arXiv subjects

Joshua Maglione

Publications and source records attributed to Joshua Maglione.

17 recordsLinked to original sources

arXiv2025

Categorification of characteristic structures

We develop a representation theory of categories as a means to explore characteristic structures in algebra. Characteristic structures play a critical role in isomorphism testing of groups and algebras, and their construction and description often rely on specific knowledge of the parent object and its automorphisms. In many cases, questions of reproducibility and comparison arise. Here we present a categorical framework that addresses these questions. We prove that every characteristic structure is the image of a functor equipped with a natural transformation. This shifts the local description in the parent object to a global one in the ambient category. Through constructions in representation theory, such as tensor products, we can combine characteristic structure across multiple categories. Our results are constructive, stated in the language of a constructive type theory, which facilitates implementations in theorem checkers.

math.GR↗

arXiv2025

Symplectic Hecke eigenbases from Ehrhart polynomials

For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular invariant valuations. We show that, in even dimensions, these functions are in fact simultaneous symplectic Hecke eigenfunctions. We leverage this and apply the theory of spherical functions and their associated zeta functions to prove analytic, asymptotic, and combinatorial results about the arithmetic functions averaging $\ell$th Ehrhart coefficients.

math.CO↗

arXiv2025

Hall-Littlewood polynomials, affine Schubert series, and lattice enumeration

We introduce multivariate rational generating series called Hall-Littlewood-Schubert ($\mathsf{HLS}_n$) series. They are defined in terms of polynomials related to Hall-Littlewood polynomials and semistandard Young tableaux. We show that $\mathsf{HLS}_n$ series provide solutions to a range of enumeration problems upon judicious substitutions of their variables. These include the problem to enumerate sublattices of a $p$-adic lattice according to the elementary divisor types of their intersections with the members of a complete flag of reference in the ambient lattice. This is an affine analog of the stratification of Grassmannians by Schubert varieties. Other substitutions of $\mathsf{HLS}_n$ series yield new formulae for Hecke series and $p$-adic integrals associated with symplectic $p$-adic groups, and combinatorially defined quiver representation zeta functions. $\mathsf{HLS}_n$ series are $q$-analogs of Hilbert series of Stanley-Reisner rings associated with posets arising from parabolic quotients of Coxeter groups of type $\mathsf{B}$ with the Bruhat order. Special values of coarsened $\mathsf{HLS}_n$ series yield analogs of the classical Littlewood identity for the generating functions of Schur polynomials.

math.CO↗

arXiv2025

Smooth cuboids in group theory

A smooth cuboid can be identified with a $3\times 3$ matrix of linear forms, with coefficients in a field $K$, whose determinant describes a smooth cubic in the projective plane. To each such matrix one can associate a group scheme over $K$. We produce isomorphism invariants of these groups in terms of their adjoint algebras, which also give information on the number of their maximal abelian subgroups. Moreover, we give a characterization of the isomorphism types of the groups in terms of isomorphisms of elliptic curves and also give a description of the automorphism group. We conclude by applying our results to the determination of the automorphism groups and isomorphism testing of finite $p$-groups of class $2$ and exponent $p$ arising in this way.

math.GR↗

arXiv2024

The Poincaré-extended ab-index

Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended ab-index, which generalizes both the ab-index and the Poincaré polynomial. For posets admitting R-labelings, we give a combinatorial description of the coefficients of the extended ab-index, proving their nonnegativity. In the case of intersection posets of hyperplane arrangements, we prove the above conjecture of the second author and Voll as well as another conjecture of the second author and Kühne. We also define the pullback ab-index generalizing the cd-index of face posets for oriented matroids. Our results recover, generalize and unify results from Billera-Ehrenborg-Readdy, Bergeron-Mykytiuk-Sottile-van Willigenburg, Saliola-Thomas, and Ehrenborg. This connection allows us to translate our results into the language of quasisymmetric functions, and-in the special case of symmetric functions-make a conjecture about Schur positivity. A proof of this conjecture now appears an appendix by Ricky Ini Liu.

math.CO↗

arXiv2024

Ehrhart polynomials, Hecke series, and affine buildings

Given a lattice polytope $P$ and a prime $p$, we define a function from the set of primitive symplectic $p$-adic lattices to the rationals that extracts the $\ell$th coefficient of the Ehrhart polynomial of $P$ relative to the given lattice. Inspired by work of Gunnells and Rodriguez-Villegas in type $\mathsf{A}$, we show that these functions are eigenfunctions of a suitably defined action of the spherical symplectic Hecke algebra. Although they depend significantly on the polytope $P$, their eigenvalues are independent of $P$ and expressed as polynomials in $p$. We define local zeta functions that enumerate the values of these Hecke eigenfunctions on the vertices of the affine Bruhat--Tits buildings associated with $p$-adic symplectic groups. We compute these zeta functions by enumerating $p$-adic lattices by their elementary divisors and, simultaneously, one Hermite parameter. We report on a general functional equation satisfied by these local zeta functions, confirming a conjecture of Vankov.

math.CO↗

arXiv2022

On the geometry of flag Hilbert-Poincaré series for matroids

We extend the definition of coarse flag Hilbert--Poincaré series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by applying geometric and combinatorial tools related to their topes. In this case, we prove that the numerators of these series are coefficient-wise bounded below by the Eulerian polynomial and equality holds if and only if all topes are simplicial. Moreover this yields a sufficient criterion for non-orientability of matroids of arbitrary rank.

math.CO↗

arXiv2022

Flag Hilbert-Poincaré series of hyperplane arrangements and their Igusa zeta functions

We introduce and study a class of multivariate rational functions associated with hyperplane arrangements, called flag Hilbert-Poincaré series. These series are intimately connected with Igusa local zeta functions of products of linear polynomials, and their motivic and topological relatives. Our main results include a self-reciprocity result for central arrangements defined over fields of characteristic zero. We also prove combinatorial formulae for a specialization of the flag Hilbert-Poincaré series for irreducible Coxeter arrangements of types $\mathsf{A}$, $\mathsf{B}$, and $\mathsf{D}$ in terms of total partitions of the respective types. We show that a different specialization of the flag Hilbert-Poincaré series, which we call the coarse flag Hilbert-Poincaré series, exhibits intriguing nonnegativity features and - in the case of Coxeter arrangements - connections with Eulerian polynomials. For numerous classes and examples of hyperplane arrangements, we determine their (coarse) flag Hilbert-Poincaré series. Some computations were aided by a SageMath package we developed.

math.CO↗

arXiv2022

Tensor Isomorphism by conjugacy of Lie algebras

We introduce an algorithm to decide isomorphism between tensors. The algorithm uses the Lie algebra of derivations of a tensor to compress the space in which the search takes place to a so-called densor space. To make the method practicable we give a polynomial-time algorithm to solve a generalization of module isomorphism for a common class of Lie modules. As a consequence, we show that isomorphism testing is in polynomial time for tensors whose derivation algebras are classical Lie algebras and whose densor spaces are 1-dimensional. The method has been implemented in the Magma computer algebra system.

math.RA↗

arXiv2020

Compatible filters with isomorphism testing

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few features with the original group: e.g. the associated Lie ring may be trivial or arbitrarily large. We determine properties of filters such that the Lie ring and group are in bijection. We prove that, under such conditions, every isomorphism between groups is induced by an isomorphism between graded Lie rings.

math.GR↗

arXiv2020

A spectral theory for transverse tensor operators

Tensors are multiway arrays of data, and transverse operators are the operators that change the frame of reference. We develop the spectral theory of transverse tensor operators and apply it to problems closely related to classifying quantum states of matter, isomorphism in algebra, clustering in data, and the design of high performance tensor type-systems. We prove the existence and uniqueness of the optimally-compressed tensor product spaces over algebras, called \emph{densors}. This gives structural insights for tensors and improves how we recognize tensors in arbitrary reference frames. Using work of Eisenbud--Sturmfels on binomial ideals, we classify the maximal groups and categories of transverse operators, leading us to general tensor data types and categorical tensor decompositions, amenable to theorems like Jordan--Hölder and Krull--Schmidt. All categorical tensor substructure is detected by transverse operators whose spectra contain a Stanley--Reisner ideal, which can be analyzed with combinatorial and geometrical tools via their simplicial complexes. Underpinning this is a ternary Galois correspondence between tensor spaces, multivariable polynomial ideals, and transverse operators. This correspondence can be computed in polynomial time. We give an implementation in the computer algebra system \textsf{Magma}.

math.SP↗

arXiv2019

Exact sequences of inner automorphisms of tensors

We produce a long exact sequence whose terms are unit groups of associative algebras that behave as inner automorphisms of a given tensor. Our sequence generalizes known sequences for associative and non-associative algebras. In a manner similar to those, our sequence facilitates inductive reasoning about, and calculation of the groups of symmetries of a tensor. The new insights these methods afford can be applied to problems ranging from understanding algebraic structures to distinguishing entangled states in particle physics.

math.RA↗

arXiv2018

Enumerating isoclinism classes of semi-extraspecial groups

We enumerate the number of isoclinism classes of semi-extraspecial $p$-groups with derived subgroup of order $p^2$. To do this, we enumerate $\text{GL}(2, p)$-orbits of sets of irreducible, monic polynomials in $\mathbb{F}_p[x]$. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

math.GR↗

arXiv2016

Efficient characteristic refinements for finite groups

Filters were introduced by J.B. Wilson in 2013 to generalize work of Lazard with associated graded Lie rings. It holds promise in improving isomorphism tests, but the formulas introduced then were impractical for computation. Here, we provide an efficient algorithm for these formulas, and we demonstrate their usefulness on several examples of $p$-groups.

math.GR↗

arXiv2015

A fast isomorphism test for groups of genus 2

Motivated by the need for efficient isomorphism tests for finite groups, we present a polynomial-time method for deciding isomorphism within a class of groups that is well-suited to studying local properties of general finite groups. We also report on the performance of an implementation of the algorithm in the computer algebra system {\sc magma}.

math.GR↗

arXiv2015

Automorphisms of small prime power groups

If $f(p, n)$ is the number of pairwise nonisomorphic groups of order $p^n$, and $g(p,n)$ is the number of groups of order $p^n$ whose automorphism group is a $p$-group, then, for $n \leq 7$, we prove that the ratio $g(p,n)/f (p,n)$ is bounded away from 1 as the prime $p$ grows to infinity. In addition, we provide some data on the number of groups whose automorphism group is a group of prime power order, for primes no larger than 11.

math.GR↗

arXiv2015

Longer nilpotent series for classical unipotent subgroups

In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing $\mathbb{N}^d$-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field $\mathbb{Z}/p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $Θ(d^2)$ with nearly all factors having $p$-bounded order.

math.GR↗