Cluster algebras, invariant theory, and Kronecker coefficients I
Copyright policy )Cluster algebras, invariant theory, and Kronecker coefficients I
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 $g_{μ,ν}^λ$ with $λ$ at most $m$ parts, we only need to count lattice points in at most $m!$ fibre (rational) polytopes inside the ${\rm g}$-vector cone, which is explicitly given.
40 pages, 20 figures, comments are welcome
- Shanghai Jiao Tong University China (People's Republic of)
- National Center for Theoretical Sciences Taiwan
unimodular Fan, diamond quiver, \(\mathsf{g}\)-vector cone, 13F60, 20C30 (Primary), 16G20, 13A50, 52B20 (Secondary), semi-invariant ring, Group Theory (math.GR), symmetric function, lattice point, Commutative Algebra (math.AC), cluster character, Kronecker coefficient, upper cluster algebra, FOS: Mathematics, Mathematics - Combinatorics, Representation Theory (math.RT), flagged Kronecker quiver, Quiver with potential, quiver with potential, Cluster algebras, Representations of finite symmetric groups, Mathematics - Rings and Algebras, Mathematics - Commutative Algebra, quiver representation, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Rings and Algebras (math.RA), Primary 20C30, 13F60, Secondary 16G20, 13A50, 52B20, Representations of quivers and partially ordered sets, Combinatorics (math.CO), Mathematics - Group Theory, Mathematics - Representation Theory, cluster algebra, Actions of groups on commutative rings; invariant theory
unimodular Fan, diamond quiver, \(\mathsf{g}\)-vector cone, 13F60, 20C30 (Primary), 16G20, 13A50, 52B20 (Secondary), semi-invariant ring, Group Theory (math.GR), symmetric function, lattice point, Commutative Algebra (math.AC), cluster character, Kronecker coefficient, upper cluster algebra, FOS: Mathematics, Mathematics - Combinatorics, Representation Theory (math.RT), flagged Kronecker quiver, Quiver with potential, quiver with potential, Cluster algebras, Representations of finite symmetric groups, Mathematics - Rings and Algebras, Mathematics - Commutative Algebra, quiver representation, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Rings and Algebras (math.RA), Primary 20C30, 13F60, Secondary 16G20, 13A50, 52B20, Representations of quivers and partially ordered sets, Combinatorics (math.CO), Mathematics - Group Theory, Mathematics - Representation Theory, cluster algebra, Actions of groups on commutative rings; invariant theory
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