Quivers, long exact sequences and Horn type inequalities

TL;DR

This paper establishes inequalities characterizing the existence of long exact sequences of finite abelian p-groups, connecting them to generalized Littlewood-Richardson coefficients and eigenvalue problems of Hermitian matrices.

Contribution

It introduces necessary and sufficient inequalities for long exact sequences of abelian p-groups and relates these to generalized Littlewood-Richardson coefficients and eigenvalue inequalities.

Findings

Derived inequalities for exact sequences of finite abelian p-groups

Connected the problem to generalized Littlewood-Richardson coefficients

Reproduced Horn's inequalities for the case m=3

Abstract

We give necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p-groups with fixed isomorphy types. This problem is related to some generalized Littlewood-Richardson coefficients that we define in this paper. We also show how this problem is related to eigenvalues of Hermitian matrices satisfying certain (in)equalities. When m=3, we recover the Horn type inequalities that solve the saturation conjecture for Littlewood-Richardson coefficients and Horn's conjecture.