Cluster Algebras, Invariant Theory, and Kronecker Coefficients I

2015/04/12 by Fei, Jiarui · 1 citation
#13A50 #13F60 #52B20 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 20C30 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary 16G20

paper · doi:10.48550/arxiv.1504.02970

Abstract

We relate the m-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. We find cluster algebra structures for these semi-invariant rings when m=2. Each \sf g-vector cone \sf G_\Diamondl of these cluster algebras controls the 2-truncated Kronecker products for all symmetric functions of degree no greater than l. As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. We also give explicit description of all \sf G_\Diamondl's. As an application, we compute some invariant rings.

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