*Tensor Isomorphism by derivations and densors*, with Peter A. Brooksbank, James B. Wilson, submitted.Abstract ###### We introduce an isomorphism test for structures having a distributive type property, such as tensors, rings, and nilpotent groups. The test applies a universal construction to generalize several existing isomorphism tests, recovering these as special cases. Whereas earlier methods exploit associative algebras and their representations, the new approach is based on Lie algebras, leading to polynomial-time isomorphism tests for new families.

*A spectral theory for transverse tensor operators*, with Uriya A. First, James B. Wilson, submitted.Abstract ###### 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 $\textsf{Magma}$.

*Compatible filters with isomorphism testing*, J. Pure Appl. Algebra, to appear.Abstract ###### 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.

*Enumerating isoclinism classes of semi-extraspecial groups*, with Mark L. Lewis, Proc. Edinb. Math. Soc. (2),**63**(2020), no. 2, 426–442.Abstract ###### 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.

*Exact sequences of inner automorphisms of tensors*, with Peter A. Brooksbank, James B. Wilson, J. Algebra,**545**(2020), 43–63.Abstract ###### 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.

*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.Abstract ###### 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.

*Efficient characteristic refinements for finite groups*, J. Symbolic Comput.,**80**(2017), part 2, 511–520.Abstract ###### 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.

*Most small $p$-groups have an automorphism of order 2*, Arch. Math. (Basel),**108**(2017), no. 3, 225–232.Abstract ###### 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 $n = 6,7$.

*Longer nilpotent series for classical unipotent groups*, J. Grp. Theory,**18**(2015), no. 4, 569–585.Abstract ###### 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.Abstract ###### 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.

- Isomorphism, derivations, and Lie representations, Groups, representations and applications: new perspectives, Isaac Newton Institute, Cambridge, United Kingdom, February 2020.
- Multilinear tools for groups, Groups, representations and applications: computational and algorithmic methods, Isaac Newton Institute, Cambridge, United Kingdom, January 2020.
- Computing order zeta functions via resolution of singularities, Buildings, Varieties, and Applications, MPI für Mathematik in den Naturwissenschaften Leipzig, Leipzig, Germany, November 2019.
- A Tensor Playground: a demonstration of TensorSpace, Tensors: Algebra-Computation-Applications, University of Colorado Boulder, Boulder, Colorado, June 2019.
- Multilinear tools through filters on groups, Groups and Geometries, University of Auckland, Auckland, New Zealand, January 2019.
- Refinements and general filters for groups, Logic and Algorithms in Group Theory, Hausdorff Institute for Mathematics, Bonn, Germany, November 2018.