Quivers, long exact sequences and Horn type inequalities

Abstract

We give necessary and sufficient inequalities for the existence of long exact sequences of m finite abelian p-groups with fixed isomorphism types. This problem is related to some generalized LittlewoodRichardson 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.

The combinatorics of quiver representations

“…Recent results of C. Chindris ( [4,5]) show that quivers can be successfully applied to get similar results for longer exact sequences.…”

“…Any σ-stable dimension vector is a nonnegative rational combination of δ 1 , δ 2 , δ 4 or a nonnegative rational combination of δ 2 , δ 3 , δ 4 . Suppose α is σ-stable and not equal to δ 1 , δ 2 , δ 3 , δ 4 .…”

“…Suppose α is σ-stable and not equal to δ 1 , δ 2 , δ 3 , δ 4 . If α is a nonnegative rational combination δ 1 , δ 2 , δ 4 , then because the support has to be path connected, it is actually a nonnegative rational combination of δ 2 , δ 4 .…”

“…If α is a nonnegative rational combination δ 1 , δ 2 , δ 4 , then because the support has to be path connected, it is actually a nonnegative rational combination of δ 2 , δ 4 . Similarly, if α is a nonnegative rational combination of δ 2 , δ 3 , δ 4 , then it must be in fact a nonnegative rational combination of δ 2 and δ 4 . Now it easily follows that the set of σ-stable dimension vectors is…”

Quivers, Long Exact Sequences and Horn Type Inequalities Ii

“…For this, we need to recall some of the terminology from [1]. Let λ(i) where the sum is taken over all partitions μ(1), .…”