a close connection with the structure theory of symmetric tensor categories.