Cluster Algebras, Invariant Theory, and Kronecker Coefficients II

Abstract

We prove that the semi-invariant ring of the standard representation space of the $l$-flagged $m$-arrow Kronecker quiver is an upper cluster algebra for any $l,m\in \mathbb{N}$. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient $g_{\mu ,\nu }^{\lambda }$ with $\lambda$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the $\mathrm{g}$-vector cone, which is explicitly given.

arXiv comment40 pages, 20 figures, comments are welcome