University of Birmingham > Talks@bham > Algebra seminar  > What is a q-reflection group?

What is a q-reflection group?

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If you have a question about this talk, please contact Simon Goodwin.

If V is a vector space over a field k, a reflection of V is a linear transformation of V which fixes a hyperplane in V pointwise. A finite group generated by reflections of V is termed a reflection group. If k=Q, R or C, reflection groups are the same as Weyl groups, Coxeter groups or complex reflection groups, respectively, and are very important in Lie theory. We will look at two key properties of a reflection group G over a field of characteristic 0:

  1. G-invariants in the polynomial ring k[V] are themselves a polynomial ring (the Chevalley-Shephard-Todd theorem);
  2. there is a special family of commuting differential-difference operators on k[V], called Dunkl operators (they give rise to a rational Cherednik algebra of G).

It is interesting to note that the Chevalley-Shephard-Todd theorem (1), which dates back to 1950s, was nontrivially extended to fields of positive characteristic in a more recent work of Serre and Kemper-Malle. In my talk, however, I will remain in characteristic 0 but will be interested in a generalisation of the C-S-T theorem where the polynomial ring is replaced with a noncommutative algebra.

I will aim to describe results of myself and Berenstein (inspired by the theory of integrable systems) which lead to a q-commuting version of Dunkl operators and Cherednik algebras (2). They are attached to a family of finite groups which are no longer reflection groups; remarkably, the q-commutative Chevalley-Shephard-Todd theorem for our groups was proved almost immediately in an independent work by Zhang and collaborators. I will describe this family of finite groups and, if time permits, will mention possible further generalisations.

This talk is part of the Algebra seminar series.

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