Research
Interests
My research interests are in asymptotic and computational algebra, algebraic combinatorics, and tensors. I develop efficient algorithms to aid in various isomorphism problems—in particular for nilpotent groups, which is a known bottleneck in the Group Isomorphism Problem. The Group Isomorphism Problem is closely related to the Tensor Isomorphism Problem, so I am also interested in tensors, their structure, and their applications to algebra. I also apply combinatorial tools to understand and compute certain $p$-adic integrals coming from zeta functions of groups and rings and Igusa's zeta function. These can be used to better understand enumerative aspects of groups, rings, and algebras such as the number of finite-index subgroups of a group.
PhD Students and Postdocs
The following are general themes of some possible projects.
- Compute and understand analytic and arithmetic properties of zeta functions of algebraic objects like groups, rings, and modules.
- Describe the poles and their orders of Igusa zeta functions associated with hyperplane arrangements.
- Develop efficient algorithms for the Group and Tensor Isomorphism Problems.
- Describe structural and algebraic properties of tensors.
I am also interested in applying for PhD Scholarships and Postdoctoral Fellowships from the Irish Research Council or Marie Skłowdowska-Curie Actions. Please contact me if you are interested in applying.
Research Publications
Preprints
- Isomorphism invariant metrics,
with
P. A. Brooksbank,
E. A. O'Brien,
J. B. Wilson,
submitted.
Within a category $\mathtt{C}$, having objects $\mathtt{C}_0$, it may be instructive to know not only that two objects are non-isomorphic, but also how far from being isomorphic they are. We introduce pseudo-metrics $d:\mathtt{C}_0 \times \mathtt{C}_0 \to [0,\infty]$ with the property that $x\cong y$ implies $d(x,y)=0$. We also give a canonical construction that associates to each isomorphism invariant a pseudo-metric satisfying that condition. This guarantees a large source of isomorphism invariant pseudo-metrics. We examine such pseudo-metrics for invariants in various categories.
- Hall–Littlewood polynomials, affine Schubert series, and lattice enumeration,
with
Christopher Voll,
submitted.
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.
- A spectral theory for transverse tensor operators,
with
Uriya A. First,
James B. Wilson,
submitted.
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 *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 $\textsf{Magma}$.
Within a category $\mathtt{C}$, having objects $\mathtt{C}_0$, it may be instructive to know not only that two objects are non-isomorphic, but also how far from being isomorphic they are. We introduce pseudo-metrics $d:\mathtt{C}_0 \times \mathtt{C}_0 \to [0,\infty]$ with the property that $x\cong y$ implies $d(x,y)=0$. We also give a canonical construction that associates to each isomorphism invariant a pseudo-metric satisfying that condition. This guarantees a large source of isomorphism invariant pseudo-metrics. We examine such pseudo-metrics for invariants in various categories.
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.
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 *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 $\textsf{Magma}$.
Published
- The Poincaré-extended $\mathbf{ab}$-index,
with
Galen Dorpalen-Barry,
Christian Stump,
to appear in J. Lond. Math. Soc. (2), with appendix by Ricky Ini Liu.
Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $\mathbf{ab}$-index, which generalizes both the $\mathbf{ab}$-index and the Poincaré polynomial. For posets admitting $R$-labelings, we give a combinatorial description of the coefficients of the extended $\mathbf{ab}$-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 $\mathbf{ab}$-index generalizing the cd-index of face posets for oriented matroids. This recovers, generalizes and unifies results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg.
- Smooth cuboids in group theory,
with
Mima Stanojkovski,
to appear in Algebra Number 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.
- Flag Hilbert–Poincaré series of hyperplane arrangements and Igusa zeta functions,
with
Christopher Voll,
to appear in Israel J. Math.
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.
- The Poincaré-extended $\mathbf{ab}$-index,
with
Galen Dorpalen-Barry,
Christian Stump,
Sémin. Lothar. Comb., 91B (2024),
extended abstract.
Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $\mathbf{ab}$-index, which generalizes both the $\mathbf{ab}$-index and the Poincaré polynomial. For posets admitting $R$-labelings, we give a combinatorial description of the coefficients of the extended $\mathbf{ab}$-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 $\mathbf{ab}$-index generalizing the cd-index of face posets for oriented matroids. This recovers, generalizes and unifies results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg.
- Ehrhart polynomials, Hecke series, and affine buildings,
with
Claudia Alfes,
Christopher Voll,
Sémin. Lothar. Comb., 91B (2024),
extended abstract.
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.
- On the geometry of flag Hilbert–Poincaré series for matroids,
with
Lukas Kühne,
Algebr. Comb., 6 (2023),
no. 3,
623–638.
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.
- Tensor Isomorphism by conjugacy of Lie algebras,
with
Peter A. Brooksbank,
James B. Wilson,
J. Algebra, 604 (2022),
790–807.
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.
- On the geometry of flag Hilbert–Poincaré series for matroids,
with
Lukas Kühne,
Sémin. Lothar. Comb., 86B (2022),
extended abstract.
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.
- Flag Hilbert–Poincaré series of hyperplane arrangements and Igusa zeta functions,
with
Christopher Voll,
Sémin. Lothar. Comb., 86B (2022),
extended abstract.
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.
- Compatible filters with isomorphism testing,
J. Pure Appl. Algebra, 225 (2021),
no. 3,
106528–106555.
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.
- Exact sequences of inner automorphisms of tensors,
with
Peter A. Brooksbank,
James B. Wilson,
J. Algebra, 545 (2020),
43–63.
We produce a long exact sequence of unit groups of associative algebras that behave as automorphisms of tensors in a manner similar to inner automorphisms for associative algebras. Analogues for Lie algebras of derivations of a tensor are also derived. These sequences, which are basis invariants of the tensor, generalize similar ones used for associative and non-associative algebras; they similarly facilitate inductive reasoning about, and calculation of the groups of symmetries of a tensor. The sequences can be used for problems as diverse as understanding algebraic structures to distinguishing entangled states in particle physics.
- Enumerating isoclinism classes of semi-extraspecial groups,
with
Mark L. Lewis,
Proc. Edinb. Math. Soc. (2), 63 (2020),
no. 2,
426–442.
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.
- Most small $p$-groups have an automorphism of order 2,
Arch. Math. (Basel), 108 (2017),
no. 3,
225–232.
Let $f(p, n)$ be the number of pairwise nonisomorphic $p$-groups of order $p^n$ , and let $g(p, n)$ be the number of groups of order $p^n$ whose automorphism group is a $p$-group. We prove that the limit, as $p$ grows to infinity, of the ratio $g(p, n) / f(p, n)$ equals $1/3$ for $n = 6,7$.
- Efficient characteristic refinements for finite groups,
J. Symbolic Comput., 80 (2017),
part 2,
511–520.
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.
- A fast isomorphism test for groups whose Lie algebra has genus 2,
with
Peter A. Brooksbank,
James B. Wilson,
J. Algebra, 473 (2017),
545–590.
Motivated by the desire for better isomorphism tests for finite groups, we present a polynomial-time algorithm for deciding isomorphism within a class of $p$-groups that is well-suited to studying local properties of general groups. We also report on the performance of an implementation of the algorithm in the computer algebra system Magma.
- Longer nilpotent series for classical unipotent groups,
J. Grp. Theory, 18 (2015),
no. 4,
569–585.
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 $\Theta(d^2)$ with nearly all factors having $p$-bounded order.
- Economical generating sets for the symmetric and alternating groups consisting of cycles of a fixed length,
with
Scott Annin,
J. Algebra Appl., 11 (2012),
no. 6,
1250110–1250118.
The symmetric group $S_n$ and the alternating group $A_n$ are groups of permutations on the set $\{0, 1, 2, \ldots , n - 1\}$ whose elements can be represented as products of disjoint cycles (the representation is unique up to the order of the cycles). In this paper, we show that whenever $n \geq k \geq 2$, the collection of all $k$-cycles generates $S_n$ if $k$ is even, and generates $A_n$ if $k$ is odd. Furthermore, we algorithmically construct generating sets for these groups of smallest possible size consisting exclusively of $k$-cycles, thereby strengthening results in [O. Ben-Shimol, The minimal number of cyclic generators of the symmetric and alternating groups, Commun. Algebra 35 (10) (2007) 3034–3037]. In so doing, our results find importance in the context of theoretical computer science, where efficient generating sets play an important role.
Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $\mathbf{ab}$-index, which generalizes both the $\mathbf{ab}$-index and the Poincaré polynomial. For posets admitting $R$-labelings, we give a combinatorial description of the coefficients of the extended $\mathbf{ab}$-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 $\mathbf{ab}$-index generalizing the cd-index of face posets for oriented matroids. This recovers, generalizes and unifies results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg.
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.
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.
Motivated by a conjecture concerning Igusa local zeta functions for intersection posets of hyperplane arrangements, we introduce and study the Poincaré-extended $\mathbf{ab}$-index, which generalizes both the $\mathbf{ab}$-index and the Poincaré polynomial. For posets admitting $R$-labelings, we give a combinatorial description of the coefficients of the extended $\mathbf{ab}$-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 $\mathbf{ab}$-index generalizing the cd-index of face posets for oriented matroids. This recovers, generalizes and unifies results from Billera–Ehrenborg–Readdy, Bergeron–Mykytiuk–Sottile–van Willigenburg, Saliola–Thomas, and Ehrenborg.
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.
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.
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.
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.
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.
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.
We produce a long exact sequence of unit groups of associative algebras that behave as automorphisms of tensors in a manner similar to inner automorphisms for associative algebras. Analogues for Lie algebras of derivations of a tensor are also derived. These sequences, which are basis invariants of the tensor, generalize similar ones used for associative and non-associative algebras; they similarly facilitate inductive reasoning about, and calculation of the groups of symmetries of a tensor. The sequences can be used for problems as diverse as understanding algebraic structures to distinguishing entangled states in particle physics.
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.
Let $f(p, n)$ be the number of pairwise nonisomorphic $p$-groups of order $p^n$ , and let $g(p, n)$ be the number of groups of order $p^n$ whose automorphism group is a $p$-group. We prove that the limit, as $p$ grows to infinity, of the ratio $g(p, n) / f(p, n)$ equals $1/3$ for $n = 6,7$.
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.
Motivated by the desire for better isomorphism tests for finite groups, we present a polynomial-time algorithm for deciding isomorphism within a class of $p$-groups that is well-suited to studying local properties of general groups. We also report on the performance of an implementation of the algorithm in the computer algebra system Magma.
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 $\Theta(d^2)$ with nearly all factors having $p$-bounded order.
The symmetric group $S_n$ and the alternating group $A_n$ are groups of permutations on the set $\{0, 1, 2, \ldots , n - 1\}$ whose elements can be represented as products of disjoint cycles (the representation is unique up to the order of the cycles). In this paper, we show that whenever $n \geq k \geq 2$, the collection of all $k$-cycles generates $S_n$ if $k$ is even, and generates $A_n$ if $k$ is odd. Furthermore, we algorithmically construct generating sets for these groups of smallest possible size consisting exclusively of $k$-cycles, thereby strengthening results in [O. Ben-Shimol, The minimal number of cyclic generators of the symmetric and alternating groups, Commun. Algebra 35 (10) (2007) 3034–3037]. In so doing, our results find importance in the context of theoretical computer science, where efficient generating sets play an important role.