Frobenius multiplicities and F-signatures of Grassmannians of two-planes
Abstract
Raedschelders, Špenko and Van den Bergh proved that the Plücker coordinate ring Cn of the Grassmannian Gr(2,n) has finite Frobenius representation type when the characteristic p satisfies p≥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 Cn, and the asymptotic proportion of every indecomposable summand, is a single rational function of p throughout the characteristic range. Its limit as p→∞ is the leading coefficient of the first free multiplicity, which we evaluate in closed form through Bernoulli numbers. For Gr(2,4) we find sp(C4)=(13p2+8)/(5(3p2+2)), and we calculate the analogous rational functions for Gr(2,5) and 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.
Related
- Betti Numbers of Odd Biisotropic Symplectic Grassmannians
- Realizability of Cohomology Classes in Orthogonal Grassmannians
- Nef Cones of Hilbert Schemes of Orthogonal Grassmannians
- Prime Fano fourfolds in classical and generalized Grassmannians
- Balancedness of Normal Bundles of Rational Curves in Grassmannians
BibTeX
@misc{lyle2026frobenius,
title = {{Frobenius multiplicities and F-signatures of Grassmannians of two-planes}},
author = {Justin Lyle},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00362},
eprint = {2610.00362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00362},
}
RIS
TY - UNPB
TI - Frobenius multiplicities and F-signatures of Grassmannians of two-planes
AU - Justin Lyle
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00362
UR - https://arxiv.org/abs/2610.00362
AB - Raedschelders, Špenko and Van den Bergh proved that the Plücker coordinate ring C_n of the Grassmannian Gr(2,n) has finite Frobenius representation type when the characteristic p satisfies p≥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→∞ is the leading coefficient of the first free multiplicity, which we evaluate in closed form through Bernoulli numbers. For Gr(2,4) we find s_p(C_4)=(13p^2+8)/(5(3p^2+2)), and we calculate the analogous rational functions for Gr(2,5) and 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.
ER -