Hilbert functions over the exterior algebra and f-vectors in higher rank II: Generalizing Linusson's count to sumsets of f-vectors
Abstract
Let E be the exterior algebra of an n-dimensional vector space over a field, let F be a graded free E-module with r generators of degrees d1 ≤ … ≤ dr, and let M range over the monomial submodules of Amata and Crupi. The Hilbert function of F/M is a shifted sum of f-vectors of simplicial complexes; the companion paper determines the convex hull of such sums, and here we count them. For r = 1 the count is Linusson's (1999); we show that no new recursion is needed for general r. Using two statistics of an f-vector -- its top nonvanishing degree and the initial degree of the corresponding ideal -- we prove that the number of achievable Hilbert functions is a product of r−1 transfer matrices, one per generator-degree gap, with entries differences of Linusson's refined counts Ep(n,k). The rank enters only as the length of a matrix product. The same holds for the proper variant. Consequences: a linear recurrence in r for constant gaps, with exact exponential asymptotics K λr when the gap is positive; polynomiality in r when all gaps vanish; a threshold beyond which the count is a plain power; and polynomiality in the number of vertices.
Related
- Hilbert functions over the exterior algebra and f-vectors in higher rank I: Amata-Crupi monomial modules, Kozlov polytopes, and r-vectors of simplicial complexes
- Schreyer Resolutions over the Exterior Algebra
- A Geometric View of Adaptive Cross Approximation via Exterior Algebra
- Plectic Lie Algebra Action on Hilbert modular varieties
- Rhaly and generalized Hilbert operators between spaces of analytic functions
BibTeX
@misc{snellman2026hilbert,
title = {{Hilbert functions over the exterior algebra and f-vectors in higher rank II: Generalizing Linusson's count to sumsets of f-vectors}},
author = {Jan Snellman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14985},
eprint = {2610.14985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14985},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hilbert functions over the exterior algebra and f-vectors in higher rank II: Generalizing Linusson's count to sumsets of f-vectors
AU - Jan Snellman
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14985
UR - https://arxiv.org/abs/2610.14985
AB - Let E be the exterior algebra of an n-dimensional vector space over a field, let F be a graded free E-module with r generators of degrees d_1 ≤ … ≤ d_r, and let M range over the monomial submodules of Amata and Crupi. The Hilbert function of F/M is a shifted sum of f-vectors of simplicial complexes; the companion paper determines the convex hull of such sums, and here we count them. For r = 1 the count is Linusson's (1999); we show that no new recursion is needed for general r. Using two statistics of an f-vector -- its top nonvanishing degree and the initial degree of the corresponding ideal -- we prove that the number of achievable Hilbert functions is a product of r−1 transfer matrices, one per generator-degree gap, with entries differences of Linusson's refined counts E^p(n,k). The rank enters only as the length of a matrix product. The same holds for the proper variant. Consequences: a linear recurrence in r for constant gaps, with exact exponential asymptotics K λ^r when the gap is positive; polynomiality in r when all gaps vanish; a threshold beyond which the count is a plain power; and polynomiality in the number of vertices.
ER -