Brauer algebra

The product of 2 basis elements A and B of the Brauer algebra with n = 12

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 ${\displaystyle K_{n}}$: 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.