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.