Searcharxiv⌕ Search

arXiv subjects

Robert Shwartz

Publications and source records attributed to Robert Shwartz.

18 recordsLinked to original sources
arXiv2026

Higher Commutativity in Finite Groups, Rigidity, Extremal bounds, and Heisenberg-Type Families

For a finite group $G$ and an integer $r\ge 2$ let $$ P_r(G):=\frac{|Hom(\mathbb Z^r,G)|}{|G|^r}, $$ where $\Hom(\mathbb Z^r,G)$ is the set of pairwise commuting $r$-tuples in $G$. This paper studies rigidity and extremal behavior of the hierarchy $\{P_r(G)\}_{r\ge2}$, together with a low-rank representation-theoretic / TQFT counting bridge. The first main direction is cyclic-index rigidity: for groups with an abelian normal subgroup $A$ and cyclic quotient of order $\omega$, under a natural fixed-subgroup hypothesis we prove the exact all-rank formula $$ P_r(G)=\frac{1}{\omega^r}+\left(1-\frac{1}{\omega^r}\right)\left(\frac{|A\cap Z(G)|}{|A|}\right)^{r-1}, $$ which yields gap and rigidity statements for non-abelian abelian extensions of prime index. The second main direction is the class-$2$ exponent-$p$ world. We develop a symplectic reduction, obtain closed formulas when $|G'|=p$, and prove a closed all-$r$ hierarchy in the $\mathbb F_q$-Heisenberg family: \[ P_r(G)=q^{-2nr}\sum_{k=0}^{\min(n,r)}L_{n,k}(q)\prod_{i=0}^{k-1}(q^r-q^i). \] In particular, inside the $\mathbb F_q$-Heisenberg family the pair $(P_2(G),P_3(G))$ already determines the isoclinism class. Combining the cyclic-index formula with the known sharp upper bound for the multiple commutativity degree gives equality and near-extremal rigidity, including a stability gap near $11/32$ for commuting triples. At the low-rank end we also prove explicit class-number formulas for $P_3(G)$ and $P_4(G)$; these recover the simple-count formulas for the untwisted Drinfeld double and the untwisted quantum triple / double-loop-groupoid algebra.

math.GR↗
arXiv2026

Higher Commutativity in Finite Groups: Exact Asymptotics and Finite Spectrum

For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting r-tuples. We prove an exact dominant asymptotic for the number of homomorphisms from the free abelian group of rank r to G. The exponential base is the maximum order of an abelian subgroup of G, and the leading coefficient is the number of abelian subgroups of that order. As a consequence, the r-th root of the higher commuting probability tends to this maximum abelian-subgroup order divided by the order of G, while the r-th root of the orbit count tends to the maximum abelian-subgroup order itself. We also prove that the associated rank-generating series is rational and has a finite Dirichlet-spectrum expansion supported on abelian subgroup indices. This spectrum yields a finite linear recurrence, a finite-rank Hankel matrix, and an inverse finite-spectrum theorem: the tail of the hierarchy determines the full abelian-index spectrum. For split abelian extensions, we express the dominant base through fixed-subgroup geometry, and for abelian acting quotients, we obtain an exact subgroup-lattice formula. In the cyclic and coprime cases, this gives closed formulas for all spectral coefficients.

math.GR↗
arXiv2023

Simply-laced mixed-sign Coxeter groups, with an associate graph is a line or a simple cycle

In 2011 Eriko Hironaka introduced an interesting generalization of Coxeter groups, motivated by studying certain mapping classes. The generalization is by labeling the vertices of a Coxeter graph either by +1 or by -1, and then generalizing the standard geometric representation of the associated Coxeter group by concerning the labels of the vertices. The group which Hironaka get by that generalization is called mixed-sign Coxeter group. In this paper we classify the simply-laced mixed-sign Coxeter groups where the associated graph is either a line or a simple cycle. We show that all the defining relations of the mixed-sign Coxeter groups with the mentioned associated graph (either a line or a simple cycle) are squares or cubes of a product of conjugates of two generators of the mixed-sign Coxeter group and are strongly connected to the labels of the vertices of the associated graph.

math.GR↗
arXiv2023

Sequences over finite fields defined by OGS and BN-pair decompositions of PSL2(q) recursively

Factorization of groups into Zappa-Szep product, or more generally into k-fold Zappa-Szep product of its subgroups, is an interesting problem, since it eases the multiplication of two elements in a group, and has recently been applied for public-key cryptography as well. We give a generalization of the k-fold Zappa-Szep product of cyclic groups, which we call OGS decomposition. It is easy to see that existence of an OGS decomposition for all the composition factors of a non-abelian group G implies the existence of an OGS for G itself. Since the composition factors of a soluble group are cyclic groups, it obviously has an OGS decomposition. Therefore, the question of the existence of an OGS decomposition is interesting for non-soluble groups. The Jordan-Holder Theorem motivates us to consider an existence of an OGS decomposition for the finite simple groups. In 1993, Holt and Rowley showed that PSL_{2}(q) and PSL_{3}(q) can be expressed as a product of cyclic groups. In this paper, we consider an OGS decomposition of PSL_{2}(q) from a point of view different than that of Holt and Rowley. We look at its connection to the BN-pair decomposition of the group. This connection leads to sequences over F_{q}, which can be defined recursively, with very interesting properties, and which are closely connected to the Dickson and to the Chebyshev polynomials. Since every finite simple group of Lie-type has $BN-pair$ decomposition, the ideas of the paper might be generalized to further simple groups of Lie-type.

math.GR↗
arXiv2022

Generalization of the basis theorem for the B-type Coxeter groups

The OGS for non-abelian groups is an interesting generalization of the basis of finite abelian groups. The definition of OGS states that every element of a group has a unique presentation as a product of some powers of specific generators of the group, in a specific given order. In case of the symmetric groups Sn there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group, which is called the standard OGS of Sn. In this paper we generalize the standard OGS of Sn to the finite classical Coxeter group Bn. We describe the exchange laws for the generalized standard OGS of Bn, and we connect it to the Coxeter length and the descent set of Bn.

math.GR↗
arXiv2022

Generalization of the basis theorem for the D-type Coxeter groups

The OGS for non-abelian groups is an interesting generalization of the basis of finite abelian groups. The definition of OGS states that every element of a group has a unique presentation as a product of some powers of specific generators of the group, in a specific given order. In case of the symmetric groups Sn there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group, which is called the standard OGS of Sn. In this paper we generalize the standard OGS of Sn to the finite classical Coxeter group Dn. We describe the exchange laws for the generalized standard OGS of Dn, and we connect it to the Coxeter length of elements of Dn.

math.GR↗
arXiv2021

Zariski pairs of conic-line arrangements of degrees 7 and 8 via fundamental groups

We find a new Zariski pair with non-isomorphic fundamental groups that consists of degree $ 8 $ conic-line arrangements. Each arrangement has three conics and two lines. We use the Zariski-van Kampen Theorem and some known Coxeter groups to determine the fundamental groups. Two examples of degree $7$ Zariski pairs that were introduced in 2014 by the last named author, are given as well. They consist of a pair of conic-line arrangements with three conics in each (and thus, each has a single line) and a pair with two conics in each (and thus, each has three lines). We were able to provide alternative proof of the fact those are indeed Zariski pairs by our methods.

math.AG↗
arXiv2020

Signed Hultman Numbers and Signed Generalized Commuting Probability in Finite Groups

Let G be a finite group. Let pi be a permutation from S{n}. We study the distribution of probabilities of equality a{1} a{2} ...a{n-1}a{n}=a{pi{1}}^{epsilon{1}} a{pi_{2}}^{epsilon{2}}...a{pi{n-1}}^{epsilon_{n-1}} a_{pi_{n}}^{epsilon{n}}, when pi varies over all the permutations in S{n}, and epsilon{i} varies over the set {+1, -1}. By the paper "Hultman Numbers and Generalized Commuting Probability in Finite Groups" (2017), The case where all epsilon{i} are +1 led to a close connection to Hultman numbers. In this paper we generalize the results, permitting epsilon{i} to be -1. We describe the spectrum of the probabilities of signed permutation equalities in a finite group G. This spectrum turns out to be closely related to the partition of 2^{n}*n! into a sum of the corresponding signed Hultman numbers.

math.GR↗
arXiv2019

Generalization of the basis theorem for alternating groups

There were defined by R. Shwartz OGS for non-abelian groups, as an interesting generalization of the basis of finite abelian groups. The definition of OGS states that that every element of a group has a unique presentation as a product of some powers of the OGS, in a specific given order. In case of the symmetric groups S_{n} there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group. The OGS presentation helps us to find the Coxeter length and the descent set of an arbitrary element of the symmetric group. Therefore, it motivates us to generalize the OGS for the alternating subgroup of the symmetric group, which we define in this paper. We generalize also the exchange laws for the alternating subgroup, and we will show some interesting properties of it.

math.GR↗
arXiv2019

OGS canonical forms and exchange laws for the I and for the A-type Coxeter groups

We consider a generalization of the fundamental theorem of finitely generated abelian groups for some non-abelian groups, which is called OGS. First, we consider the dihedral group, which is a non-abelian extension of an abelian group by an involution. Then, we focus on a special case, where the abelian group is cyclic, which is the two-generated Coxeter group I{2}(m). We mention interesting connections between the reduced Coxeter presentation and a particular OGS canonical presentation, which we call the standard OGS canonical presentation. These connections motivate us to offer a generalization of the standard OGS to the A-type Coxeter group, which can be considered as the dual family to the I-type Coxeter groups. The n-1 generated A-type Coxeter groups can be considered as the symmetric group S{n} for an arbitrary n. We mention the standard and the dual-standard OGS of S{n}, where, The standard OGS canonical form of S{n} has a special interest in combinatorics, since in 2001, R. M. Adin, and Y. Roichman has proved that sum of the exponents in the canonical form is coincide with the major-index of the permutation, which is equi-disributed with the Coxeter length. In this paper we extend the results of Adin and Roichman very significantly, where we show interesting properties of the exchange laws, we define standard OGS elementary factorization, which connects between the standard OGS and the descent set of a permutation. Then, by using the standard OGS elementary factorization, we find a new explicit formula for the Coxeter length of an element of S{n}, and we give a new algorithm for the standard OGS canonical form and the descent set of the inverse element of an arbitrary element of S{n}.

math.CO↗
arXiv2014

Hultman Numbers and Generalized Commuting Probability in Finite Groups

Let $G$ be a finite group and $π$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}} \] when $π$ varies over all the permutations in $S_{n}$. The probability \[ Pr_π(G)=Pr(a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{π_{1}}a_{π_{2}}\cdots a_{π_{n-1}}a_{π_{n}}) \] is identical to $Pr_{1}^ω(G)$, with \[ ω=a_{1}a_{2}...a_{n-1}a_{n}a_{π_{1}}^{-1}a_{π_{2}}^{-1}\cdots a_{π_{n-1}}^{-1}a_{π_{n}}^{-1}, \] as it is defined in \cite{DasNath1} and \cite{NathDash1}. The notion of commutativity degree, or the probability of a permutation equality $a_{1}a_{2}=a_{2}a_{1}$, for which $n=2$ and $π=\langle2\;\;1\rangle$, was introduced and assessed by P. Erdös and P. Turan in \cite{ET} in 1968 and by W. H. Gustafson in \cite{G} in 1973. In \cite{G} Gustafson establishes a relation between the probability of $a_{1},a_{2}\in G$ commuting and the number of conjugacy classes in $G$. In this work we define several other parameters, which depend only on a certain interplay between the conjugacy classes of $G$, and compute the probabilities of general permutation equalities in terms of these parameters. It turns out that this probability, for a permutation $π$, depends only on the number $c(Gr(π))$ of the alternating cycles in the cycle graph $Gr(π)$ of $π$. The cycle graph of a permutation was introduced by V. Bafna and P. A. Pevzner in \cite{BP}. We describe the spectrum of the probabilities of permutation equalities in a finite group as $π$ varies over all the elements of $S_{n}$. This spectrum turns-out to be closely related to the partition of $n!$ into a sum of the corresponding Hultman numbers.

math.GR↗
arXiv2014

Weighted Coxeter graphs and generalized geometric representations of Coxeter Groups

We introduce the notion of weighted Coxeter graph and associate to it a certain generalization of the standard geometric representation of a Coxeter group. We prove sufficient conditions for faithfulness and non-faithfulness of such a representation. In the case when the weighted Coxeter graph is balanced we discuss how the generalized geometric representation is related to the numbers game played on the Coxeter graph.

math.CO↗
arXiv2010

Classification of Fundamental Groups of Galois Covers of Surfaces of Small Degree Degenerating to Nice Plane Arrangements

Let $X$ be a surface of degree $n$, projected onto $\mathbb{CP}^2$. The surface has a natural Galois cover with Galois group $S_n.$ It is possible to determine the fundamental group of a Galois cover from that of the complement of the branch curve of $X.$ In this paper we survey the fundamental groups of Galois covers of all surfaces of small degree $n \leq 4$, that degenerate to a nice plane arrangement, namely a union of $n$ planes such that no three planes meet in a line. We include the already classical examples of the quadric, the Hirzebruch and the Veronese surfaces and the degree $4$ embedding of $\mathbb{CP}^1 \times \mathbb{CP}^1,$ and also add new computations for the remaining cases: the cubic embedding of the Hirzebruch surface $F_1$, the Cayley cubic (or a smooth surface in the same family), for a quartic surface that degenerates to the union of a triple point and a plane not through the triple point, and for a quartic $4$-point. In an appendix, we also include the degree $8$ surface $\mathbb{CP}^1\times \mathbb{CP}^1$ embedded by the $(2,2)$ embedding, and the degree $2n$ surface embedded by the $(1,n)$ embedding, in order to complete the classification of all embeddings of $\mathbb{CP}^1 \times \mathbb{CP}^1,$ which was begun in \cite{15}.

math.AG↗
arXiv2008

On the excedance sets of colored permutations

We define the excedence set and the excedance word on $G_{r,n}$, generalizing a work of Ehrenborg and Steingrimsson and use the inclusion-exclusion principle to calculate the number of colored permutations having a prescribed excedance word. We show some symmetric properties as Log concavity and unimodality of a specific sequence of excedance words.

math.CO↗
arXiv2008

Recursions for Excedance number in some permutations groups

The excedance number for S_n is known to have an Eulerian distribution. Nevertheless, the classical proof uses descents rather than excedances. We present a direct recursive proof which seems to be folklore and extend it to the colored permutation groups G_r,n. The generalized recursion yields some interesting connection to Stirling numbers of the second kind. We also show some logconcavity result concerning a variant of the excedance number. Finally, we show that the generating function of the excedance number defined on G_r,n is symmetric.

math.CO↗
arXiv2007

Major Indices and Perfect Bases for Complex Reflection Groups

It is shown that, under mild conditions, a complex reflection group $G(r,p,n)$ may be decomposed into a set-wise direct product of cyclic subgroups. This property is then used to extend the notion of major index and a corresponding Hilbert series identity to these and other closely related groups.

math.CO↗
arXiv2004

Ordered Generating Systems of Finite Non-Abelian Groups

In this paper we define Ordered Generating System for finite non-abelian groups, which is a generalization of the basis theorem for finite abelian groups. We prove the following: If each composition factor of a group G has Ordered Generating System, then G has Ordered Generating System as well. Hence, it remains to prove that every finite simple group has Ordered Generating System. In this paper we prove that the sporadic group of Mathieu has the property of Ordered Generating System.

math.GR↗