Brauer algebra

In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer^[1] in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality.

The Brauer algebra ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ is a ${\displaystyle \mathbb {Z} [\delta ]}$-algebra depending on the choice of a positive integer ${\displaystyle n}$. Here ${\displaystyle \delta }$ is an indeterminate, but in practice ${\displaystyle \delta }$ is often specialised to the dimension of the fundamental representation of an orthogonal group ${\displaystyle O(\delta )}$. The Brauer algebra has the dimension

${\displaystyle \dim {\mathfrak {B}}_{n}(\delta )={\frac {(2n)!}{2^{n}n!}}=(2n-1)!!=(2n-1)(2n-3)\cdots 5\cdot 3\cdot 1}$

A basis of ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ consists of all pairings on a set of ${\displaystyle 2n}$ elements ${\displaystyle X_{1},...,X_{n},Y_{1},...,Y_{n}}$ (that is, all perfect matchings of a complete graph ${\displaystyle K_{2n}}$: any two of the ${\displaystyle 2n}$ elements may be matched to each other, regardless of their symbols). The elements ${\displaystyle X_{i}}$ are usually written in a row, with the elements ${\displaystyle Y_{i}}$ beneath them.

The product of two basis elements ${\displaystyle A}$ and ${\displaystyle B}$ is obtained by concatenation: first identifying the endpoints in the bottom row of ${\displaystyle A}$ and the top row of ${\displaystyle B}$ (Figure AB in the diagram), then deleting the endpoints in the middle row and joining endpoints in the remaining two rows if they are joined, directly or by a path, in AB (Figure AB=nn in the diagram). Thereby all closed loops in the middle of AB are removed. The product ${\displaystyle A\cdot B}$ of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by ${\displaystyle \delta ^{r}}$ where ${\displaystyle r}$ is the number of deleted loops. In the example ${\displaystyle A\cdot B=\delta ^{2}AB}$.

${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ can also be defined as the ${\displaystyle \mathbb {Z} [\delta ]}$-algebra with generators ${\displaystyle s_{1},\ldots ,s_{n-1},e_{1},\ldots ,e_{n-1}}$ satisfying the following relations:

• Relations of the symmetric group:

${\displaystyle s_{i}^{2}=1}$

${\displaystyle s_{i}s_{j}=s_{j}s_{i}}$ whenever ${\displaystyle |i-j|>1}$

${\displaystyle s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}}$

• Almost-idempotent relation:

${\displaystyle e_{i}^{2}=\delta e_{i}}$

• Commutation:

${\displaystyle e_{i}e_{j}=e_{j}e_{i}}$

${\displaystyle s_{i}e_{j}=e_{j}s_{i}}$

whenever${\displaystyle |i-j|>1}$

• Tangle relations

${\displaystyle e_{i}e_{i\pm 1}e_{i}=e_{i}}$

${\displaystyle s_{i}s_{i\pm 1}e_{i}=e_{i\pm 1}e_{i}}$

${\displaystyle e_{i}s_{i\pm 1}s_{i}=e_{i}e_{i\pm 1}}$

• Untwisting:

${\displaystyle s_{i}e_{i}=e_{i}s_{i}=e_{i}}$:

${\displaystyle e_{i}s_{i\pm 1}e_{i}=e_{i}}$

In this presentation ${\displaystyle s_{i}}$ represents the diagram in which ${\displaystyle X_{k}}$ is always connected to ${\displaystyle Y_{k}}$ directly beneath it except for ${\displaystyle X_{i}}$ and ${\displaystyle X_{i+1}}$ which are connected to ${\displaystyle Y_{i+1}}$ and ${\displaystyle Y_{i}}$ respectively. Similarly ${\displaystyle e_{i}}$ represents the diagram in which ${\displaystyle X_{k}}$ is always connected to ${\displaystyle Y_{k}}$ directly beneath it except for ${\displaystyle X_{i}}$ being connected to ${\displaystyle X_{i+1}}$ and ${\displaystyle Y_{i}}$ to ${\displaystyle Y_{i+1}}$.

The Brauer algebra is a subalgebra of the partition algebra.

The Brauer algebra ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ is semisimple if ${\displaystyle \delta \in \mathbb {C} -\{0,\pm 1,\pm 2,\dots ,\pm n\}}$.^[2][3]

The subalgebra of ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ generated by the generators ${\displaystyle s_{i}}$ is the group algebra of the symmetric group ${\displaystyle S_{n}}$.

The subalgebra of ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ generated by the generators ${\displaystyle e_{i}}$ is the Temperley-Lieb algebra ${\displaystyle TL_{n}(\delta )}$.^[4]

The Brauer algebra is a cellular algebra.

For a pairing ${\displaystyle A}$ let ${\displaystyle n(A)}$ be the number of closed loops formed by identifying ${\displaystyle X_{i}}$ with ${\displaystyle Y_{i}}$ for any ${\displaystyle i=1,2,\dots ,n}$: then the Jones trace ${\displaystyle {\text{Tr}}(A)=\delta ^{n(A)}}$ obeys ${\displaystyle {\text{Tr}}(AB)={\text{Tr}}(BA)}$ i.e. it is indeed a trace.

Brauer-Specht modules are finite-dimensional modules of the Brauer algebra. If ${\displaystyle \delta }$ is such that ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ is semisimple, they form a complete set of simple modules of ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$.^[4] These modules are parametrized by partitions, because they are built from the Specht modules of the symmetric group, which are themselves parametrized by partitions.

For ${\displaystyle 0\leq \ell \leq n}$ with ${\displaystyle \ell \equiv n{\bmod {2}}}$, let ${\displaystyle B_{n,\ell }}$ be the set of perfect matchings of ${\displaystyle n+\ell }$ elements ${\displaystyle X_{1},\dots ,X_{n},Y_{1},\dots ,Y_{\ell }}$, such that ${\displaystyle Y_{j}}$ is matched with one of the ${\displaystyle n}$ elements ${\displaystyle X_{1},\dots ,X_{n}}$. For any ring ${\displaystyle k}$, the space ${\displaystyle kB_{n,\ell }}$ is a left ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$-module, where basis elements of ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ act by graph concatenation. (This action can produce matchings that violate the restriction that ${\displaystyle Y_{1},\dots ,Y_{\ell }}$ cannot match with one another: such graphs must be modded out.) Moreover, the space ${\displaystyle kB_{n,\ell }}$ is a right ${\displaystyle S_{\ell }}$-module.^[5]

Given a Specht module ${\displaystyle V_{\lambda }}$ of ${\displaystyle kS_{\ell }}$, where ${\displaystyle \lambda }$ is a partition of ${\displaystyle \ell }$ (i.e. ${\displaystyle |\lambda |=\ell }$), the corresponding Brauer-Specht module of ${\displaystyle {\mathfrak {B}}_{n}(\delta )}$ is

${\displaystyle W_{\lambda }=kB_{n,|\lambda |}\otimes _{kS_{|\lambda |}}V_{\lambda }\qquad {\big (}|\lambda |\leq n,|\lambda |\equiv n{\bmod {2}}{\big )}}$

A basis of this module is the set of elements ${\displaystyle b\otimes v}$, where ${\displaystyle b\in B_{n,|\lambda |}}$ is such that the ${\displaystyle |\lambda |}$ lines that end on elements ${\displaystyle Y_{j}}$ do not cross, and ${\displaystyle v}$ belongs to a basis of ${\displaystyle V_{\lambda }}$.^[5] The dimension is

${\displaystyle \dim(W_{\lambda })={\binom {n}{|\lambda |}}(n-|\lambda |-1)!!\dim(V_{\lambda })}$

i.e. the product of a binomial coefficient, a double factorial, and the dimension of the corresponding Specht module, which is given by the hook length formula.

Let ${\displaystyle V=\mathbb {R} ^{d}}$ be a Euclidean vector space of dimension ${\displaystyle d}$, and ${\displaystyle O(V)=O(d,\mathbb {R} )}$ the corresponding orthogonal group. Then write ${\displaystyle B_{n}(d)}$ for the specialisation ${\displaystyle \mathbb {R} \otimes _{\mathbb {Z} [\delta ]}{\mathfrak {B}}_{n}(\delta )}$ where ${\displaystyle \delta }$ acts on ${\displaystyle \mathbb {R} }$ by multiplication with ${\displaystyle d}$. The tensor power ${\displaystyle V^{\otimes n}:=\underbrace {V\otimes \cdots \otimes V} _{n{\text{ times}}}}$ is naturally a ${\displaystyle B_{n}(d)}$-module: ${\displaystyle s_{i}}$ acts by switching the ${\displaystyle i}$th and ${\displaystyle (i+1)}$th tensor factor and ${\displaystyle e_{i}}$ acts by contraction followed by expansion in the ${\displaystyle i}$th and ${\displaystyle (i+1)}$th tensor factor, i.e. ${\displaystyle e_{i}}$ acts as

${\displaystyle v_{1}\otimes \cdots \otimes v_{i-1}\otimes {\Big (}v_{i}\otimes v_{i+1}{\Big )}\otimes \cdots \otimes v_{n}\mapsto v_{1}\otimes \cdots \otimes v_{i-1}\otimes \left(\langle v_{i},v_{i+1}\rangle \sum _{k=1}^{d}(w_{k}\otimes w_{k})\right)\otimes \cdots \otimes v_{n}}$

where ${\displaystyle w_{1},\ldots ,w_{d}}$ is any orthonormal basis of ${\displaystyle V}$. (The sum is in fact independent of the choice of this basis.)

This action is useful in a generalisation of the Schur-Weyl duality: if ${\displaystyle d\geq n}$, the image of ${\displaystyle B_{n}(d)}$ inside ${\displaystyle \operatorname {End} (V^{\otimes n})}$ is the centraliser of ${\displaystyle O(V)}$ inside ${\displaystyle \operatorname {End} (V^{\otimes n})}$, and conversely the image of ${\displaystyle O(V)}$ is the centraliser of ${\displaystyle B_{n}(d)}$.^[2] The tensor power ${\displaystyle V^{\otimes n}}$ is therefore both an ${\displaystyle O(V)}$- and a ${\displaystyle B_{n}(d)}$-module and satisfies

${\displaystyle V^{\otimes n}=\bigoplus _{\lambda }U_{\lambda }\boxtimes W_{\lambda }}$

where ${\displaystyle \lambda }$ runs over a subset of the partitions such that ${\displaystyle |\lambda |\leq n}$ and ${\displaystyle |\lambda |\equiv n{\bmod {2}}}$, ${\displaystyle U_{\lambda }}$ is an irreducible ${\displaystyle O(V)}$-module, and ${\displaystyle W_{\lambda }}$ is a Brauer-Specht module of ${\displaystyle B_{n}(d)}$.

It follows that the Brauer algebra has a natural action on the space of polynomials on ${\displaystyle V^{n}}$, which commutes with the action of the orthogonal group.

If ${\displaystyle \delta }$ is a negative even integer, the Brauer algebra is related by Schur-Weyl duality to the symplectic group ${\displaystyle {\text{Sp}}_{-\delta }(\mathbb {C} )}$, rather than the orthogonal group.

The walled Brauer algebra ${\displaystyle {\mathfrak {B}}_{r,s}(\delta )}$ is a subalgebra of ${\displaystyle {\mathfrak {B}}_{r+s}(\delta )}$. Diagrammatically, it consists of diagrams where the only allowed pairings are of the types ${\displaystyle X_{i\leq r}-X_{j>r}}$, ${\displaystyle Y_{i\leq r}-Y_{j>r}}$, ${\displaystyle X_{i\leq r}-Y_{j\leq r}}$, ${\displaystyle X_{i>r}-Y_{j>r}}$. This amounts to having a wall that separates ${\displaystyle X_{i\leq r},Y_{i\leq r}}$ from ${\displaystyle X_{i>r},Y_{i>r}}$, and requiring that ${\displaystyle X-Y}$ pairings cross the wall while ${\displaystyle X-X,Y-Y}$ pairings don't.^[6]

The walled Brauer algebra is generated by ${\displaystyle \{s_{i}\}_{1\leq i\leq r+s-1,i\neq r}\cup \{e_{r}\}}$. These generators obey the basic relations of ${\displaystyle {\mathfrak {B}}_{r+s}(\delta )}$ that involve them, plus the two relations^[7]

${\displaystyle e_{r}s_{r+1}s_{r-1}e_{r}s_{r-1}=e_{r}s_{r+1}s_{r-1}e_{r}s_{r+1}\quad ,\quad s_{r-1}e_{r}s_{r+1}s_{r-1}e_{r}=s_{r+1}e_{r}s_{r+1}s_{r-1}e_{r}}$

(In ${\displaystyle {\mathfrak {B}}_{r+s}(\delta )}$, these two relations follow from the basic relations.)

For ${\displaystyle \delta }$ a natural integer, let ${\displaystyle V}$ be the natural representation of the general linear group ${\displaystyle GL_{\delta }(\mathbb {C} )}$. The walled Brauer algebra ${\displaystyle {\mathfrak {B}}_{r,s}(\delta )}$ has a natural action on ${\displaystyle V^{\otimes r}\otimes (V^{*})^{\otimes s}}$, which is related by Schur-Weyl duality to the action of ${\displaystyle GL_{\delta }(\mathbb {C} )}$.^[6]

• Birman–Wenzl algebra, a deformation of the Brauer algebra.