ABSTRACT:
The original “shuffle conjecture” of Haglund, Haiman, Loehr, Ulyanov, and Remmel predicted a striking combinatorial formula for the bigraded character of the diagonal coinvariant algebra in type A, in terms of some fascinating parking functions statistics. I will start by explaining this formula, as well as the ideas that went into my recent proof of this conjecture with Anton Mellit, namely the construction of a new algebra which has many elements in common with DAHA’s, and which has been expected to have a geometric construction. I will then explain a new result with Eugene Gorsky and Mellit, in which we construct this algebra using the torus-equivariant K theory of a certain smooth subscheme of the flag Hilbert scheme, which parametrizes flags of ideals in C[x,y] of finite codimension.