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Chris McDaniel

Publications and source records attributed to Chris McDaniel.

17 recordsLinked to original sources
arXiv2026

Some Remarks on Hessians and Wronskians

The purpose of this note is to elaborate on the apparent connection between Wronskians and Hessians. More generally, to a given subspace of homogeneous bivariate forms over the complex numbers, we associate two determinantal polynomials called the $W$-polynomial and the $\hat{W}$-polynomial. We give expansion and factorization formulas for these polynomials, and study their behavior under change of coordinates and duality. As an application, we give another proof of Iarrobino's theorem on the strong Lefschetz property for standard graded Artinian Gorenstein algebras in codimension two.

math.AC↗
arXiv2025

A Hessian criterion for totally nonnegative Toeplitz matrices and a theorem of Cattani

To a homogeneous real bivariate form we associate a family of Toeplitz matrices, as well as a family of auxiliary homogeneous real bivariate forms called higher Hessian polynomials. We show that for a given form, certain positivity properties of its higher Hessian polynomials imply total nonnegativity of its Toeplitz matrices. One of our main result turns out to be equivalent to a special case of a deep theorem from Hodge theory due to E. Cattani. Our other result uses a recent theorem of S. Karp and K. Purbhoo related to problems in real Schubert calculus.

math.AC↗
arXiv2024

Lattice Paths, Lefschetz Properties, and Almkvist's Conjecture in Two Variables

We study a certain two-parameter family of non-standard graded complete intersections $A(m,n)$. In case $n=2$, we show that $A(m,2)$ has the strong Lefschetz property and the complex Hodge-Riemann property if and only if $m$ is even. This supports a strengthening of a conjecture of Almkvist on the unimodality of the Hilbert function of $A(m,n)$.

math.AC↗
arXiv2023

Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Oriented Artinian Gorenstein Algebras in Codimension Two

A (standard graded) oriented Artinian Gorenstein algebra over the real numbers is uniquely determined by a real homogeneous polynomial called its Macaulay dual generator. We study the mixed Hodge-Riemann relations on oriented Artinian Gorenstein algebras for which we give a signature criterion on the higher mixed Hessian matrices of its Macaulay dual generator. Inspired by recent work of Brändén and Huh, we introduce a class of homogeneous polynomials in two variables called $i$-Lorentzian polynomials, and show that these are exactly the Macaulay dual generators of oriented Artinian Gorenstein algebras in codimension two satisfying mixed Hodge-Riemann relations up to degree $i$ on the positive orthant of linear forms. We further show that the set of $i$-Lorentzian polynomials of degree $d$ are in one-to-one correspondence with the set of totally nonnegative Toeplitz matrices of size depending on $i$ and $d$. A corollary is that all normally stable polynomials, i.e. polynomials whose normalized coefficients form a PF sequence, are $i$-Lorentzian. Another corollary is an analogue of Whitney's theorem for Toeplitz matrices, which appears to be new: the closure of the set of totally positive Toeplitz matrices, in the Euclidean space of all real matrices of a given size, is equal to the set of totally nonnegative Toeplitz matrices.

math.AC↗
arXiv2022

Artinian algebras and Jordan type

The Jordan type of an element $\ell$ of the maximal ideal of an Artinian k-algebra A acting on an A-module M of k-dimension n, is the partition of n given by the Jordan block decomposition of the multiplication map $m_\ell$ on M. In general the Jordan type has more information than whether the pair $(\ell,M)$ is strong or weak Lefschetz. We develop basic properties of the Jordan type and their loci for modules over graded or local Artinian algebras. We as well study the relation of generic Jordan type of $A$ to the Hilbert function of $A$. We introduce and study a finer invariant, the Jordan degree type. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We as well propose open problems.

math.AC↗
arXiv2021

Cohomological Blow Ups of Graded Artinian Gorenstein Algebras Along Surjective Maps

We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.

math.AC↗
arXiv2021

Principal Radical Systems, Lefschetz Properties and Perfection of Specht Ideals of Two-Rowed Partitions

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results in particular give a self-contained proof of Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by Etingof-Gorsky-Losev over the complex numbers using rational Cherednik algebras.

math.AC↗
arXiv2021

Connected Sums of Graded Artinian Gorenstein Algebras and Lefschetz Properties

A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties.

math.AC↗
arXiv2020

Free Extensions and Jordan type

Free extensions of commutative Artinian algebras were introduced by T. Harima and J. Watanabe. The Jordan type of a multiplication map $m$ by a nilpotent element of an Artinian algebra is the partition determining the sizes of the blocks in a Jordan matrix for $m$. We show that a free extension of the Artinian algebra $A$ with fibre $B$ is a deformation of the usual tensor product. This has consequences for the generic Jordan types of $A,B$ and $C$, showing that the Jordan type of $C$ is at least that of the usual tensor product in the dominance order. We give applications to algebras of relative coinvariants of linear group actions on a polynomial ring.

math.AC↗
arXiv2019

Artinian Gorenstein algebras that are free extensions over ${\sf k}[t]/(t^n)$, and Macaulay duality

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras $C$ over a base $A$ with fibre $B$. The free extensions are deformations of the usual tensor product, when $C$ is also Gorenstein, so are $A$ and $B$, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator $F$ for $C$ as a homogeneous "polynomial" in $T$ and the dual variables for $B$, and given the dual generator for $B$, we give sufficient conditions on $F$ that ensure that $C$ is a free extension of $A={\sf k}[t]/(t^n)$ with fiber $B$. We give examples that explore the sharpness of the statements. We also consider a special set of coinvariant algebras $C$ which are free extensions of $A$, but which do not satisfy the sufficient conditions of our main result.

math.AC↗
arXiv2018

Free extensions and Lefschetz properties, with an application to rings of relative coinvariants

Graded Artinian algebras can be regarded as algebraic analogues of cohomology rings (in even degrees) of compact topological manifolds. In this analogy, a free extension of a base ring with a fiber ring corresponds to a fiber bundle over a manifold. For rings, as with manifolds, it is a natural question to ask: to what extent do properties of the base and the fiber carry over to the extension? For example, if the base and fiber both satisfy a strong Lefschetz property, can we conclude the same for the extension? Or, more generally, can one determine the generic Jordan type for the extension given the generic Jordan types of the base and fiber? We address these questions using the relative coinvariant rings as prototypical models. We show that if $V$ is a vector space and if the subgroup $W$ of the general linear group Gl(V), is a non-modular finite reflection group and $K\subset W$ is a non parabolic reflection subgroup, then the relative coinvariant ring $R^K_W$ cannot have a linear element of strong Lefschetz Jordan type. However, we give examples where these rings $R^K_W$, some with non-unimodal Hilbert functions, nevertheless have (non-homogeneous) elements of strong Lefschetz Jordan type. Some of these examples give rise to open questions concerning Lefschetz properties of certain algebras $A(m,n)$, related to combinatorial questions proposed and partially answered by G. Almqvist.

math.AC↗
arXiv2016

Some Remarks on Watanabe's Bold Conjecture

At the 2015 Workshop on Lefschetz Properties of Artinian Algebras, Junzo Watanabe conjectured that every graded Artinian complete intersection algebra with the standard grading can be embedded into another such algebra cut out by quadratic generators. We verify this conjecture in the case where the defining polynomials split into linear factors.

math.AC↗
arXiv2014

Morse Theory on 1-Skeleta

Guillemin and Zara gave necessary and sufficient conditions under which Morse theoretic techniques could be used to construct an additive basis for the equivariant cohomology of a 1-skeleton that is either 3-independent or GKM. We show that their conditions remain valid for all 1-skeleta, 3-independent, GKM, or otherwise.

math.CO↗
arXiv2013

Generalized 1-skeleta and a lifting result

In their paper "1-skeleta, Betti numbers, and equivariant cohomology" Guillemin and Zara described some beautiful constructions enabling them to use Morse theory on a certain class 1-skeleta including 1-skeleta of simple polytopes. In this paper we extend some of the notions and constructions from that paper to a larger class of 1-skeleta that includes 1-skeleta of projected simple polytopes. As an application of these ideas we prove a lifting result for 1-skeleta, which yields a characterization of 1-skeleta coming from projected simple polytopes.

math.CO↗