Littlewood-Richardson coefficients as a signed sum of Kostka numbers

TL;DR

This paper presents a novel representation of Littlewood-Richardson coefficients as signed sums of Kostka numbers, providing a polynomial-time algorithm and connecting these concepts in representation theory and combinatorics.

Contribution

It introduces a new formulation expressing LR coefficients as signed sums of Kostka numbers and demonstrates a polynomial-time decision algorithm for them.

Findings

LR coefficients can be expressed as signed sums of Kostka numbers

A polynomial-time algorithm for computing LR coefficients is developed

Steinberg's formula is proved using Kostant's partition function

Abstract

Littlewood-Richardson (LR) coefficients and Kostka Numbers appear in representation theory and combinatorics related to $GL_n$. It is known that Kostka numbers can be represented as special Littlewood-Rischardson coefficient. In this paper, we show how one can represent LR coefficient as a signed sum of Kostka numbers, and use the formulation to give a polynomial time algorithm for the same, hence showing that they belong to the same class of decision problems. As a corollary, we will prove Steinberg's formula using Kostant's partition function.