A polyhedral formula for n×2×2 Kronecker coefficients via cluster algebras

2026/07/28 by Jiarui Fei, Chenxin Xue
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Abstract

We study the triple-invariant algebra \Bbbk[\Bbbk³⊗\Bbbk²⊗\Bbbk²]^{U₃× U₂× U₂}. A quotient slice and the induced logarithmic top form determine a signed Markov chart, realized as the fiber ζ=-1 of an ordinary cluster family. We prove \mathscr U_gen=\mathcal M_u[u_Δ], where \mathcal M_u is a specialized middle cluster algebra and u_Δ is the discriminant of weight (220;22;22). Its theta cone has a sixteen-element Hilbert basis. Pairing its positive- and negative-degree generators reduces each triple-weight space to a single discriminant level determined by the weight. Counting the resulting two-dimensional slice gives an explicit nonnegative finite-sum formula. Determinant reduction extends the formula to all n×2×2 Kronecker coefficients.

Citations

• Valuative independence and cluster theta reciprocity
• Skein and cluster algebras of punctured surfaces
• Vector partition functions and Kronecker coefficients
• Khovanskii bases, higher rank valuations and tropical geometry
• Cluster Algebras, Invariant Theory, and Kronecker Coefficients I
• Canonical bases for cluster algebras
• Geometry of canonical bases and mirror symmetry
• Birational geometry of cluster algebras
• Quasipolynomial formulas for the Kronecker coefficients indexed by two two-row shapes (extended abstract)
• Cluster algebras III: Upper bounds and double Bruhat cells
• Cluster algebras I: Foundations
• The Kronecker product of Schur functions indexed by two-row shapes or hook shapes

Related

• Cluster Algebras, Invariant Theory, and Kronecker Coefficients I
• Cluster algebraic interpretation of generalized Markov numbers and their matrixizations
• Cluster algebras IV: Coefficients
• Vector partition functions and Kronecker coefficients
• Cluster algebras IV: Coefficients