20th June 2019: Diagonal coinvariants via DAHA

Ivan Cherednik (UNC Chapel Hill)

Title:  Diagonal coinvariants via DAHA

Abstract: 

The initial (and difficult) problem was to

prove that (n+1)^(n-1) is the dimension of the algebra  of polynomials in terms of two sets of variables x_1,…,x_n  and y_1,…,y_n divided by the ideal generated by diagonally  S_n-invariant polynomials without the constant term. It was managed by Haiman based on results by Bridgeland, King and Reid. For more general root systems of rank n the  formula (h+1)^n  can be expected for the Coxeter number h, but it fails. However this formula gives the right answer, if the problem is restated using rational DAHA (Iain Gordon) and  q,t-DAHA (the speaker). I.e. if we don’t assume x to commute with y any longer. In the q,t-approach, the Weyl algebras at (h+1)-th root of unity q appeared sufficient, a specialization of the simplest  non-trivial “perfect” DAHA module. Such  modules are closely related to Verlinde algebras and have many other applications. Combinatorially, we arrive at “trigonometric diagonal coinvariants”, where new turns are expected.