Graded relations on crossed products
We classify crossed product gradings for arbitrary groups and fields up to several equivalence relations in terms of group actions and their orbits.
math.RA↗
arXiv subjects
Ofir Schnabel
Publications and source records attributed to Ofir Schnabel.
Quotient gradings and the intrinsic fundamental group
Quotient grading classes are essential participants in the computation of the intrinsic fundamental group $π_1(A)$ of an algebra $A$. In order to study quotient gradings of a finite-dimensional semisimple complex algebra $A$ it is sufficient to understand the quotient gradings of twisted gradings. We establish the graded structure of such quotients using Mackey's obstruction class. Then, for matrix algebras $A=M_n(\mathbb{C})$ we tie up the concepts of braces, group-theoretic Lagrangians and elementary crossed products. We also manage to compute the intrinsic fundamental group of the diagonal algebras $A=\mathbb{C} ^4$ and $A=\mathbb{C} ^5$.
math.RA↗
Units of twisted group rings and their correlations to classical group rings
This paper is centered around the classical problem of extracting properties of a finite group $G$ from the ring isomorphism class of its integral group ring $\mathbb{Z} G$. This problem is considered via describing the unit group $\mathcal{U}( \mathbb{Z} G)$ generically for a finite group. Since the $`90s$ several well known generic constructions of units are known to generate a subgroup of finite index in $\mathcal{U}(\mathbb{Z } G)$ if $\mathbb{Q} G$ does not have so-called exceptional simple epimorphic images, e.g. $M_2 (\mathbb{Q})$. However it remained a major open problem to find a {\it generic} construction under the presence of the latter type of simple images. In this article we obtain such generic construction of units. Moreover, this new construction also exhibits new properties, such as providing generically free subgroups of large rank. As an application we answer positively for several classes of groups recent conjectures on the rank and the periodic elements of the abelianisation $\mathcal{U}(\mathbb{Z} G)^{ab}$. To obtain all this, we investigate the group ring $R \Gamma$ of an extension $\Gamma$ of some normal subgroup $N$ by a group $G$, over a domain $R$. More precisely, we obtain a direct sum decomposition of the (twisted) group algebra of $\Gamma$ over the fraction field $F$ of $R$ in terms of various twisted group rings of $G$ over finite extensions of $F$. Furthermore, concrete information on the kernel and cokernel of the associated projections is obtained. Along the way we also launch the investigations of the unit group of twisted group rings and of $\mathcal{U}( R\Gamma)$ via twisted group rings.
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Categories and weak equivalences of graded algebras
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. Each group grading on an algebra can be weakly equivalent to G-gradings for many different groups G, however it turns out that there is one distinguished group among them called the universal group of the grading. In this paper we study categories and functors related to the notion of weak equivalence of gradings. In particular, we introduce an oplax 2-functor that assigns to each grading its support and show that the universal grading group functor has neither left nor right adjoint.
math.RA↗
On weak equivalences of gradings
When one studies the structure (e.g. graded ideals, graded subspaces, radicals, ...) or graded polynomial identities of graded algebras, the grading group itself does not play an important role, but can be replaced by any other group that realizes the same grading. Here we come to the notion of weak equivalence of gradings: two gradings are weakly equivalent if there exists an isomorphism between the graded algebras that maps each graded component onto a graded component. The following question arises naturally: when a group grading on a finite dimensional algebra is weakly equivalent to a grading by a finite group? It turns out that this question can be reformulated purely group theoretically in terms of the universal group of the grading. Namely, a grading is weakly equivalent to a grading by a finite group if and only if the universal group of the grading is residually finite with respect to a special subset of the grading group. The same is true for all the coarsenings of the grading if and only if the universal group of the grading is hereditarily residually finite with respect to the same subset. We show that if $n\geqslant 349$, then on the full matrix algebra $M_n(F)$ there exists an elementary group grading that is not weakly equivalent to any grading by a finite (semi)group, and if $n\leqslant 3$, then any elementary grading on $M_n(F)$ is weakly equivalent to an elementary grading by a finite group.
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The Herzog-Schönheim Conjecture for small groups and harmonic subgroups
We prove that the Herzog-Schönheim Conjecture holds for any group $G$ of order smaller than $1440$. In other words we show that in any non-trivial coset partition $\{g_i U_i\}_{i=1}^n $ of $G$ there exist distinct $1 \leq i, j \leq n$ such that $[G:U_i]=[G:U_j]$. We also study interaction between the indices of subgroups having cosets with pairwise trivial intersection and harmonic integers. We prove that if $U_1$,...,$U_n$ are subgroups of $G$ which have pairwise trivially intersecting cosets and $n \leq 4$ then $[G:U_1]$,...,$[G:U_n]$ are harmonic integers.
math.GR↗
A different approach to the Fraenkel Conjecture for low $n$ values
We present a new approach to deal with Fraenkel's conjecture, which describes how the integers can be partitioned into sets of rational Beatty sequences, in the case where the numerators of the moduli are equal. We use this approach to give a new proof of the known $n=4$ case when the numerators are equal.
math.NT↗
Twisted group ring isomorphism problem
We propose and study a variation of the classical isomorphism problem for group rings in the context of projective representations. We formulate several weaker conditions following from our notion and give all logical connections between these condition by studying concrete examples. We introduce methods to study the problem and provide results for various classes of groups, including abelian groups, groups of central type, $p$-groups of order $p^4$ and groups of order $p^2q^2$, where $p$ and $q$ denote different primes.
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Groups of central type, maximal Connected Gradings and Intrinsic Fundamental Groups of Complex Semisimple Algebras
Maximal connected grading classes of a finite-dimensional algebra $A$ are in one-to-one correspondence with Galois covering classes of $A$ which admit no proper Galois covering and therefore are key in computing the intrinsic fundamental group $π_1(A)$. Our first concern here is the algebras $A=M_n(\mathbb{C})$. Their maximal connected gradings turn out to be in one-to-one correspondence with the Aut$(G)$-orbits of non-degenerate classes in $H^2(G,\C^*)$, where $G$ runs over all groups of central type whose orders divide $n^2$. We show that there exist groups of central type $G$ such that $H^2(G,\C^*)$ admits more than one such orbit of non-degenerate classes. We compute the family $Λ$ of positive integers $n$ such that there is a unique group of central type of order $n^2$, namely $C_n\times C_n$. The family $Λ$ is of square-free integers and contains all prime numbers. It is obtained by a full description of all groups of central type whose orders are cube-free. We establish the maximal connected gradings of all finite dimensional semisimple complex algebras using the fact that such gradings are determined by dimensions of complex projective representations of finite groups. In some cases we give a description of the corresponding fundamental groups.
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Simple twisted group algebras of dimension $p^4$ and their semi-centers
For simple twisted group algebra over a group $G$, if $G^{\shortmid}$ is Hall subgroup of $G$ then the semi-center is simple. Simple twisted groups algebras correspond to groups of central type. We classify all groups of central type of order $p^4$ where $p$ is prime and use this to show that for odd primes $p$ there exists a unique group $G$ of order $p^4$ such that there exists simple twisted group algebra over $G$ with a commutative semi-center. Moreover, if $1< |G|< 64$, then the semi-center of simple twisted group algebras over $G$ is non-commutative and this bounds are strict.
math.RA↗
Classification of groups of central type of order $p^4$
The classification of abelian groups of central type is well known. However, the description of non-abelian groups of central type which are known to be solvable, is far from being understood. In this paper we classify all groups of central type of order $p^4$ where $p$ is a prime number.
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On the reducibility of exact covering systems
There exist irreducible exact covering systems (ECS). These are ECS which are not a proper split of a coarser ECS. However, an ECS admiting a maximal modulus which is divisible by at most two distinct primes, primely splits a coarser ECS. As a consequence, if all moduli of an ECS $A$, are divisible by at most two distinct primes, then $A$ is natural. That is, $A$ can be formed by iteratively splitting the trivial ECS.
math.CO↗
A graph-theoretic approach for comparing dimensions of components in simply-graded algebras
Any simple group-grading of a finite dimensional complex algebra induces a natural family of digraphs. We prove that $|E\circ E^{\text{op}}\cup E^{\text{op}}\circ E|\geq |E|$ for any digraph $Γ=(V,E)$ without parallel edges, and deduce that for any simple group-grading, the dimension of the trivial component is maximal.
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