Algebra and Combinatorics Seminar

Date: March 1, 2019

Time: 3:00PM - 4:00PM

Location: BLOC 628

Speaker: Benjamin Briggs, University of Utah

Title: Reflection Groups and Derivations

Abstract: If you start with a polynomial ring (say over the complex numbers) and you factor out by the ideal generated by symmetric polynomials (of positive degree), then you get a very interesting ring. For example, it is isomorphic to the cohomology ring of a flag manifold.

How many derivations does this ring have (i.e. what is the dimension of the space of C-linear derivations)? The ring is also graded: how many derivations does it have in each degree? These are tricky to count, but it turns out there is a surprisingly nice formula. You get this formula by writing down a free resolution of the module of derivations, which for some reason turns out to be periodic.

You can do all this by messing around with symmetric polynomials (but the combinatorics get quite complicated). It turns out though that this all works for certain reflection groups (all the real reflection groups included, and some complex reflection groups). I'll talk about this too, mainly focusing on the symmetric group example.