Hook formulas for skew shapes I. $q$-analogues and bijections

TL;DR

This paper provides algebraic and combinatorial proofs of Naruse's hook-length formula for skew shapes, introduces two new $q$-analogues for skew Schur functions and reverse plane partitions, and establishes explicit bijections for these objects.

Contribution

It offers the first algebraic and combinatorial proofs of Naruse's formula and introduces novel $q$-analogues along with explicit bijections for skew shapes.

Findings

Two new $q$-analogues of Naruse's formula for skew shapes.

Explicit bijections between skew shapes, $q$-analogues, and integer arrays.

Proofs using factorial Schur functions and generalized Hillman--Grassl correspondence.

Abstract

The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse's formula, by using factorial Schur functions and a generalization of the Hillman--Grassl correspondence, respectively. The main new results are two different $q$-analogues of Naruse's formula: for the skew Schur functions, and for counting reverse plane partitions of skew shapes. We establish explicit bijections between these objects and families of integer arrays with certain nonzero entries, which also proves the second formula.