The odd Littlewood-Richardson rule

TL;DR

This paper introduces a new odd analogue of Schur functions called plactic Schur functions, proves their equivalence with existing types, and establishes an odd Littlewood-Richardson rule, connecting combinatorics and polytope models.

Contribution

It introduces the plactic Schur functions and proves their equivalence with previous odd Schur functions, establishing an odd Littlewood-Richardson rule and linking it to polytope models.

Findings

Plactic Schur functions coincide with odd Kostka and symmetrization-based Schur functions.

An odd Littlewood-Richardson rule is established.

The rule is expressed using the Knutson-Tao hive polytope model.

Abstract

In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via the odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson-Tao hive model.