# 16th ARTIN Meeting

## University of Leeds, Friday March 14 - Saturday March 15, 2008

## Abstracts of Talks

**Maud De Visscher (City)**: The blocks of the Brauer algebra.

The Brauer algebra was introduced by Brauer in the study of the representation theory of the orthogonal and symplectic groups. Even over the complex numbers, this algebra is not semisimple in general. In this talk, I will give a description of the blocks of the Brauer algebra in terms of a suitable action of the Weyl group of type D. This is joint work with Anton Cox and Paul Martin.

**Volydmyr Mazorchuk (Glasgow)**: Quadratic duality and applications.

For a positively graded algebra A we construct a functor from the derived category of graded A-modules to the derived category of graded modules over the quadratic dual A^! of A. This functor is an equivalence of certain bounded subcategories if and only if the algebra A is Koszul. In the latter case the functor gives the classical Koszul duality. The approach I will talk about uses the category of linear complexes of projective A-modules. Its advantage is that the Koszul duality functor is given in a nice and explicit way for computational applications. The applications I am going to discuss are Koszul dualities between certain functors on the regular block of the category O, which lead to connections between different categorifications of certain knot invariants. (Joint work with S.Ovsienko and C.Stroppel.)

**Hugh Morton (Liverpool) **: Mutants with extra symmetry.

The Homfly polynomial of any satellite of a knot K is determined by a combination of quantum invariants of K derived from irreducible modules over the quantum group sl(N,q). Two closely related knots K, K' which are mutants in the sense of Conway always share the same Homfly polynomial, and that of any 2-string satellite. In general their 3-string satellites can have different polynomials. If, however, the mutants have certain extra symmetry then all k-string satellites with k < 6 share the same Homfly polynomial. I will discuss how this result relates to features of the symmetric and exterior squares of irreducible sl(N,q) modules whose Young diagrams have k cells.

**Alison Parker (Leeds) **: Some properties of the symplectic blob algebra.

The symplectic blob algebra is a diagram algebra which is a natural
generalization of the Temperley-Lieb algebra and the blob algebra. These
three diagram algebras are all finite dimensional quotients of Hecke
algebras of types A, B and C^{~} respectively, and hence may be
used to study these algebras.

While decomposition numbers are well-understood for the Temperley-Lieb and blob algebras, they are, as yet, not known for the symplectic blob algebra. I will discuss some of the reasons why the symplectic blob algebra is qualitatively different but also strikingly similar to these other two algebras. I will also discuss some known results in this direction. For instance, all these algebras are cellular and, in fact, usually quasi-hereditary. Our results so far point to an "alcove-geometry" for the symplectic blob algebra and raise many interesting questions.

This is joint work with R. M. Green and P. P. Martin.

Information about the meeting.