Motivation and reference for Brauer algebras

Question

I am looking for a good reference and motivation for Brauer monoid and Brauer algebras. Kindly help me with some suggestions. Thanks.

Answer 1

For motivation I would advise starting with Brauer's original paper. You'll need a JSTOR login though: https://www.jstor.org/stable/1968843?origin=crossref&seq=1#metadata_info_tab_contents

Answer 2

I think the motivation comes from classical invariant theory or the grand program launched around 1845 by Cayley in "On the theory of linear transformations" and "On linear transformations". In the latter article, Cayley stated the main goal as follows:

In fact the question may be proposed, " To find all the derivatives of any number of functions, which have the property of preserving their form unaltered after any linear transformations of the variables."

A particular case of this problem can be stated in more modern terms as describing the $G$-invariant elements of $$ V^{\otimes n}\otimes {V^{\vee}}^{\otimes n}\simeq {\rm Hom}(V^{\otimes n},V^{\otimes n})\ . $$ Here $G$ is a group acting linearly on the vector space $V$. This action automatically and naturally defines a resulting action on the dual $V^{\vee}$ and on all tensor products made of $V$'s and $V^{\vee}$'s. The case of equal number $n$ of each factors, i.e., ${\rm Hom}(V^{\otimes n},V^{\otimes n})$ comes with an algebra structure using composition of maps as multiplication. The keyword here is "describing" which can be meant to say: find a linearly generating set given explicitly in terms of combinatorial objects like Brauer's pictures. Finding a basis would be better, but this is hard, unless the dimension of $V$ is large enough with respect to $n$ which results in killing linear relations. Finding such a description is often, following Weyl, called the First Fundamental Theorem of Classical Invariant Theory (for the particular setting at hand), while finding the linear relations is called the Second Fundamental Theorem of Classical Invariant Theory. Brauer's algebra is what appears when $G$ is the orthogonal group of $V$ equipped with some nondegenerate symmetric bilinear form. In his original article, what Brauer did is find an elegant graphical formulation of the FFT for $O(V)$. I don't know who first proved the FFT for $O(V)$, see my ramblings in the MO posts:

Invariants for the exceptional complex simple Lie algebra $F_4$

and

The $GL(N)$ chicken versus the $SL(N)$ egg, the Erlangen Program and relations between FFTs

Note that the SFT for $O(V)$ has only been worked out recently by Lehrer and Zhang (see this article), instead of in the 19th century.