Frobenius multiplicities and F-signatures of Grassmannians of two-planes

Abstract

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.