20 research outputs found
A geometric characterisation of the blocks of the Brauer algebra
We give a geometric description of the blocks of the Brauer algebra
in characteristic zero as orbits of the Weyl group of type .
We show how the corresponding affine Weyl group controls the representation
theory of the Brauer algebra in positive characteristic, with orbits
corresponding to unions of blocks.Comment: 26 pages, 24 figure
Quiver presentations and isomorphisms of Hecke categories and Khovanov arc algebras
We prove that the extended Khovanov arc algebras are isomorphic to the basic
algebras of anti-spherical Hecke categories for maximal parabolics of symmetric
groups. We present these algebras by quiver and relations and provide the full
submodule lattices of Verma modules
The anti-spherical Hecke categories for Hermitian symmetric pairs
We calculate the -Kazhdan--Lusztig polynomials for Hermitian symmetric
pairs and prove that the corresponding anti-spherical Hecke categories
categories are standard Koszul. We prove that the combinatorial invariance
conjecture can be lifted to the level of graded Morita equivalences between
subquotients of these Hecke categories
On the blocks of semisimple algebraic groups and associated generalized Schur algebras
In this paper we give a new proof for the description of the blocks of any semisimple simply connected algebraic group when the characteristic of the field is greater than 5. The first proof was given by Donkin and works in arbitrary characteristic. Our new proof has two advantages. First we obtain a bound on the length of a minimum chain linking two weights in the same block. Second we obtain a sufficient condition on saturated subsets π of the set of dominant weights which ensures that the blocks of the associated generalized Schur algebra are simply the intersection of the blocks of the algebraic group with the set π. However, we show that this is not the case in general for the symplectic Schur algebras, disproving a conjecture of Renner
Extensions of modules for SL(2,K)
AbstractIn this paper, we consider the induced modules ∇ and the Weyl modules Δ for the algebraic group G=SL(2,K) where K is an algebraically closed field of characteristic p>0. We determine the G-modules Hi(G1,∇(s)⊗∇(t)) for all i⩾0, where G1 is the first Frobenius kernel of G. We then use it to find the Ext1-spaces between twisted tensor products of Weyl modules and induced modules for G. Moreover, we describe explicitly the non-split extensions corresponding to ∇'s
Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra
We determine the decomposition numbers for the Brauer and walled Brauer
algebra in characteristic zero in terms of certain polynomials associated to
cap and curl diagrams (recovering a result of Martin in the Brauer case). We
consider a second family of polynomials associated to such diagrams, and use
these to determine projective resolutions of the standard modules. We then
relate these two families of polynomials to Kazhdan-Lusztig theory via the work
of Lascoux-Sch\"utzenberger and Boe, inspired by work of Brundan and Stroppel
in the cap diagram case.Comment: 32 pages, 22 figure