Algebra Seminar

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.