arXiv subjects

Chayim Lowen

Publications and source records attributed to Chayim Lowen.

6 recordsLinked to original sources

arXiv2026

Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements

The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik--Solomon algebra. It is known that the defining ideal of the Orlik--Solomon algebra has a quadratic Gr\"obner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik--Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik--Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over $\mathbb{Q}$, and of any rank $\geq 3$. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik--Terao algebras coming from non-supersolvable arrangements.

math.CO↗

arXiv2026

Matroid flat counts can have many peaks

We disprove Rota's conjecture that the counts of flats in a matroid according to rank form a unimodal sequence. Furthermore, we show that this sequence can have arbitrarily many peaks. The construction starts by finding a generalized theta graph for which log-concavity fails severely. By taking direct sums, we break log-concavity in many places. We then use Whittle's $q$-lift construction to produce a matroid whose flat counts have many peaks.

math.CO↗

arXiv2026

Matroid flat counts are not unimodal

We give counterexamples to Rota's 1970 conjecture that the sequence counting flats of varying rank in a matroid is unimodal. More specifically, inspired by Larson's recent disproof of the stronger log-concavity conjecture of Mason, we explain a mechanism which turns failures of log-concavity for flats into failures of unimodality under suitable conditions.

math.CO↗

arXiv2026

Structural properties of Bia{\l}ynicki-Birula decompositions

We investigate several aspects of the Bialynicki-Birula decomposition of a smooth complete $\mathbb{G}_m$-variety with finite fixed locus. Our results include novel characterizations of when the Bialynicki-Birula decomposition is filterable or forms a stratification, showing that these properties are invariant under reversing the $\mathbb{G}_m$-action. We additionally classify the smooth projective toric varieties for which the Bialynicki-Birula decomposition either may or must be a stratification. Our study of $\mathbb{G}_m$-convexity and $\mathbb{G}_m$-rigidity -- properties recently introduced by Buch--Chaput--Perrin -- answers several questions posed in their $\textit{Equivariant rigidity of Richardson varieties}$. In particular, assuming only filterability of the decomposition, we show that the Bialynicki-Birula cell closures are determined by their $\mathbb{G}_m$-equivariant Chow classes.

math.AG↗

arXiv2026

On the asymptotic behavior of finite hyperfields

Hobby has recently shown that almost all finite hyperfields of even order fail to be the quotient of a field. Using a probabilistic argument, we extend this result to all orders: a finite hyperfield is almost always non-quotient. This confirms a conjecture of Baker--Jin. We show that in almost every finite hyperfield the sum of any four or more nonzero elements contains 0. We also give a precise asymptotic for the number of finite hyperfields on a given finite abelian group.

math.RA↗

arXiv2025

Simultaneous generating sets for flags

We prove that any triple of complete flags in $\mathbb R^d$ admits a common generating set of size $\lfloor 5d/3\rfloor$ and that this bound is sharp. This result extends the classical linear-algebraic fact -- a consequence of the Bruhat decomposition of $\text{GL}_d(\mathbb R)$ -- that any pair of complete flags in $\mathbb R^d$ admits a common generating set of size $d$. We also deduce an analogue for $m$-tuples of flags with $m>3$.

math.CO↗