Morita equivalences of Ariki–Koike algebras

We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle. Received: 22 March 2000; in final form: 19 September 2001 / Published online: 29 April 2002     

On monomial linearisation and supercharacters of pattern subgroups

Column closed pattern subgroups U of the finite upper unitriangular groups U_n(q) are defined as sets of matrices in U_n(q) having zeros in a prescribed set of columns besides the diagonal ones. We explain Jedlitschky's construction of monomial linearisation in his thesis and apply this to CU yielding a generalisation of Yan's coadjoint cluster representations. Then we give a complete classification of the resulting supercharacters,by describing the resulting orbits and determining the Hom-spaces between orbit modules.     

The (Q, q)-Schur Algebra

In this paper we use the Hecke algebra of type B to define anew algebra S which is an analogue of the q-Schur algebra. Weshow that S has ‘generic’ basis which is independentof the choice of ring and the parameters q and Q. We then constructWeyl modules for S and obtain, as factor modules, a family ofirreducible S-modules defined over any field. 1991 MathematicsSubject Classification: 16G99, 20C20, 20G05.     

Brauer algebras, symplectic Schur algebras and Schur-Weyl duality

In this paper we prove the Schur-Weyl duality between the symplectic group and the Brauer algebra over an arbitrary infinite field . We show that the natural homomorphism from the Brauer algebra to the endomorphism algebra of the tensor space as a module over the symplectic similitude group (or equivalently, as a module over the symplectic group ) is always surjective. Another surjectivity, that of the natural homomorphism from the group algebra for to the endomorphism algebra of as a module over , is derived as an easy consequence of S. Oehms's results [S. Oehms, J. Algebra (1) 244 (2001), 19-44].

     

Harish-Chandra Vertices and Steinberg's Tensor Product Theorems for Finite General Linear Groups

Representations of Hecke and q-Schur algebras are closely relatedto those of finite general linear groups G in non-describingcharacteristics. Such a relationship can be described by certainfunctors. Using these functors, we determine the Harish-Chandravertices and sources of certain indecomposable G-modules. TheGreen correspondence is investigated in this context. As a furtherapplication of our theory, we establish Steinberg's tensor producttheorems for irreducible representations of G in non-describingcharacteristics. 1991 Mathematics Subject Classification: 20C20,20C33, 20G05, 20G40.     

The Quantized Walled Brauer Algebra and Mixed Tensor Space

In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (1994). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring R of rank independent of R. As an application, we prove one side of Schur–Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the R-endomorphism ring of mixed tensor space is, for all choices of R and the parameter q, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra U of the general linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the U-invariants in the ring of R-linear endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$ .