行列式簇K多项式的组合推导研究
Towards combinatorial derivations of K-polynomials for determinantal varieties
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中文总结 AI 辅助
本文针对行列式簇的K多项式组合推导问题，提出对秩不超过k的复矩阵簇希尔伯特级数相关公式给出直接组合证明的目标，在k=1或k=min{m,n}-1时通过Littlewood–Richardson表征集上的显式符号反转对合完成了证明。
中文摘要 AI 辅助
设$\boldsymbol{\frak X}_k\boldsymbol{\textsf{Mat}}_{m,n}$为秩不超过$k$的$m\times n$复矩阵簇。$\frak X_k$希尔伯特级数的幂级数与有理表达式已通过几何论证得到，联立这些表达式可得到推广经典柯西恒等式与对偶柯西恒等式的一族公式。我们提出对$0<k<\text{min}\{m,n\}$情形下的这些公式给出直接组合证明的问题。当$k=1$或$k=\text{min}\{m,n\}-1$时，我们通过在特定Littlewood–Richardson（李特尔伍德-理查森）表征集上构造显式符号反转对合给出了此类证明。
英文摘要
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0 < k < \min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood--Richardson tableaux.
