发表机构
Auburn University（奥本大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出了格拉斯曼簇Gr(2,n)的Plücker坐标环的Frobenius重数矩阵的非递归公式，证明其元素为p的多项式，从而得到F-签名等为p的有理函数，并给出Gr(2,4)等的具体表达式。
AI 中文摘要
Raedschelders、Špenko和Van den Bergh证明了当特征p满足p≥max{n-2,3}时，格拉斯曼簇Gr(2,n)的Plücker坐标环C_n具有有限Frobenius表示类型，并确定了其Frobenius前推的不可分解直和项（直到非零重数）。我们给出了Frobenius重数矩阵每个元素的非递归有限公式，并证明每个元素是p的次数至多为2n-3的多项式。因此，C_n的F-签名以及每个不可分解直和项的渐近比例在整个特征范围内是p的单一有理函数。当p→∞时，其极限是第一个自由重数的首项系数，我们通过Bernoulli数以闭式形式求值。对于Gr(2,4)，我们得到s_p(C_4)=(13p^2+8)/(5(3p^2+2))，并在计算机辅助下计算了Gr(2,5)和Gr(2,6)的类似有理函数。我们还从Riemann-Roch权重和两个残差矩阵构造了完整特征多项式，其元素在p中的次数至多为n-1。在后续工作中，我们利用这项工作证明秩二行列式环具有FFRT，并计算它们的F-签名。
英文摘要
Raedschelders, Špenko and Van den Bergh proved that the Plücker coordinate ring $C_n$ of the Grassmannian $\operatorname{Gr}(2,n)$ has finite Frobenius representation type when the characteristic $p$ satisfies $p\ge\max\{n-2,3\}$, and they determined the indecomposable summands of its Frobenius pushforwards up to nonzero multiplicity. We give nonrecursive finite formulas for every entry of the Frobenius multiplicity matrix and prove that each entry is a polynomial in $p$ of degree at most $2n-3$. Consequently the $F$-signature of $C_n$, and the asymptotic proportion of every indecomposable summand, is a single rational function of $p$ throughout the characteristic range. Its limit as $p\to\infty$ is the leading coefficient of the first free multiplicity, which we evaluate in closed form through Bernoulli numbers. For $\operatorname{Gr}(2,4)$ we find $s_p(C_4)=(13p^2+8)/(5(3p^2+2))$, and we calculate the analogous rational functions for $\operatorname{Gr}(2,5)$ and $\operatorname{Gr}(2,6)$ with computer assistance. We also construct the full characteristic polynomial from the Riemann-Roch weights and two residual matrices whose entries have degree at most $n-1$ in $p$. In the sequel, we use this work to show rank-two determinantal rings have FFRT, and to calculate their $F$-signatures.
