Hilbert functions over the exterior algebra and f-vectors in higher rank II: Generalizing Linusson's count to sumsets of f-vectors
Jan Snellman
math.CO
Oct 10, 2026 · v1
TL;DR
The combinatorial content of Sections 5–9 is formalised in Lean 4 with Mathlib, sorry-free, and shipped as ancillary files.
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 $d_1 \le \dots \le d_r$, 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.
Problem
Count the achievable Hilbert functions of quotients F/M, where F is a graded free module of rank r over the exterior algebra and M is a monomial submodule. Each such Hilbert function is a shifted sum of f-vectors of simplicial complexes. The rank-one case is Linusson's 1999 count.
Approach
Two statistics of an f-vector are introduced: its top nonvanishing degree and the initial degree of the corresponding ideal. Their joint distribution is expressed through differences of Linusson's refined counts E^p(n,k). Amata–Crupi lex submodules give a canonical representative for each Hilbert function. This turns the sumset count into a count of chained tuples with a nearest-neighbour constraint. The combinatorial results are formalised in Lean 4 with Mathlib, relying only on standard axioms, and checked computationally in Python, SageMath, Macaulay2 and Julia.
Results
The count equals a product of r−1 transfer matrices, one per generator-degree gap, and the same holds for the proper variant. Consequences include a linear recurrence in r with exponential asymptotics Kλ^r for a positive constant gap, polynomiality in r when all gaps vanish, a threshold beyond which the count is a plain power, and polynomiality in the number of vertices.
