Brauer Algebra Research Articles

A modified Brauer algebra as centralizer algebra of the unitary group

The centralizer algebra of the action of U ( n ) U(n) on the real tensor powers ⊗ R r V \otimes _\mathbb {R}^r V of its natural module, V = C n V=\mathbb {C}^n , is described by means of a modification in the multiplication of the signed Brauer algebras. The relationships of this algebra with the invariants for U ( n ) U(n) and with the decomposition of ⊗ R r V \otimes _\mathbb {R}^r V into irreducible submodules is considered.

On the semi-simplicity of the cyclotomic Brauer algebras

In this paper, a criterion on the semi-simplicity of the cyclotomic Brauer algebras is given. This result can be considered as a generalization of Wenzl's theorem on Brauer algebras [Ann. of Math. 128 (1988) 173–193], which was known as Hanlon–Wales conjecture before.

A New Quantum Analog of the Brauer Algebra

We introduce a new algebra B_l(z,q) depending on two nonzero complex parameters such that B_l(q^n,q) at q=1 coincides with the Brauer algebra B_l(n). We establish an analog of the Brauer-Schur-Weyl duality where the action of the new algebra commutes with the representation of the twisted deformation U'_q(o_n) of the enveloping algebra U(o_n) in the tensor power of the vector representation.

Self-adjunctions and matrices

It is shown that the multiplicative monoids of Temperley–Lieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction is found in a category whose arrows are matrices, and the functor adjoint to itself is based on the Kronecker product of matrices. This self-adjunction underlies the orthogonal group case of Brauer's representation of the Brauer centralizer algebras.

Representations of G-Brauer Algebras

A new class of G-Brauer algebras which generalizes signed Brauer algebras has been introduced. The structure and representations of such algebras have been studied.

On tensor categories of Lie Type EN, N≠9

Let V be the minuscule representation of the Lie algebra g(E N) , N=6,7. We show that the centralizer of the action of the quantum group U q g(E N) on V ⊗ n is generated by the R-matrices and one additional element, appearing in the ( N−1)st tensor power; a similar description can be given for the classical limit. An analogous statement is true for a certain direct summand of V ⊗ n for U q g(E N) with N>5, N≠9 ; here V is the module whose highest weight is the fundamental weight labeled by the vertex furthest from the triple point of the graph E N . Moreover, we obtain a 2-parameter family of braid representations generalizing a q-version of the Brauer centralizer algebras.

The geometry of self-adjunction

This paper is a companion to another paper where it is shown that the multiplicative monoids of Temperley-Lieb algebras are isomorphic to monoids of endomorphisms in categories where an endofunctor is adjoint to itself. Such a self-adjunction underlies the orthogonal group case of Brauer's representation of the Brauer centralizer algebras. The present paper provides detailed proofs of results on the presentation of various monoids of diagrams by generators and relations, on which the other paper depends.

Representations for Lie superalgebras of type C

Let G =osp(2,2n) be the classical Lie superalgebra of type C of rank n +1. Let λ be a partition with λ 1 ⩽ n . Then λ labels a finite-dimensional irreducible G -module, V ( λ ). We describe the character of V ( λ ) in terms of tableaux. This tableaux description of characters enable us to decompose T =⊗ f V , the f -fold tensor product of the natural representation of G , into its irreducible submodules and to show that the centralizer algebra of G on T is isomorphic to the Brauer algebra B f (2−2 n ) for n > f .

GENERALIZED BRAUER ALGEBRAS

ABSTRACT Using some ideas of Brauer, we introduce what we call generalized Brauer algebras and, as a special case, Brauer orders. We show that many well-known classes of so-called crossed product algebras, and in particular, the well-known crossed product orders, can be obtained as special instances of our construction. We prove several results showing when Brauer orders are Azumaya, maximal, hereditary or Gorenstein.

A characteristic free approach to Brauer algebras

Brauer algebras arise in representation theory of orthogonal or symplectic groups. These algebras are shown to be iterated inflations of group algebras of symmetric groups. In particular, they are cellular (as had been shown before by Graham and Lehrer). This gives some information about block decomposition of Brauer algebras.

Cellular Algebras: Inflations and Morita Equivalences

Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra. The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper. The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

A Brauer Algebra Theoretic Proof of Littlewood's Restriction Rules

LetUbe a complex vector space endowed with an orthogonal or symplectic form, and letGbe the subgroup ofGL(U) of all the symmetrics of this form (resp.O(U) orSp(U)); ifMis an irreducibleGL(U)-module, the Littlewood's restriction rule describes theG-moduleM|GL(U)G. In this paper we give a new representation-theoretic proof of this formula: realizingMin a tensor powerU⊗fand using Schur's duality, we reduce to the problem of describing the restriction to an irreducibleSf-module of an irreducible module for the centralizer algebra of the action ofGonU⊗f; the latter is a quotient of the Brauer algebra, and we know the kernel of the natural epimorphism, whence we deduce the Littlewood's restriction rule.

Partition Algebras are Cellular

The partition algebra P(q) is a generalization both of the Brauer algebra and the Temperley–Lieb algebra for q-state n-site Potts models, underpining their transfer matrix formulation on the arbitrary transverse lattices. We prove that for arbitrary field k and any element q∈ k the partition algebra P(q) is always cellular in the sense of Graham and Lehrer. Thus the representation theory of P(q) can be determined by applying the developed general representation theory on cellular algebras and symmetric groups. Our result also provides an explicit structure of P(q) for arbitrary field and implies the well-known fact that the Brauer algebra D(q) and the Temperley–Lieb algebra TL(q) are cellular.

Signed brauer's algebras as centralizer algebras

Signed brauer's algebras as centralizer algebras

The induced representations of Brauer algebras and the Clebsch-Gordan coefficients of

Induced representations of from with are discussed. The induction coefficients (IDCs) or the outer-product reduction coefficients of with up to a normalization factor are derived by using the linear equation method. Weyl tableaux for the corresponding Gel'fand basis of SO(n) are defined. The assimilation method for obtaining Clebsch-Gordan coefficients of SO(n) in the Gel'fand basis for no modification rule involved couplings from IDCs of Brauer algebras is proposed. Some isoscalar factors of for the resulting irrep with are tabulated.

Tensor product representations for orthosymplectic Lie superalgebras

We derive a general result about commuting actions on certain objects in braided rigid monoidal categories. This enables us to define an action of the Brauer algebra on the tensor space V ⊗ k which commutes with the action of the orthosymplectic Lie superalgebra spo ( V) and the orthosymplectic Lie color algebra spo ( V, β). We use the Brauer algebra action to compute maximal vectors in V ⊗ k and to decompose V ⊗ k into a direct sum of submodules T λ . We compute the characters of the modules T λ , give a combinatorial description of these characters in terms of tableaux, and model the decomposition of V ⊗ k into the submodules T λ with a Robinson-Schensted-Knuth-type insertion scheme.

Signed brauer's algebras

Signed brauer's algebras

Polynomial representations, algebraic monoids, and Schur algebras of classical type

First we study Zariski-closed subgroups of general linear groups over an infinite field with the property that their polynomial representation theory is graded in a natural way. There are “Schur algebras” associated with such a group and their representations completely determine the polynomial representations of the original subgroup. Moreover, the polynomial representations of the subgroup are equivalent with the rational representations of a certain algebraic monoid associated with the subgroup, and the aforementioned Schur algebras are the linear duals of the graded components of the coordinate bialgebra on the monoid. Then we study the monoids and Schur algebras associated with the classical groups. We obtain some structural information on the monoids. Then we characterize the associated Schur algebras as centralizer algebras for the Brauer algebras, in characteristic zero.

A new Young diagrammatic method for Kronecker products of and

A new simple Young diagrammatic method for Kronecker products of O(n) and Sp(2m) is proposed based on representation theory of Brauer algebras. A general procedure for the decomposition of tensor products of representations for O(n) and Sp(2m) is outlined, which is similar to that for U(n) known as the Littlewood rules together with trace contractions from a Brauer algebra and some modification rules given by King.

Representation of $q$-analogue of rational Brauer algebras

Representation of $q$-analogue of rational Brauer algebras