Speaker: Greta Panova, University of Pennsylvania
Title: Hook formulas for skew shapes: combinatorics, asymptotics and beyond
Abstract: The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for skew shapes. In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using "excited diagrams" [ Ikeda-Naruse, Kreiman, Knutson-Miller-Yong]. We will show combinatorial and aglebraic proof of this formula, leading to a bijection between SSYTs or reverse plane partitions of skew shape and certain integer arrays that gives two q-analogues of the formula. We will also show how these formulas can be proven via non-intersecting lattice paths interpretations, and show various applications connecting Dyck paths and alternating permutations. We show how excited diagrams give asymptotic results for the number of skew Standard Young Tableaux in various regimes of convergence for both partitions. We will also show a multivariate versions of the hook formula with consequences to exact product formulas for certain skew SYTs and lozenge tilings with multivariate weights. Joint work with A. Morales and I. Pak.