Brauer代数的中心维数问题

Brauer代数B_n(t)是一种在表示论,数学物理中重要的带一个参数t的有限维代数.当t取普通值时它们的结构已经了解得比较清楚,例如,不可约表示分类.当t取某些特殊值时有关它们还仍有些问题未探明.本文讨论任意参数时Brauer代数的中心的维数问题.主要结论是当t取某些特殊值时,Brauer代数中心的维数必定大于或等于t取普通值时它们的维数.     

A cell filtration of mixed tensor space

We construct a cellular basis of the walled Brauer algebra which has similar properties as the Murphy basis of the group algebra of the symmetric group. In particular, the restriction of a cell module to a certain subalgebra can be easily described via this basis. Furthermore, the mixed tensor space possesses a filtration by cell modules—although not by cell modules of the walled Brauer algebra itself, but by cell modules of its image in the algebra of endomorphisms of mixed tensor space.     

Brauer代数的图子式(英文)

本文研究Brauer代数的根基问题.利用图子式的方法,获得了Gavarini的猜想对Brauer代数B1n是成立的结果.     

On the Center of the Brauer Algebra

Using the determination of conjugacy classes in an earlier paper, we study the center of the Brauer algebra. In the case of finite groups, conjugacy class sums determine the center of the group algebra. In the case of the Brauer algebra the corresponding class sums only yield a basis of the centralizer of the symmetric group in the Brauer algebra. However, we exhibit an explicit algorithm to determine conditions for a centralizer element to be central and show how to compute a basis for the center using these methods. We will outline how this can be used to compute blocks over fields of arbitrary characteristic. We will also show that similar methods can be applied for computing a basis of the center of the walled Brauer algebra.     

Automorphisms of Green Orders and Their Derived Categories

In an earlier paper, Raphaël Rouquier and the author introduced the group of self-equivalences of a derived category. In the case of a Brauer tree algebra, we determined a nontrivial homomorphism of the Artin braid group to this group of self-equivalences. The class of Brauer tree algebras include blocks of finite group rings over a large enough field with cyclic defect groups. In the present paper we give an integral version of this homomorphism. Moreover, we identify some interesting arithmetic subgroups with natural groups of self-equivalences of the derived category.     

Braid Action on Derived Category Nakayama Algebras

We construct an action of a braid group associated to a complete graph on the derived category of a certain symmetric Nakayama algebra which is also a Brauer star algebra with no exceptional vertex. We connect this action with the affine braid group action on Brauer star algebras defined by Schaps and Zakay–Illouz. We show that for Brauer star algebras with no exceptional vertex, the action is faithful.     

Cyclotomic Extensions of Diagram Algebras

A uniform approach to cyclotomic extensions of diagram algebras is given, focussing on cellular structures. Cyclotomic Temperley–Lieb algebras, cyclotomic Brauer algebras and cyclotomic walled Brauer algebras are discussed as examples.     

Quantum Yang-Baxter module algebras

LetH be a quantum group over a commutative ringR. We introduce the concept of quantum Yang-BaxterH-module algebra, generalizing the notion ofH-dimodule algebra in the case whereH is commutative, cocommutative and faithfully projective. After discussing some examples, we introduceH-Azumaya algebras. The set of quivalence classes ofH-Azumaya algebras can be made into a group, called the Brauer group of the quantum groupH. This group is a generalization of the Brauer-Long group.This author wishes to thank the Department of Mathematics, UIA, for its hospitality and financial support during the time when most of this paper was written.     

A bideterminant basis for a reductive monoid

We use the rational tableaux introduced by Stembridge to give a bideterminant basis for a normal reductive monoid and for its variety of noninvertible elements. We also obtain a bideterminant basis for the full coordinate ring of the general linear group and for all its truncations with respect to saturated sets. Finally, we deduce an alternative proof of the double centraliser theorem for the rational Schur algebra and the walled Brauer algebra over an arbitrary infinite base field which was first obtained by Dipper, Doty and Stoll.     

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.     

Characteristic maps for the Brauer algebra

The classical characteristic map associates symmetric functions to characters of the symmetric groups. There are two natural analogues of this map involving the Brauer algebra. The first of them relies on the action of the orthogonal or symplectic group on a space of tensors, while the second is provided by the action of this group on the symmetric algebra of the corresponding Lie algebra. We consider the second characteristic map both in the orthogonal and symplectic case, and calculate the images of central idempotents of the Brauer algebra in terms of the Schur polynomials. The calculation is based on the Okounkov–Olshanski binomial formula for the classical Lie groups. We also reproduce the hook dimension formulas for representations of the classical groups by deriving them from the properties of the primitive idempotents of the symmetric group and the Brauer algebra.     

Spherical objects of the multiplicity free Brauer tree algebra with two edges

We explicitly describe all spherical objects, as well as tilting objects, of the multiplicity free Brauer tree algebra with two edges.     

The Brauer-Clifford group for locally finite (S,H)-Azumaya algebras

In this article, we define the notion of Brauer-Clifford group for H-locally finite (S, H)-Azumaya algebras, when H is a cocommutative Hopf algebra and S is an H-locally finite commutative H-module algebra over a commutative noetherian ring k. This is the situation that arises in applications with connections to algebraic geometry. This Brauer-Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.     

Derivees logarithmiques pour une s-derivation algebrique

We use the results on the minimal basis of the centre of an Iwahori–Hecke algebra from our earlier work, as well as some additional results on the minimal basis, to describe the image and kernel of the Brauer homomorphism for Iwahori–Hecke algebras defined by L. Jones (Jones, L. Centres of Generic Hecke Algebras; Ph.D. Thesis; University of Virginia, 1987.).

     

Brauer and Jones tied monoids

We introduce a ramified monoid, attached to each Brauer–type monoid, that is, to the symmetric group, to the Jones and Brauer monoids among others. Ramified monoids correspond to a class of tied monoids arising from knot theory and are interesting in themselves. The ramified monoid attached to the symmetric group is the Coxeter-like version of the so–called tied braid monoid. We give a presentation of the ramified monoid attached to the Brauer monoid. Also, we introduce and study two tied-like monoids that cannot be described as ramified monoids. However, these monoids can also be regarded as tied versions of the Jones and Brauer monoids.     

Yetter—Drinfel'd H-Azumaya Monoids in Closed Categories

When C is a symmetric closed category with equalizers and coequalizers and H is a Hopf algebra in C, the category of Yetter—Drinfeld H-modules is a braided monoidal category.We develop a categorical version of the results in (10) constructing a Brauer group BQ(C,H) and studying its functorial properties.     

On presentations of Brauer-type monoids

We obtain presentations for the Brauer monoid, the partial analogue of the Brauer monoid, and for the greatest factorizable inverse submonoid of the dual symmetric inverse monoid. In all three cases we apply the same approach, based on the realization of all these monoids as Brauer-type monoids.