Abstracts for the Seminar
 Discrete Geometry and Combinatorics
 Spring 2018

Speaker:  Greta Panova, University of Pennsylvania
Title: Hook formulas for skew shapes
Time: 2:30 PM, Monday, March 19, 2018
Place:  Malott 206

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] coming from Schubert calculus.

We will show several combinatorial and algebraic proofs 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. These formulas have further combinatorial meaning and another proof as non-intersecting lattice paths interpretations. Multivariate versions of the hook formula lead also to exact product formulas for certain skew SYTs and evaluations of Schubert polynomials. They are directly related to lozenge tilings with multivariate weights, which also appear to have interesting behavior in the limit.

Joint work with A. Morales and I. Pak

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