A Combinatorial Formula and a Hilbert Scheme Realization of a Rank-One Virasoro Irregular Conformal Block

Abstract

For general central charge, we give a combinatorial formula and a geometric realization of the formal expansion of the rank-one Virasoro irregular conformal block of type $(0,0,1)$ at its irregular singular point. We realize the normalized block as a coefficientwise matrix element on the localized equivariant Hilbert--Fock space associated with Hilbert schemes of points on $\mathbb C^2$. At $c=1$, we prove the author's conjecture and express the parameter-independent coefficients in terms of skew-tableau numbers and Littlewood--Richardson coefficients, establishing their nonnegativity and integrality. The proof combines sectorial asymptotics of screening integrals with Laguerre--Jack evaluations. We express the screening kernel using the flow of a degree-raising differential operator and compute the Jack coefficients of its exponential from the one-box Pieri rule and the shifted Jack coefficient formula of Okounkov and Olshanski.