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Diagrammatical description of Br,s p q
Aside from the definition of Br,s p q as a factor-algebra of Br,s, there is also a convenient graphical presentation for a basis of Br,s p q in terms of the so-called walled diagrams, which are defined as follows. Let pu pu pu and pd pd pd pbe two sets, each consisting of r ` r,s “ r Y s r,s “ r Y s d are placed under s nodes aligned horizontally on the plane. The nodes in the set pr,s u u d the nodes in the set pr,s and a vertical wall separates the first r nodes pr (pr ) in the upper (lower) row from the last s nodes pus (pds). A walled diagram d is a bijection between the set pur,s Y pdr,s and visualised by placing the edges between the corresponding points in the following way:
1. edges connecting nodes between pur,s and pdr,s do not cross the wall (we call them prop-agating lines),
2. edges connecting nodes between pur,s and pur,s and between pdr,s and pdr,s cross the wall (we call them arcs).
Let be a complex parameter. As a vector space, the walled Brauer algebra Br,sp q is identified with the C-linear span of the walled diagrams. The product of two basis elements d2d1 is obtained by placing d1 above d2 and identifying the nodes of the top row of d2 with the corresponding nodes in the bottom row of d1. Let ` be the number of closed loops so obtained. The product d1d2 is given by ` times the resulting diagram with loops omitted. The following walled diagrams represent the generators si (the vertical dotted line rep-resents the wall).
Fusion procedure for the walled Brauer algebra
A fusion procedure gives a construction of the maximal family of pairwise orthogonal minimal idempotents in the algebra.
The fusion procedure (for the symmetric group) originates in the work of Jucys [19], see also the subsequent works [Ch, Na, GP]. A simplified version of the fusion procedure for the symmetric group involving the consecutive evaluation was suggested by Molev in [Mo]. Later the analogues of this simplified fusion procedure were suggested for the Hecke algebra [IMOs], for the Brauer algebra [IM, IMO], for the complex reflection groups of type Gpm, 1, nq, for the cyclotomic Hecke algebras [OP1, OP2], for the cyclotomic Brauer algebras [C], for the Birman–Murakami–Wenzl algebras [IMO2]. In [OP3] this fusion procedure was applied for the calculation of weights of certain Markov traces on the cyclotomic Hecke algebras.
The fusion procedure is closely related with the inductive approach to the representation theory of towers of algebras, see [OV] for the symmetric groups, and the generalizations for the Hecke algebra [IO], for the cyclotomic Hecke algebra [OP4], complex reflection groups [OP5] and Brauer algebras [IO2].
Simple Uqs`p2|1q-modules
Every finite-dimensional irreducible module over the Lie superalgebra s`pn|1q can be de-formed into an irreducible module over Uqs`pn|1q, see [PT]. The Lie superalgebras s`pn|1q belong to the class of the simple complex Lie superalgebras, classified by Kac [K1]-[K3]. Following his terminology we use notations ”typical” and ”atypical” to classify Uqs`p2|1q-modules (see also [S]). In the sequel we use notations from [ST]. We consider a subcategory of Uqs`p2|1q-modules with k eigenvalues of the form q´n for n P Z. The subcategory is closed under tensor products. The simple finite-dimensional Uqs`p2|1q-modules can be labeled as (see [ST] and references therein) Zs,r↵, , ↵ “,˘1, s • 1, r P Z. (1.7) The module Zs,r↵, contains a unique heighest-weight vector |↵, s,, r y–0, such that.
E|↵, s,, r y–0 “ 0, C|↵, s,, r y–0 “ 0.
K|↵, s,, r y–0 “ ↵qs´1|↵, s,, r y–0.
k|↵, s,, r y–0 “ q´r|↵, s,, r y–0.
Table of contents :
1 Walled Brauer algebra
1.1 Definition
1.2 Normal form and reduction algorithm
1.3 Modules over Br,s p!q
1.3.1 Diagrammatical description of Br,s p!q
1.3.2 Br,s p!q-modules
1.3.4 Annihilator ideal
1.3.7 Gelfand-Zetlin basis
1.4 Jucys–Murphy elements
1.5 Orthogonal primitive idempotents
1.5.1 Algebraic background
1.5.2 Fusion procedure for the walled Brauer algebra
2 Schur-Weyl duality between Uqs`p2|1q and quantum walled Brauer algebra qBm,n
2.1 The Hopf algebra Uqs`p2|1q
2.1.1 Definition of Uqs`p2|1q
2.1.2 Simple Uqs`p2|1q-modules
2.1.3 Uqs`p2|1q-action on simple modules
2.1.4 Ext1 spaces for atypical modules
2.1.5 Projective Uqs`p2|1q-modules
2.2 The mixed tensor product
2.2.2 The centralizer of Uqs`p2|1q on the mixed tensor product
2.3 Quantum walled Brauer algebra
2.3.1 Definition
2.3.3 Cell modules
2.4 Modules over Xm,n
2.4.1 Modules in the decomposition of the mixed tensor product
2.4.2 The restriction functors
2.5 The mixed tensor product as a bimodule
2.5.1 Bimodule decomposition
2.5.3 Verification
Appendix
A: partial order on pB r,s
B: multiplication by generators
C: atypical part of the bimodule .
