arXiv subjects

Heiko Dietrich

Publications and source records attributed to Heiko Dietrich.

34 records · Page 2Linked to original sources

arXiv2022

The isomorphism problem for plain groups is in $Σ_3^{\mathsf{P}}$

Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{é}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of \emph{plain groups}, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in $Σ_3^{\mathsf{P}}$. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).

math.GR↗

arXiv2021

Classification of four qubit states and their stabilisers under SLOCC operations

We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ on the Hilbert space $\mathcal{H}_4 = (\mathbb{C}^2)^{\otimes 4}$. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$-orbits on $\mathcal{H}_4$. It follows that an element of $(\mathbb{C}^2)^{\otimes 4}$ is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in $\mathrm{\mathop{SL}}(2,\mathbb{C})^4$. We also present a complete and irredundant classification of elements and stabilisers up to the action of ${\rm Sym}_4\ltimes\mathrm{\mathop{SL}}(2,\mathbb{C})^4$ where ${\rm Sym}_4$ permutes the four tensor factors of $(\mathbb{C}^2)^{\otimes 4}$.

quant-ph↗

arXiv2021

Galois trees in the graph of $p$-groups of maximal class

The investigation of the graph $\mathcal{G}_p$ associated with the finite $p$-groups of maximal class was initiated by Blackburn (1958) and became a deep and interesting research topic since then. Leedham-Green and McKay (1976-1984) introduced skeletons of $\mathcal{G}_p$, described their importance for the structural investigation of $\mathcal{G}_p$ and exhibited their relation to algebraic number theory. Here we go one step further: we partition the skeletons into so-called Galois trees and study their general shape. In the special case $p \geq 7$ and $p \equiv 5 \bmod 6$, we show that they have a significant impact on the periodic patterns of $\mathcal{G}_p$ conjectured by Eick, Leedham-Green, Newman and O'Brien (2013). In particular, we use Galois trees to prove a conjecture by Dietrich (2010) on these periodic patterns.

math.GR↗

arXiv2021

Polynomial-time isomorphism testing of groups of most finite orders

[PLEASE SEE COMMENT] We consider the isomorphism problem for finite abelian groups and finite meta-cyclic groups. We prove that for a dense set of positive integers $n$, isomorphism testing for abelian groups of black-box type of order $n$ can be done in time polynomial in $\log n$. We also prove that for a dense set of orders $n$ with given prime factors, one can test isomorphism for coprime meta-cyclic groups of black-box type of order $n$ in time polynomial in $\log n$. Prior methods for these two classes of groups have running times exponential in $\log n$.

math.GR↗

arXiv2021

Group isomorphism is nearly-linear time for most orders

We show that there is a dense set $\ourset\subseteq \mathbb{N}$ of group orders and a constant $c$ such that for every $n\in \ourset$ we can decide in time $O(n^2(\log n)^c)$ whether two $n\times n$ multiplication tables describe isomorphic groups of order $n$. This improves significantly over the general $n^{O(\log n)}$-time complexity and shows that group isomorphism can be tested efficiently for almost all group orders $n$. We also show that in time $O(n^2 (\log n)^c)$ it can be decided whether an $n\times n$ multiplication table describes a group; this improves over the known $O(n^3)$ complexity. Our complexities are calculated for a deterministic multi-tape Turing machine model. We give the implications to a RAM model in the promise hierarchy as well.

cs.CC↗

arXiv2021

A note on étale representations from nilpotent orbits

A linear étale representation of a complex algebraic group $G$ is given by a complex algebraic $G$-module $V$ such that $G$ has a Zariski-open orbit on $V$ and $\dim G=\dim V$. A current line of research investigates which étale representations can occur for reductive algebraic groups. Since a complete classification seems out of reach, it is of interest to find new examples of étale representations for such groups. The aim of this note is to describe two classical constructions of Vinberg and of Bala & Carter for nilpotent orbit classifications in semisimple Lie algebras, and to determine which reductive groups and étale representations arise in these constructions. We also explain in detail the relation between these two~constructions.

math.RT↗

arXiv2020

Universal Covers of Finite Groups

Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $\tilde H$ of a finite group $H$ by a direct sum of isomorphic simple $\mathbb{Z}_p H$-modules such that $H$ and $\tilde H$ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of $H$. Defining this covering group requires a study of the representation module, as introduced by Gaschütz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. An important application of our results is that they can be used to compute, for a given epimorphism $G\to H$ and simple $\mathbb{Z}_p H$-module $V$, the largest quotient of $G$ that maps onto $H$ with kernel isomorphic to a direct sum of copies of $V$. For this we also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group $H$, assuming a confluent rewriting system for $H$. To represent the corresponding group extensions on the computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.

math.GR↗

arXiv2020

Generation of finite groups with cyclic Sylow subgroups

Slattery (2007) described computational methods to enumerate, construct, and identify finite groups of squarefree order. We generalise Slattery's result to the class of finite groups that have cyclic Sylow subgroups and provide an implementation for the computer algebra system GAP.

math.GR↗

arXiv2020

On two group functors extending Schur multipliers

Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions and Schur multipliers. In this work we relate $K$ and $\tilde K$ to a group functor $τ$ arising in the construction of the non-abelian exterior square of a group. In contrast to $\tilde K$, there exist efficient algorithms for constructing $τ$, especially for polycyclic groups. Supported by computations with the computer algebra system GAP, we investigate when $K(G,3)$ is a quotient of $τ(G)$, and when $τ(G)$ and $\tilde K(G,3)$ are isomorphic.

math.GR↗

arXiv2019

Constructing cocyclic Hadamard matrices of order 4p

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.

math.CO↗

arXiv2019

Computing the real Weyl group

Let g be a semisimple Lie algebra over the real numbers. We describe an explicit combinatorial construction of the real Weyl group of g with respect to a given Cartan subalgebra. An efficient computation of this Weyl group is important for the classification of regular semisimple subalgebras, real carrier algebras, and real nilpotent orbits associated with g; the latter have various applications in theoretical physics.

math.RT↗

arXiv2018

Isomorphism testing of groups of cube-free order

A group $G$ has cube-free order if no prime to the third power divides $|G|$. We describe an algorithm that given two cube-free groups $G$ and $H$ of known order, decides whether $G\cong H$, and, if so, constructs an isomorphism $G\to H$. If the groups are input as permutation groups, then our algorithm runs in time polynomial in the input size, improving on the previous super-polynomial bound. An implementation of our algorithm is provided for the computer algebra system {\sf GAP}.

math.GR↗

arXiv2017

Small partial Latin squares that embed in an infinite group but not into any finite group

Suppose that $Y_1,Y_2,Y_3$ are finite sets and $P\subseteq Y_1\times Y_2\times Y_3$. We say that $P$ embeds in a group $G$ if there exist injective maps $ϕ_i\colon Y_i\rightarrow G$ for $i=1,2,3$ such that $ϕ_1(y_1)ϕ_2(y_2)=ϕ_3(y_3)$ for each $(y_1,y_2,y_3)\in P$. Hirsch and Jackson asked for the cardinality of the smallest $P$ that embeds in some infinite group but not into any finite group. We prove that the answer to their question is 12. Moreover, we show that there are 50 examples of cardinality 12, up to equivalence, and each of them embeds in the (infinite) Baumslag group $G=\langle a,b \mid b=[b,b^a]\rangle$. Our proof uses computations to answer questions about finitely presented groups which are known to be algorithmically undecidable in general.

math.GR↗

arXiv2016

Nilpotent orbits in real symmetric pairs and stationary black holes

In the classification of stationary solutions in extended supergravities with symmetric scalar manifolds, the nilpotent orbits of a real symmetric pair play an important role. In this paper we discuss two approaches to determining the nilpotent orbits of a real symmetric pair. We apply our methods to an explicit example, and thereby classify the nilpotent orbits of SL_2(R)^4 acting on the fourth tensor power of the natural 2-dimensional SL_2(R)-module. This makes it possible to classify all stationary solutions of the so-called STU-supergravity model.

math.RT↗

arXiv2016

Groups in which every non-abelian subgroup is self-centralizing

We study groups having the property that every non-abelian subgroup contains its centralizer. We describe various classes of infinite groups in this class, and address a problem of Berkovich regarding the classification of finite $p$-groups with the above property.

math.GR↗

arXiv2012

A computational approach to the Kostant-Sekiguchi correspondence

Let g be a real form of a simple complex Lie algebra. Based on ideas of Djokovic and Vinberg, we describe an algorithm to compute representatives of the nilpotent orbits of g using the Kostant-Sekiguchi correspondence. Our algorithms are implemented for the computer algebra system GAP and, as an application, we have built a database of nilpotent orbits of all real forms of simple complex Lie algebras of rank at most 8. In addition, we consider two real forms g and g' of a complex simple Lie algebra g^c with Cartan decompositions g= k+p and g'=k'+p'. We describe an explicit construction of an isomorphism g -> g', respecting the given Cartan decompositions, which fails if and only if g and g' are not isomorphic. This isomorphism can be used to map the representatives of the nilpotent orbits of g to other realisations of the same algebra.

math.RA↗