Involutions on standard Young tableaux and divisors on metric graphs

TL;DR

This paper explores a bijection between rectangular standard Young tableaux and divisor classes on metric graphs, providing explicit formulas and revealing how tableau operations correspond to graph symmetries and divisor dualities.

Contribution

It introduces an explicit formula for divisors associated with tableaux and links tableau operations to graph symmetries and divisor dualities.

Findings

Evacuation of tableaux corresponds to reflecting the metric graph.

Conjugation of tableaux corresponds to Riemann-Roch dual of the divisor.

Explicit formula for computing $v_0$-reduced divisors from tableaux.

Abstract

We elaborate upon a bijection discovered by Cools, Draisma, Payne, and Robeva between the set of rectangular standard Young tableaux and the set of equivalence classes of chip configurations on certain metric graphs under the relation of linear equivalence. We present an explicit formula for computing the $v_0$-reduced divisors (representatives of the equivalence classes) associated to given tableaux, and use this formula to prove (i) evacuation of tableaux corresponds (under the bijection) to reflecting the metric graph, and (ii) conjugation of the tableaux corresponds to taking the Riemann-Roch dual of the divisor.