Cluster algebras, invariant theory, and Kronecker coefficients II

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∈N. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequen...

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Bibliographic Details
Published inAdvances in mathematics (New York. 1965) Vol. 341; pp. 536 - 582
Main Author Fei, Jiarui
Format Journal Article
LanguageEnglish
Published Elsevier Inc 07.01.2019
Subjects
Kronecker coefficient
Lattice point
Quiver Representation
Quiver with potential
Semi-invariant ring
Upper cluster algebra
Upper cluster algebra
Lattice point
Semi-invariant ring
Quiver Representation
Quiver with potential
Kronecker coefficient
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Summary: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∈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 gμ,νλ with λ at most m parts, we only need to count lattice points in at most m! fiber (rational) polytopes inside the g-vector cone, which is explicitly given.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2018.10.042