UBC Discrete Math Seminar: Mikhail Mazin

Abstract: The classical q,t-Catalan polynomials can be computed as enumerations of subpartitions of staircase partitions with respect to the area and dinv statistics. The triangular q,t-Catalan polynomials are a generalization where the staircase partitions are replaced with more general triangular partitions. The celebrated Shuffle Theorems provide a connection between the (triangular) q,t-Catalan polynomials and the action of the so-called Negut's elements of the elliptic Hall algebra on symmetric functions. In turn, this allows one to generalize the q,t-Catalan polynomials beyond the case of triangular partitions. However, the formula obtained from the elliptic Hall algebra action is not a positive sum of monomials, but rather a complicated sum of rational functions. Nevertheless, it is known that for any partition lambda one obtains a polynomial with integer coefficients, and its specialization at t=1 counts subpartitions of lambda with respect to area. Furthermore, computer computations suggest that if lambda is concave, then the coefficients of the polynomial are non-negative. A natural question is then to generalize the second statistic, dinv, to the case of concave partitions.

In this talk I will introduce the Negut's operators, Shuffle Theorems, and the generalized q,t-Catalan polynomials. I will then turn to the proposed geometric model: the Hilbert schemes of monomial ideals, focusing mostly on the three series of cases mentioned above.