Brauer algebra

Structure

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

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

Diagrammatic definition

A basis of \mathfrak{B}n(\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\cdot B of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by \delta^r where r is the number of deleted loops. In the example A\cdot B = \delta^{2} AB.

Generators and relations

\mathfrak{B}n(\delta) can also be defined as the \mathbb{Z}[\delta]-algebra with generators s_1,\ldots,s{n-1}, e_1, \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 = \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\pm 1} e_i = e_i :s_i s_{i\pm 1} e_i = e_{i\pm 1} e_i :e_i s_{i\pm 1} s_i = e_i e_{i\pm 1}
• Untwisting: :s_i e_i = e_i s_i = e_i: :e_i s_{i\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}.

Basic properties

The Brauer algebra is a subalgebra of the partition algebra.

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

The subalgebra of \mathfrak{B}_n(\delta) generated by the generators s_i is the group algebra of the symmetric group S_n.

The subalgebra of \mathfrak{B}_n(\delta) generated by the generators e_i is the Temperley-Lieb algebra TL_n(\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,\dots,n: then the Jones trace \text{Tr}(A) = \delta^{n(A)} obeys \text{Tr}(AB)=\text{Tr}(BA) i.e. it is indeed a trace.

Representations

Brauer-Specht modules

Brauer-Specht modules are finite-dimensional modules of the Brauer algebra. If \delta is such that \mathfrak{B}_n(\delta) is semisimple, they form a complete set of simple modules of \mathfrak{B}_n(\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\leq \ell \leq n with \ell\equiv n\bmod 2, let B_{n,\ell} be the set of perfect matchings of n+\ell elements X_1,\dots ,X_n,Y_1,\dots ,Y_\ell, such that Y_j is matched with one of the n elements X_1,\dots ,X_n. For any ring k, the space kB_{n,\ell} is a left \mathfrak{B}n(\delta)-module, where basis elements of \mathfrak{B}n(\delta) act by graph concatenation. (This action can produce matchings that violate the restriction that Y_1,\dots ,Y\ell cannot match with one another: such graphs must be modded out.) Moreover, the space kB{n,\ell} is a right S_\ell-module.

Given a Specht module V_\lambda of kS_\ell, where \lambda is a partition of \ell (i.e. |\lambda|=\ell), the corresponding Brauer-Specht module of \mathfrak{B}n(\delta) is : 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 b\otimes v, where b\in B_{n,|\lambda|} is such that the |\lambda| lines that end on elements Y_j do not cross, and v belongs to a basis of V_\lambda. The dimension is : \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

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

: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 w_1,\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\geq n, the image of B_n(d) inside \operatorname{End}(V^{\otimes n}) is the centraliser of O(V) inside \operatorname{End}(V^{\otimes n}), and conversely the image of O(V) is the centraliser of B_n(d). The tensor power V^{\otimes n} is therefore both an O(V)- and a B_n(d)-module and satisfies :V^{\otimes n} = \bigoplus_{\lambda} U_\lambda \boxtimes W_\lambda where \lambda runs over a subset of the partitions such that |\lambda|\leq n and |\lambda|\equiv n \bmod 2, U_\lambda is an irreducible O(V)-module, and W_\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 \delta is a negative even integer, the Brauer algebra is related by Schur-Weyl duality to the symplectic group \text{Sp}_{-\delta}(\mathbb{C}), rather than the orthogonal group.

Walled Brauer algebra

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

The walled Brauer algebra is generated by {s_i}{1\leq i\leq r+s-1,i\neq r} \cup{e_r}. These generators obey the basic relations of \mathfrak{B}{r+s}(\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 \mathfrak{B}_{r+s}(\delta), these two relations follow from the basic relations.)

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

References

References

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