Brauer algebra#Walled Brauer algebra

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

:$\backslash dim\backslash mathfrak\{B\}\_n(\backslash delta)\; =\; \backslash frac\{(2n)!\}\{2^n\; n!\}\; =\; (2n-1)!!\; =\; (2n-1)(2n-3)\backslash cdots\; 5\backslash cdot\; 3\backslash cdot\; 1$

Image:Brauer algebra.jpg

A basis of $\backslash mathfrak\{B\}\_n(\backslash delta)$ consists of all pairings on a set of $2n$ elements $X\_1,\; ...,\; X\_n,\; Y\_1,\; ...,\; Y\_n$ (that is, all perfect matchings of a complete graph $K\_\{2n\}$: any two of the $2n$ elements may be matched to each other, regardless of their symbols). The elements $X\_i$ are usually written in a row, with the elements $Y\_i$ beneath them.

The product of two basis elements $A$ and $B$ is obtained by concatenation: first identifying the endpoints in the bottom row of $A$ and the top row of $B$ (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 $A\backslash cdot\; B$ of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by $\backslash delta^r$ where $r$ is the number of deleted loops. In the example $A\backslash cdot\; B\; =\; \backslash delta^\{2\}\; AB$.

$\backslash mathfrak\{B\}\_n(\backslash delta)$ can also be defined as the $\backslash mathbb\{Z\}[\backslash delta]$-algebra with generators $s\_1,\backslash ldots,s\_\{n-1\},\; e\_1,\; \backslash ldots,\; e\_\{n-1\}$ satisfying the following relations:

• Relations of the symmetric group:

:$s\_i^2\; =\; 1$

:$s\_i\; s\_j\; =\; s\_j\; s\_i$ whenever $|i-j|>1$

:$s\_i\; s\_\{i+1\}\; s\_i\; =\; s\_\{i+1\}\; s\_i\; s\_\{i+1\}$

• Almost-idempotent relation:

:$e\_i^2\; =\; \backslash delta\; e\_i$

• Commutation:

:$e\_i\; e\_j\; =\; e\_j\; e\_i$

:$s\_i\; e\_j\; =\; e\_j\; s\_i$

:whenever$|i-j|>1$

• Tangle relations

:$e\_i\; e\_\{i\backslash pm\; 1\}\; e\_i\; =\; e\_i$

:$s\_i\; s\_\{i\backslash pm\; 1\}\; e\_i\; =\; e\_\{i\backslash pm\; 1\}\; e\_i$

:$e\_i\; s\_\{i\backslash pm\; 1\}\; s\_i\; =\; e\_i\; e\_\{i\backslash pm\; 1\}$

• Untwisting:

:$s\_i\; e\_i\; =\; e\_i\; s\_i\; =\; e\_i$:

:$e\_i\; s\_\{i\backslash pm\; 1\}\; e\_i\; =\; e\_i$

In this presentation $s\_i$ represents the diagram in which $X\_k$ is always connected to $Y\_k$ directly beneath it except for $X\_i$ and $X\_\{i+1\}$ which are connected to $Y\_\{i+1\}$ and $Y\_i$ respectively. Similarly $e\_i$ represents the diagram in which $X\_k$ is always connected to $Y\_k$ directly beneath it except for $X\_i$ being connected to $X\_\{i+1\}$ and $Y\_i$ to $Y\_\{i+1\}$.

The Brauer algebra is a subalgebra of the partition algebra.

The Brauer algebra $\backslash mathfrak\{B\}\_n(\backslash delta)$ is semisimple if $\backslash delta\backslash in\backslash mathbb\{C\}-\backslash \{0,\backslash pm\; 1,\backslash pm\; 2,\backslash dots,\backslash pm\; n\backslash \}$.

The subalgebra of $\backslash mathfrak\{B\}\_n(\backslash delta)$ generated by the generators $s\_i$ is the group algebra of the symmetric group $S\_n$.

The subalgebra of $\backslash mathfrak\{B\}\_n(\backslash delta)$ generated by the generators $e\_i$ is the Temperley-Lieb algebra $TL\_n(\backslash delta)$.

The Brauer algebra is a cellular algebra.

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

Brauer-Specht modules are finite-dimensional modules of the Brauer algebra.

If $\backslash delta$ is such that $\backslash mathfrak\{B\}\_n(\backslash delta)$ is semisimple,

they form a complete set of simple modules of $\backslash mathfrak\{B\}\_n(\backslash delta)$. 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 $0\backslash leq\; \backslash ell\; \backslash leq\; n$ with $\backslash ell\backslash equiv\; n\backslash bmod\; 2$, let $B\_\{n,\backslash ell\}$ be the set of perfect matchings of $n+\backslash ell$ elements $X\_1,\backslash dots\; ,X\_n,Y\_1,\backslash dots\; ,Y\_\backslash ell$, such that $Y\_j$ is matched with one of the $n$ elements $X\_1,\backslash dots\; ,X\_n$. For any ring $k$, the space $kB\_\{n,\backslash ell\}$ is a left $\backslash mathfrak\{B\}\_n(\backslash delta)$-module, where basis elements of $\backslash mathfrak\{B\}\_n(\backslash delta)$ act by graph concatenation. (This action can produce matchings that violate the restriction that $Y\_1,\backslash dots\; ,Y\_\backslash ell$ cannot match with one another: such graphs must be modded out.) Moreover, the space $kB\_\{n,\backslash ell\}$ is a right $S\_\backslash ell$-module.

Given a Specht module $V\_\backslash lambda$ of $kS\_\backslash ell$, where $\backslash lambda$ is a partition of $\backslash ell$ (i.e. $|\backslash lambda|=\backslash ell$), the corresponding Brauer-Specht module of $\backslash mathfrak\{B\}\_n(\backslash delta)$ is

:$$

W_{\lambda} = kB_{n,|\lambda|} \otimes_{kS_

A basis of this module is the set of elements $b\backslash otimes\; v$, where $b\backslash in\; B\_\{n,|\backslash lambda|\}$ is such that the $|\backslash lambda|$ lines that end on elements $Y\_j$ do not cross, and $v$ belongs to a basis of $V\_\backslash lambda$. The dimension is

:$$

\dim(W_\lambda) = \binom{n}

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

:$v\_1\backslash otimes\; \backslash cdots\backslash otimes\; v\_\{i-1\}\backslash otimes\backslash Big(v\_i\backslash otimes\; v\_\{i+1\}\backslash Big)\backslash otimes\backslash cdots\; \backslash otimes\; v\_n\; \backslash mapsto\; v\_1\backslash otimes\; \backslash cdots\backslash otimes\; v\_\{i-1\}\backslash otimes\backslash left(\backslash langle\; v\_i,\; v\_\{i+1\}\backslash rangle\; \backslash sum\_\{k=1\}^d\; (w\_k\backslash otimes\; w\_k)\backslash right)\backslash otimes\backslash cdots\; \backslash otimes\; v\_n$

where $w\_1,\backslash ldots,w\_d$ is any orthonormal basis of $V$. (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 $d\backslash geq\; n$, the image of $B\_n(d)$ inside $\backslash operatorname\{End\}(V^\{\backslash otimes\; n\})$ is the centraliser of $O(V)$ inside $\backslash operatorname\{End\}(V^\{\backslash otimes\; n\})$, and conversely the image of $O(V)$ is the centraliser of $B\_n(d)$. The tensor power $V^\{\backslash otimes\; n\}$ is therefore both an $O(V)$- and a $B\_n(d)$-module and satisfies

:$V^\{\backslash otimes\; n\}\; =\; \backslash bigoplus\_\{\backslash lambda\}\; U\_\backslash lambda\; \backslash boxtimes\; W\_\backslash lambda$

where $\backslash lambda$ runs over a subset of the partitions such that $|\backslash lambda|\backslash leq\; n$ and $|\backslash lambda|\backslash equiv\; n\; \backslash bmod\; 2$,

$U\_\backslash lambda$ is an irreducible $O(V)$-module, and $W\_\backslash lambda$ is a Brauer-Specht module of

$B\_n(d)$.

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

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

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

The walled Brauer algebra is generated by $\backslash \{s\_i\backslash \}\_\{1\backslash leq\; i\backslash leq\; r+s-1,i\backslash neq\; r\}\; \backslash cup\backslash \{e\_r\backslash \}$. These generators obey the basic relations of $\backslash mathfrak\{B\}\_\{r+s\}(\backslash delta)$ that involve them, plus the two relations

:$$

e_rs_{r+1}s_{r-1}e_r s_{r-1} = e_rs_{r+1}s_{r-1}e_r s_{r+1} \quad , \quad s_{r-1}e_rs_{r+1}s_{r-1}e_r = s_{r+1}e_rs_{r+1}s_{r-1}e_r

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

For $\backslash delta$ a natural integer, let $V$ be the natural representation of the general linear group $GL\_\backslash delta(\backslash mathbb\{C\})$.

The walled Brauer algebra $\backslash mathfrak\{B\}\_\{r,s\}(\backslash delta)$ has a natural action on $V^\{\backslash otimes\; r\}\backslash otimes\; (V^*)^\{\backslash otimes\; s\}$, which is related by Schur-Weyl duality to the action of $GL\_\backslash delta(\backslash mathbb\{C\})$.

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

{{Reflist|refs=

{{cite journal | last1=Bulgakova | first1=D. V. | last2=Ogievetsky | first2=O. | title=Fusion procedure for the walled Brauer algebra | journal=Journal of Geometry and Physics | date=2019-11-24 | volume=149 | page=103580 | doi=10.1016/j.geomphys.2019.103580 | arxiv=1911.10537v1 | s2cid=208267893 }}

{{cite arXiv | last1=Cox | first1=Anton | last2=Visscher | first2=De | last3=Doty | first3=Stephen | last4=Martin | first4=Paul | title=On the blocks of the walled Brauer algebra | date=2007-09-06 | class=math.RT | eprint=0709.0851v1 }}

{{citation | last1=Benkart | first1=Georgia | last2=Moon | first2=Dongho | title=Representations of Algebras and Related Topics | chapter=Tensor product representations of Temperley-Lieb algebras and Chebyshev polynomials | publisher=American Mathematical Society | publication-place=Providence, Rhode Island | date=2005-04-26 | doi=10.1090/fic/045/05 | pages=57–80| isbn=9780821834152 }}

{{cite arXiv | last1=Halverson | first1=Tom | last2=Jacobson | first2=Theodore N. | title=Set-partition tableaux and representations of diagram algebras | date=2018-08-24 | class=math.RT | eprint=1808.08118v2 }}

{{cite arXiv | last=Martin | first=Paul P | title=The decomposition matrices of the Brauer algebra over the complex field | date=2009-08-11 | class=math.RT | eprint=0908.1500v1 }}

{{Citation | last1=Brauer | first1=Richard | authorlink = Richard Brauer | title=On Algebras Which are Connected with the Semisimple Continuous Groups | jstor=1968843 | publisher=Annals of Mathematics | series=Second Series | year=1937 | journal=Annals of Mathematics | issn=0003-486X | volume=38 | issue=4 | pages=857–872 | doi=10.2307/1968843}}

{{Citation | last1=Wenzl | first1=Hans | title=On the structure of Brauer's centralizer algebras | jstor=1971466 | mr=951511 | year=1988 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=128 | issue=1 | pages=173–193 | doi=10.2307/1971466}}

}}

Category:Representation theory

Category:Diagram algebras