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.

Structure

The Brauer algebra is a -algebra depending on the choice of a positive integer . Here is an indeterminate, but in practice is often specialised to the dimension of the fundamental representation of an orthogonal group . The Brauer algebra has the dimension

Diagrammatic definition

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

A basis of consists of all pairings on a set of elements (that is, all perfect matchings of a complete graph : any two of the elements may be matched to each other, regardless of their symbols). The elements are usually written in a row, with the elements beneath them.

The product of two basis elements and is obtained by concatenation: first identifying the endpoints in the bottom row of and the top row of (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 of the basis elements is then defined to be the basis element corresponding to the new pairing multiplied by where is the number of deleted loops. In the example .

Generators and relations

can also be defined as the -algebra with generators satisfying the following relations:

whenever
  • Commutation:
whenever
  • Tangle relations
  • Untwisting:
:

In this presentation represents the diagram in which is always connected to directly beneath it except for and which are connected to and respectively. Similarly represents the diagram in which is always connected to directly beneath it except for being connected to and to .

Basic properties

The Brauer algebra is a subalgebra of the partition algebra.

The Brauer algebra is semisimple if .[2][3]

The subalgebra of generated by the generators is the group algebra of the symmetric group .

The subalgebra of generated by the generators is the Temperley-Lieb algebra .[4]

The Brauer algebra is a cellular algebra.

For a pairing let be the number of closed loops formed by identifying with for any : then the Jones trace obeys i.e. it is indeed a trace.

Representations

Brauer-Specht modules

Brauer-Specht modules are finite-dimensional modules of the Brauer algebra. If is such that is semisimple, they form a complete set of simple modules of .[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 with , let be the set of perfect matchings of elements , such that is matched with one of the elements . For any ring , the space is a left -module, where basis elements of act by graph concatenation. (This action can produce matchings that violate the restriction that cannot match with one another: such graphs must be modded out.) Moreover, the space is a right -module.[5]

Given a Specht module of , where is a partition of (i.e. ), the corresponding Brauer-Specht module of is

A basis of this module is the set of elements , where is such that the lines that end on elements do not cross, and belongs to a basis of .[5] The dimension is

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 be a euclidean vector space of dimension , and the corresponding orthogonal group. Then write for the specialisation where acts on by multiplication with . The tensor power is naturally a -module: acts by switching the th and th tensor factor and acts by contraction followed by expansion in the th and th tensor factor, i.e. acts as

where is any orthonormal basis of . (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 , the image of inside is the centraliser of inside , and conversely the image of is the centraliser of .[2] The tensor power is therefore both an - and a -module and satisfies

where runs over a subset of the partitions such that and , is an irreducible -module, and is a Brauer-Specht module of .

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

If is a negative even integer, the Brauer algebra is related by Schur-Weyl duality to the symplectic group , rather than the orthogonal group.

Walled Brauer algebra

The walled Brauer algebra is a subalgebra of . Diagrammatically, it consists of diagrams where the only allowed pairings are of the types , , , . This amounts to having a wall that separates from , and requiring that pairings cross the wall while pairings don't.[6]

The walled Brauer algebra is generated by . These generators obey the basic relations of that involve them, plus the two relations[7]

(In , these two relations follow from the basic relations.)

For a natural integer, let be the natural representation of the general linear group . The walled Brauer algebra has a natural action on , which is related by Schur-Weyl duality to the action of .[6]

See also

References

  1. Brauer, Richard (1937), "On Algebras Which are Connected with the Semisimple Continuous Groups", Annals of Mathematics, Second Series, Annals of Mathematics, 38 (4): 857–872, doi:10.2307/1968843, ISSN 0003-486X, JSTOR 1968843
  2. 1 2 Benkart, Georgia; Moon, Dongho (2005-04-26), "Tensor product representations of Temperley-Lieb algebras and Chebyshev polynomials", Representations of Algebras and Related Topics, Providence, Rhode Island: American Mathematical Society, pp. 57–80, doi:10.1090/fic/045/05, ISBN 9780821834152
  3. Wenzl, Hans (1988), "On the structure of Brauer's centralizer algebras", Annals of Mathematics, Second Series, 128 (1): 173–193, doi:10.2307/1971466, ISSN 0003-486X, JSTOR 1971466, MR 0951511
  4. 1 2 Halverson, Tom; Jacobson, Theodore N. (2018-08-24). "Set-partition tableaux and representations of diagram algebras". arXiv:1808.08118v2 [math.RT].
  5. 1 2 Martin, Paul P (2009-08-11). "The decomposition matrices of the Brauer algebra over the complex field". arXiv:0908.1500v1 [math.RT].
  6. 1 2 Cox, Anton; Visscher, De; Doty, Stephen; Martin, Paul (2007-09-06). "On the blocks of the walled Brauer algebra". arXiv:0709.0851v1 [math.RT].
  7. Bulgakova, D. V.; Ogievetsky, O. (2019-11-24). "Fusion procedure for the walled Brauer algebra". Journal of Geometry and Physics. 149: 103580. arXiv:1911.10537v1. doi:10.1016/j.geomphys.2019.103580. S2CID 208267893.
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