Hall–Littlewood polynomials and vector bundles on the Hilbert scheme

Let E be the bundle defined by applying a polynomial functor to the tautological bundle on the Hilbert scheme of n points in the complex plane. By a result of Haiman [5], the Čech cohomology groups Hi(E)Hi(E) vanish for all i>0i>0. It follows that the equivariant Euler characteristic with respect to the standard two-dimensional torus action has nonnegative integer coefficients in the torus variables z1,z2z1,z2, because they count the dimensions of the weight spaces of H0(E)H0(E). We derive a formula for this Euler characteristic using residue formulas for the Euler characteristic coming from the description of the Hilbert scheme as a quiver variety [13] and [14]. We evaluate this expression using Jing's Hall–Littlewood vertex operator with parameter z1z1[7], and a new vertex operator formula given in Proposition 1 below. We conjecture that the summand in this formula is a polynomial in z1z1 with nonnegative integer coefficients, a special case of which was known to Lascoux and Schützenberger [8].