Bijective proofs of skew Schur polynomial factorizations

TL;DR

This paper provides bijective proofs for factorizations of skew Schur polynomials into group characters, using combinatorial models like Gelfand-Tsetlin patterns and perfect matchings, extending previous determinant-based proofs.

Contribution

It introduces bijective combinatorial proofs for skew Schur polynomial factorizations, generalizing earlier determinant-based results and applying graph-theoretic techniques.

Findings

Bijective proofs for skew Schur polynomial factorizations.

Extension of factorizations to skew shapes.

Application of Ciucu's theorem with symmetric edge weights.

Abstract

In a recent paper, Ayyer and Behrend present for a wide class of partitions factorizations of Schur polynomials with an even number of variables where half of the variables are the reciprocals of the others into symplectic and/or orthogonal group characters, thereby generalizing results of Ciucu and Krattenthaler for rectangular shapes. Their proofs proceed by manipulations of determinants underlying the characters. The purpose of the current paper is to provide bijective proofs of such factorizations. The quantities involved have known combinatorial interpretations in terms of Gelfand-Tsetlin patterns of various types or half Gelfand-Tsetlin patterns, which can in turn be transformed into perfect matchings of weighted trapezoidal honeycomb graphs. An important ingredient is then Ciucu's theorem for graphs with reflective symmetry. However, before being able to apply it, we need to employ a certain averaging procedure in order to achieve symmetric edge weights. This procedure is based on a "randomized" bijection, which can however also be turned into a classical bijection. For one type of Schur polynomial factorization, we also need an additional graph operation that almost doubles the underlying graph. Finally, our combinatorial proofs reveal that the factorizations under consideration can in fact also be generalized to skew shapes as discussed at the end of the paper.