Matrix pairs over discrete valuation rings determine Littlewood-Richardson fillings

TL;DR

This paper establishes a connection between matrix pairs over discrete valuation rings and Littlewood-Richardson fillings, providing a new combinatorial invariant for matrix orbit classification.

Contribution

It constructs a Littlewood-Richardson filling invariant from matrix pairs and relates it to algebraic combinatorics and semicanonical matrices.

Findings

Constructed Littlewood-Richardson fillings as invariants of matrix pairs.

Linked algebraic properties of matrices to combinatorial Littlewood-Richardson fillings.

Provided a method to derive combinatorial invariants from matrix orbit data.

Abstract

Let M and N be two r x r matrices over a discrete valuation ring of characteristic zero. The orders (with respect to a uniformizing parameter) of the invariant factors of M form a partition of non-negative integers, called the invariant partition of M. Let the invariant partition of M be mu, of N be nu, and of the product MN be lambda. In this paper we construct a Littlewood-Richardson filling of the skew shape lambda/mu with content nu, and show that this filling is an invariant of the orbit of the pair (M,N) with respect to a natural group action on the pair. We relate the algebraic combinatorics of Littlewood-Richardson fillings to a special semicanonical matrix in the orbit of (M,N), from which the Littlewood-Richardson filling, and other combinatorial invariants may be obtained.