Open Collections

Hook Formulas for Skew Shapes

Banff International Research Station for Mathematical Innovation and Discovery

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 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. We will also exhibit other curious phenomena emerging from there techniques like product formulas for various classes of skew SYTs and relations with reduced decompositions of permutations. Joint work with Alejandro Morales and Igor Pak.

Mathematics; Combinatorics; Group theory and generalizations

video/mp4

Author affiliation: University of Pennsylvania and Institute for Advanced Study

2017-12-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63794

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/