Blocks of the Brauer category over the complex field

TL;DR

This paper investigates the structure of blocks in the Brauer category over the complex numbers, revealing their correspondence with semi-infinite wedge spaces and detailing their composition in the non-semisimple case.

Contribution

It establishes a novel connection between blocks of the Brauer category and semi-infinite wedge spaces, providing a classification of blocks in the non-semisimple case.

Findings

Each weight space corresponds to a block or union of two blocks.

Every block contains infinitely many irreducible representations.

All blocks can be characterized via semi-infinite wedge space correspondence.

Abstract

Let $\mathcal B(\delta)$ be the Brauer category over the complex field $\mathbb C$ with the parameter $\delta$. In non-semisimple case, $\delta$ is an integer, and each weight space of $(\frac{\delta}2-1)$th semi-infinite wedge space corresponds to either a single block or a union of two different blocks of $\mathcal B(\delta)$-lfdmod, the category of the locally finite-dimensional representations of $\mathcal B(\delta)$. Furthermore, each block contains an infinite number of irreducible representations of $\mathcal B(\delta)$, and all blocks of $\mathcal B(\delta)$-lfdmod can be obtained in this way