AI 中文总结
研究通过簇代数得到\(n\times2\times2\)克罗内克系数的多面体公式，利用商切片等确定有符号马尔可夫图，证明相关等式，通过对生成元配对等得到非负有限和公式，并用行列式约化扩展到所有系数。
AI 中文摘要
我们研究三重不变代数\(\Bbbk[\Bbbk^3\otimes\Bbbk^2\otimes\Bbbk^2]^{U_3\times U_2\times U_2}\)。一个商切片和诱导的对数顶形式确定一个有符号马尔可夫图，它作为普通簇族中\(\zeta = -1\)的纤维实现。我们证明\(\mathscr U_{\mathrm{gen}}=\mathcal M_u[u_\Delta]\)，其中\(\mathcal M_u\)是一个特殊的中间簇代数，\(u_\Delta\)是权重为\((220;22;22)\)的判别式。其theta锥有一个十六元希尔伯特基。对其正负度生成元配对将每个三重权重空间简化为一个由权重确定的单一判别式水平。计算所得二维切片给出一个明确的非负有限和公式。行列式约化将该公式扩展到所有\(n\times2\times2\)克罗内克系数。
英文摘要
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 $ζ=-1$, the intersection of its initial Laurent ring with the three adjacent Laurent rings equals $\mathscr U$, and \[ \mathscr U=\mathcal M_u[u_Δ], \] where $\mathcal M_u$ is the middle algebra. Its theta functions are indexed by a cone with a sixteen-element Hilbert basis. Multiplication by $u_Δ$ 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.
Comments29 pages, comments are welcome. v2. minor correction
