A Polyhedral Formula for $n\times2\times2$ Kronecker Coefficients via Cluster Algebras

Abstract

Let \[ \Bbbk[\Bbbk^3\otimes\Bbbk^2\otimes\Bbbk^2]^{U_3\times U_2\times U_2}. \] We construct an ordinary cluster family with Markov principal part. At $\zeta=-1$, the intersection of its initial Laurent ring with the three adjacent Laurent rings equals $\mathscr U$, and \[ \mathscr U=\mathcal M_u[u_\Delta], \] where $\mathcal M_u$ is the middle algebra. Its theta functions are indexed by a cone with a sixteen-element Hilbert basis. Multiplication by $u_\Delta$ pairs the Hilbert generators of mutable degrees $1$ and $-1$ and reduces each triple-weight space to the slice $\ell=\ell_0$. Counting the lattice points in this slice gives a finite sum with nonnegative summands. Determinant reduction extends the formula to all $n\times2\times2$ Kronecker coefficients.