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

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

Diagrammatic definition edit

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 ${\displaystyle A\cdot B=\delta ^{2}AB}$ .

Generators and relations edit

${\displaystyle {\mathfrak {B}}_{n}(\delta )}$  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 ${\displaystyle Y_{i+1}}$ .

Basic properties edit

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 edit

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.

Schur-Weyl duality edit

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.