### School of Mathematics and Statistics

### Jacqui Ramagge

#### Jucys-Murphy elements and affine Hecke algebras.

*Friday 20th February, 12-1pm, Carslaw 273.*

Given a group *G* with a *BN*--pair, the representation theory of its
associated Iwahori--Hecke algebra is an important tool in the study of the
decomposition of the induced representation *1*_{B}^{G}.

Seminormal representations of Iwahori--Hecke algebras are characterized by
the existence of a basis of simultaneous eigenvectors for a particular set
of elements, called Jucys-Murphy elements, of the algebra. Hoefsmit (1974)
showed that in types **A**, **B** and **D** all irreducible
representations of the Iwahori--Hecke algebras are seminormal generically.
He used this to classify all irreducible representations of these algebras.

Throughout the study of seminormal representations, the Jucys-Murphy
elements and their properties have remained a little mysterious and
surpirsing. For example, they form an abelian subalgebra of the
Iwahori-Hecke algebra and, un type **A**, the centre of the
Iwahori-Hecke algebra is precisely the algebra of symmetric polynomials in
the Jucys-Murphy elements. In joint work with Arun Ram, we show that there
are algebra homomorphisms from affine Hecke algebras to Iwahori-Hecke
algebras of types **A**, **B** and **D** which demystify the
Jucys-Murphy elements. From this point of view the properties of the
Jucys-Murphy elements are self-evidient and, to a certain extent, generic
for types **A**, **B** and **D**.