The Brauer Category and Invariant Theory

TL;DR

This paper introduces a category of Brauer diagrams, establishes relations and functors to classical group representations, and generalizes fundamental theorems of invariant theory, leading to new algebraic presentations.

Contribution

It provides a new categorical framework for Brauer diagrams, extends invariant theory theorems, and introduces additional relations to the Brauer algebra for better algebraic understanding.

Findings

Established a presentation with seven relations for the Brauer category.

Constructed tensor functors to orthogonal and symplectic group representations.

Derived new presentations for endomorphism algebras of tensor powers.

Abstract

A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, and a presentation of the category is given; specifically, we prove that seven relations among its four generating homomorphisms suffice to deduce all equations among the morphisms. Full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group $O(V)$ or the symplectic group $Sp(V)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain new presentations for the endomorphism algebras of the module $V^{\otimes r}$. These are obtained by appending to the standard presentation of the Brauer algebra of degree $r$ one additional relation. This relation stipulates the vanishing of an element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if $\dim V = 2n$, the element is precisely the central idempotent in the Brauer subalgebra of degree $n + 1$, which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, our generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras.