Document Zbl 1403.13037

Cluster algebras, invariant theory, and Kronecker coefficients. I. (English) Zbl 1403.13037

Adv. Math. 310, 1064-1112 (2017).

In the present paper, the author relates the \(m\)-truncated Kronecker products of symmetric functions to the semi-invariant rings of a family of quiver representations. The author finds cluster algebra structures for these semi-invariant rings when \(m=2\). Each \(g\)-vector cone \(G_{\diamond _{l}}\) of these cluster algebras controls the \(2\)-truncated Kronecker products for all symmetric functions of degree not greater than \(l\). As a consequence, each relevant Kronecker coefficient is the difference of the number of the lattice points inside two rational polytopes. The author also gives explicit description of all \(G_{\diamond _{l}}\)’s. As an application, the author computes some invariant rings.

Reviewer: Nan Ji-Zhu (Dalian)

Cited in 1 Review

Cited in 3 Documents

MSC:

13F60	Cluster algebras
16G20	Representations of quivers and partially ordered sets
13A50	Actions of groups on commutative rings; invariant theory
20C30	Representations of finite symmetric groups
52B20	Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)

Keywords:

symmetric function; cluster algebra; quiver representation; Quiver with potential; lattice point; diamond quiver; \(\mathsf{g}\)-vector cone; unimodular Fan; flagged Kronecker quiver; semi-invariant ring; cluster character; Kronecker coefficient

Software:

Normaliz

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Full Text: DOI arXiv

References:

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