Permutation Groups Research Articles

Symmetry, gauge freedoms, and the interpretability of sequence-function relationships.

Quantitative models that describe how biological sequences encode functional activities are ubiquitous in modern biology. One important aspect of these models is that they commonly exhibit gauge freedoms, i.e., directions in parameter space that do not affect model predictions. In physics, gauge freedoms arise when physical theories are formulated in ways that respect fundamental symmetries. However, the connections that gauge freedoms in models of sequence-function relationships have to the symmetries of sequence space have yet to be systematically studied. Here we study the gauge freedoms of models that respect a specific symmetry of sequence space: the group of position-specific character permutations. We find that gauge freedoms arise when model parameters transform under redundant irreducible matrix representations of this group. Based on this finding, we describe an "embedding distillation" procedure that enables analytic calculation of the number of independent gauge freedoms, as well as efficient computation of a sparse basis for the space of gauge freedoms. We also study how parameter transformation behavior affects parameter interpretability. We find that in many (and possibly all) nontrivial models, the ability to interpret individual model parameters as quantifying intrinsic allelic effects requires that gauge freedoms be present. This finding establishes an incompatibility between two distinct notions of parameter interpretability. Our work thus advances the understanding of symmetries, gauge freedoms, and parameter interpretability in sequence-function relationships.

Permutation of National Identification Number for a Better Security in Communication Channel

This study proposes a new method to enhance the security of transmitting national identification number via various communication channels like SMS, email, and Box office. It introduces a permutation technique utilizing the Gamma 1 Non-Deranged Permutation group, ensuring the confidentiality of the information. By employing 11 factorial (11!) permutations, the national identification number is transformed into a mixed format for unintended recipients, yet remains easily decrypted by the intended recipient. This approach mitigates concerns regarding data interception and unauthorized access, providing a robust solution for secure information transmission across diverse media channels.

Entanglement-invertible channels

In a well-known result [R. Werner, J. Phys. A: Math. Gen. 34(35), 7081 (2001)], Werner classified all tight quantum teleportation and dense coding schemes, showing that they correspond to unitary error bases. Here tightness is a certain dimensional restriction: the quantum system to be teleported and the entangled resource must be of dimension d, and the measurement must have d2 outcomes. Here we generalise this classification so as to remove the dimensional restriction altogether, thereby resolving an open problem raised in that work. In fact, we classify not just teleportation and dense coding schemes, but entanglement-reversible channels. These are channels between finite-dimensional C*-algebras which are reversible with the aid of an entangled resource state, generalising ordinary reversibility of a channel. We show that such channels correspond to families of linear maps which are bi-isometric with respect to a duality defined by the resource state. In particular, in Werner’s classification, a bijective correspondence between tight teleportation and dense coding schemes was shown: swapping Alice and Bob’s operations turns a teleportation scheme into a dense coding scheme and vice versa. We observe that this property generalises ordinary invertibility of a channel; we call it entanglement-invertibility. We show that entanglement-invertible channels are precisely the quantum bijections previously studied in noncommutative topology [B. Musto et al., J. Math. Phys. 59(8), 081706 (2018)], and therefore admit a classification in terms of Wang’s quantum permutation group [S. Wang, Commun. Math. Phys. 195, 195–211 (1998)].

Transitivity in finite general linear groups

It is known that the notion of a transitive subgroup of a permutation group G extends naturally to subsets of G. We consider subsets of the general linear group GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} acting transitively on flag-like structures, which are common generalisations of t-dimensional subspaces of Fqn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_q^n$$\\end{document} and bases of t-dimensional subspaces of Fqn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_q^n$$\\end{document}. We give structural characterisations of transitive subsets of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} using the character theory of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} and interpret such subsets as designs in the conjugacy class association scheme of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document}. In particular we generalise a theorem of Perin on subgroups of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} acting transitively on t-dimensional subspaces. We survey transitive subgroups of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document}, showing that there is no subgroup of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} with 1<t<n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<t<n$$\\end{document} acting transitively on t-dimensional subspaces unless it contains SL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SL}\\,}}(n,q)$$\\end{document} or is one of two exceptional groups. On the other hand, for all fixed t, we show that there exist nontrivial subsets of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} that are transitive on linearly independent t-tuples of Fqn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_q^n$$\\end{document}, which also shows the existence of nontrivial subsets of GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document} that are transitive on more general flag-like structures. We establish connections with orthogonal polynomials, namely the Al-Salam–Carlitz polynomials, and generalise a result by Rudvalis and Shinoda on the distribution of the number of fixed points of the elements in GL(n,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{GL}\\,}}(n,q)$$\\end{document}. Many of our results can be interpreted as q-analogs of corresponding results for the symmetric group.

Discrete global symmetries: gauging and twisted compactification

Discrete global symmetries of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs are studied via two operations: gauging and twisted compactification. We consider gauging of discrete symmetries in several well-known 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs, including SU(n) SQCD with 2n flavors, theories of class S\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{S} $$\\end{document} of type A2n−1, and Argyres-Douglas theories of type (AN, AN), as well as propose new 4d SCFTs as a result. The wreathing technique, which involves gauging a subgroup of the automorphism group of the quiver diagram of the corresponding 3d mirror theory, is exploited. This allows us to understand several properties of discretely gauged theories, including moduli spaces and how discrete gauging affects the mixed ’t Hooft anomaly between the 1-form symmetry and the 0-form flavor symmetry. Many examples are viewed through the lens of the Argyres-Seiberg duality and its generalization. We also examine discrete gauging of SU(2) SQCD with 4 flavors by various ℤ2 and ℤ2 × ℤ2 subgroups of the permutation group S4 using the superconformal index. Regarding compactification, we propose a magnetic quiver for 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SU(n) SQCD with 2n flavors compactified on a circle with a ℤ2 twist. The twisted compactification by non-invertible symmetries of the 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM theory with gauge group SU(N) is revisited. The non-invertible symmetry naturally gives rise to a ℤk action on the scalar fields parametrizing the moduli space. Upon examining the ℤk invariant chiral ring of the Higgs branch, we find that, in addition to the largest branch of the moduli space that is expected to be captured by the ABJ(M) theory, there exist in general nilpotent operators that lead to a branch of the moduli space which is a radical ideal.

Searching linear structures of permutation groups with quantum computer

Searching linear structures of permutation groups with quantum computer

On the canonicity of the singularities of quotients of the Fulton-MacPherson compactification

We prove that quotients of the Fulton-MacPherson compactification of configuration spaces of smooth projective varieties of dimension &gt; 1 &gt;1 by permutation groups have canonical singularities.

The isomorphism problem for oligomorphic groups with weak elimination of imaginaries

AbstractIn Kechris et al. [J. Symb. Log. 83 (2018), no. 3, 1190–1203], it was asked if equality on the reals is sharp as a lower bound for the complexity of topological isomorphism between oligomorphic groups. We prove that under the assumption of weak elimination of imaginaries, this is indeed the case. Our methods are model theoretic and they also have applications on the classical problem of reconstruction of isomorphisms of permutation groups from (topological) isomorphisms of automorphisms groups. As a concrete application, we give an explicit description of for any vector space of dimension over a finite field, in affinity with the classical description for finite‐dimensional spaces due to Schreier and van der Waerden.

Skew-morphisms of elementary abelian 𝑝-groups

Abstract A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) \sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y) for all x , y ∈ G x,y\in G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with ⟨ σ ⟩ \langle\sigma\rangle forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups Z p n \mathbb{Z}_{p}^{n} . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X if p = 2 p=2 and either Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X or ( Z p n ) X ≅ Z p n − 1 (\mathbb{Z}_{p}^{n})_{X}\cong\mathbb{Z}_{p}^{n-1} if 𝑝 is an odd prime. As an application, for n ≤ 3 n\leq 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) \mathrm{AGL}(n,p) and enumerate the number of skew-morphisms of Z p n \mathbb{Z}_{p}^{n} .

Classifying primitive solvable permutation groups of rank 5 and 6

Abstract Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G 0 G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G 0 G_{0} in Ω, including the trivial orbit { α } \{\alpha\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.

Complete Contraction of Tensor and Its Application

This article presents a mathematical model for tensor complete contraction utilizing permutation group theory. The model helps in identifying the index class that dictates the complete contraction of even-order tensors. Additionally, the concept of a complete contraction closed loop is introduced along with a method for calculating the number of completely contracted images of any even-order tensor under any &amp;lambda; type permutation. The model is applied to investigate the algebraic structure of global conformal invariants on three-dimensional hypersurfaces. Moreover, it is shown that the generalized Willmore functional is the sole global conformal invariant in three-dimensional hypersurfaces when the difference is a constant multiple.

Spreading primitive groups of diagonal type do not exist

The synchronization hierarchy of finite permutation groups consists of classes of groups lying between $2$ -transitive groups and primitive groups. This includes the class of spreading groups, which are defined in terms of sets and multisets of permuted points, and which are known to be primitive of almost simple, affine or diagonal type. In this paper, we prove that in fact no spreading group of diagonal type exists. As part of our proof, we show that all non-abelian finite simple groups, other than six sporadic groups, have a transitive action in which a proper normal subgroup of a point stabilizer is supplemented by all corresponding two-point stabilizers.

Type Synthesis of a 3DOF Wrist Applying the Coupled-Input Cable-Driven Parallel Robot

Abstract The spatial cable-driven parallel robots (CDPRs) with low DOF (degree-of-freedom; n &amp;lt; 6) are like the physiological structure of bone and muscles, which are suitable for designing humanoid joints. Therefore, the type synthesis of the CDPR is of great interest for the design of new humanoid wrist joints. In this paper, we present a type synthesis of the coupled-input CDPRs to design a 3DOF wrist. Coupled input means that one actuator controls more than one cable. First, the Yamanouchi symbols of the coupled-input CDPRs are listed using the permutation group. In addition, two winding methods for the cable and the actuator are defined in the coupled-input CDPRs. Finally, a topology configuration of the coupled-input CDPR suitable for the 3DOF wrist model is determined based on a comparative analysis of the workspaces of a class of coupled-input CDPRs. It is shown that type synthesis of the coupled-input CDPRs is an effective way to innovate low DOF CDPRs.

A Concrete Model for the Quantum Permutation Group on 4 Points

In 2019, Jung–Weber gave an example of a concrete magic unitary M, which defines a C * -algebraic model of the quantum permutation group S 4 + . We show with the help of a computer that there exist no polynomials up to degree 50 separating the entries of M from the generators of C ( S 4 + ) . This indicates that the magic unitary M might already define a faithful model of S 4 + .

Towards an elementary formulation of the Riemann hypothesis in terms of permutation groups

Towards an elementary formulation of the Riemann hypothesis in terms of permutation groups

Group-subgroup subfactors revisited

For all Frobenius groups and a large class of finite multiply transitive permutation groups, we show that the corresponding group-subgroup subfactors are completely characterized by their principal graphs. The class includes all the sharply k -transitive permutation groups for k=2,3,4 , and in particular the Mathieu group M_{11} of degree 11.

Cliques in derangement graphs for innately transitive groups

Abstract Given a permutation group 𝐺, the derangement graph of 𝐺 is the Cayley graph with connection set the derangements of 𝐺. The group 𝐺 is said to be innately transitive if 𝐺 has a transitive minimal normal subgroup. Clearly, every primitive group is innately transitive. We show that, besides an infinite family of explicit exceptions, there exists a function f : N → N f\colon\mathbb{N}\to\mathbb{N} such that, if 𝐺 is innately transitive of degree 𝑛 and the derangement graph of 𝐺 has no clique of size 𝑘, then n ≤ f ⁢ ( k ) n\leq f(k) . Motivation for this work arises from investigations on Erdős–Ko–Rado type theorems for permutation groups.

Demazure Product of Permutations and Hopping

The Demazure product (also called the 0-Hecke product or greedy product) is an associative operation on Coxeter groups with interesting properties and important applications. In this paper, we study permutation groups and present a way to compute the Demazure product of two permutations using only their one-line notation and not relying on reduced words. The algorithm starts from their usual product and then applies a new operator, which we call the hopping operator. We also give an analogous result for the group of signed permutations.

Determination of the Conjugacy Classes and Character Table of the Full Non-Rigid Group of Hexachlorocyclopropane Chemical Compound Via Wreath Product of Pair of Permutation Groups

The use of full non-rigid (f.NRG) molecules group theory to study the internal dynamics of molecular structures of chemical compounds is trending in the research space. In this paper, we use computational method to compute the group elements and group table of the Hexachlorocyclopropane molecular (algebraic) structure and thereafter determine the order and conjugacy classes of the group and finally the corresponding symmetry for each permutation group. We considered the point group of the compound which turns out to be isomorphic to the Wreath Products C3wrC2, where Cn denotes a cyclic group of order n. We found that the group has order 18 and 9 conjugacy classes. Investigating the solubility of the components of Hexachlorocyclopropane chemical compound can be considered for further study.

On computing the closures of solvable permutation groups

Let [Formula: see text] be an integer. It is proved that the m-closure of a given solvable permutation group of degree n can be constructed in time [Formula: see text].