Computation of dilated Kronecker coefficients

Abstract

Abstract. The computation of Kronecker coefficients is a challenging problem with a variety of applications. In this paper we present an approach based on methods from symplectic geometry and residue calculus. We outline a general algorithm for the problem and then illustrate its effectiveness in several interesting examples. Significantly, our algorithm does not only compute individual Kronecker coefficients, but also symbolic formulas that are valid on an entire polyhedral chamber. As a byproduct, we are able to compute several Hilbert series.

Two classes of minimal generic fundamental invariants for tensors

“…Example 4.10. By (4.11), we have 2, 2, 3 = e (4) 1 ⊗ e (6) 1 ⊗ e (6) 1 + e (4) 1 ⊗ e (6) 2 ⊗ e (6) 3 + e (4) 1 ⊗ e (6) 3 ⊗ e (6) 5 + e (4) 2 ⊗ e (6) 4 ⊗ e (6) 1 + e (4) 2 ⊗ e (6) 5 ⊗ e (6) 3 + e (4) 2 ⊗ e (6) 6 ⊗ e (6) 5 + e (4) 3 ⊗ e (6) 1 ⊗ e (6) 2 + e (4) 3 ⊗ e (6) 2 ⊗ e (6) 4 + e (4) 3 ⊗ e (6) 3 ⊗ e (6) 6 + e (4) 4 ⊗ e (6) 4 ⊗ e (6) 2 + e (4) 4 ⊗ e (6) 5 ⊗ e (6) 4 + e (4) 4 ⊗ e (6) 6 ⊗ e (6) 6 ∈ C 4 ⊗ C 6 ⊗ C 6 . (4.12)…”

“…Specially, let 2, 2, 3 I 0 be the triple labeling defined in (4.21). Then we have val H ( 2, 2, 3 I 0 ) = det e (4) 1 , e (4) 2 , e (4) 3 , e (4)…”

“…• det e (4) 1 , e (4) 2 , e (4) 3 , e (4) 4 × det e (6) 1 , e 6) 4 , e (6) 2 , e (6) 5 , e (6) 3 , e (6) 6…”

Convexity and Thimm’s Trick

“…The branching cone. We begin by recalling the branching cone, as described in [5]. Let G be a compact, connected Lie group.…”

“…This set is called the branching cone for the subgroup K ≤ G [5]. Since it is a polyhedral cone, C K,G is defined as a subset of t * K × t * G by a finite list of inequalities, (23)…”