Matrix Pairs over Valuation Rings and R-Valued Littlewood-Richardson Fillings

TL;DR

This paper extends Littlewood-Richardson fillings to real-valued diagrams, linking them to matrix pairs over valuation rings and exploring continuous deformations of these generalized fillings.

Contribution

It introduces a new framework for real-valued Littlewood-Richardson fillings and establishes a bijection with matrix pairs over valuation rings, connecting to classical combinatorial bijections.

Findings

Real-valued Littlewood-Richardson fillings are realized as invariants of matrix pairs.

A bijection between matrix pair fillings and classical fillings is established.

Continuous deformations of generalized fillings are interpreted through matrix calculations.

Abstract

We extend the theory of Littlewood-Richardson fillings (defined over the non-negative integers) to include diagrams with rows and boxes of real-valued length. We realize such fillings as invariants of matrix pairs over rings with a real-valued valuation. We then prove a bijection between pairs of fillings determined by a matrix pair, and connect this to the classical row-switching bijection for Littlewoo-Richardson fillings. We utilize matrix calculations to interpret continuous deformations of generalized Littlewood-Richardson fillings.