Symmetric Group Research Articles

Super Multiset RSK and a Mixed Multiset Partition Algebra

Through dualities on representations on tensor powers and symmetric powers respectively, the partition algebra and multiset partition algebra have been used to study long-standing questions in the representation theory of the symmetric group. In this paper we extend this story to exterior powers, introducing the mixed multiset partition algebra as well as a generalization of the Robinson-Schensted-Knuth algorithm to two-row arrays of multisets with elements from two alphabets. From this algorithm, we obtain enumerative results that reflect representation-theoretic decompositions of this algebra. Furthermore, we use the generalized RSK algorithm to describe the decomposition of a polynomial ring in sets of commuting and anti-commuting variables as a module over both the general linear group and the symmetric group.

Duality analysis in a symmetric group and its application to random tensor network models

Abstract The Ising model is the simplest model for describing many-body effects in classical statistical mechanics. A duality analysis leads to its critical point under several assumptions. The Ising model has Z2-symmetry. The basis of duality analysis is a nontrivial relationship between low- and high-temperature expansions. However, discrete Fourier transformation automatically determines the hidden relationship. The duality analysis can naturally extend to systems with various degrees of freedom, Zq symmetry, and random spin systems. Furthermore, we obtained the duality relation in a series of permutation models in the present study by considering the symmetric group Sq and its Fourier transformation. The permutation model in a symmetric group is closely related to random quantum circuits and random tensor network models, which are frequently discussed in quantum computing. It also relates to the holographic principle, a property of string theories and quantum gravity. We provide a systematic approach using duality analysis to examine the phase transition in these models.

Lineup polytopes of products of simplices

Consider a real point configuration \bm{A} of size n and an integer r \leq n . The vertices of the r -lineup polytope of \bm{A} correspond to the possible orderings of the top r points of the configuration obtained by maximizing a linear functional. The motivation behind the study of lineup polytopes comes from the representability problem in quantum chemistry. In that context, the relevant point configurations are the vertices of hypersimplices and the integer points contained in an inflated regular simplex. The central problem consists in providing an inequality representation of lineup polytopes as efficiently as possible. In this article, we adapt the developed techniques to the quantum information theory setup. The appropriate point configurations become the vertices of products of simplices. A particular case is that of lineup polytopes of cubes, which form a type B analog of hypersimplices, where the symmetric group of type A naturally acts. To obtain the inequalities, we center our attention on the combinatorics and the symmetry of products of simplices to obtain an algorithmic solution. Along the way, we establish relationships between lineup polytopes of products of simplices with the Gale order, standard Young tableaux, and the resonance arrangement.

Permutation-equivariant quantum convolutional neural networks

Abstract The Symmetric group $S_{n}$ manifests itself in large classes of quantum systems as the invariance of certain characteristics of a quantum state with respect to permuting the qubits. Subgroups of $S_{n}$ arise, among many other contexts, to describe label symmetry of classical images with respect to spatial transformations, such as reflection or rotation. Equipped with the formalism of geometric quantum machine learning, in this study we propose the architectures of equivariant quantum convolutional neural networks (EQCNNs) adherent to $S_{n}$ and its subgroups. We demonstrate that a careful choice of pixel-to-qubit embedding order can facilitate easy construction of EQCNNs for small subgroups of $S_{n}$. Our novel EQCNN architecture corresponding to the full permutation group $S_{n}$ is built by applying all possible QCNNs with equal probability, which can also be conceptualized as a dropout strategy in quantum neural networks. For subgroups of $S_{n}$, our numerical results using MNIST datasets show better classification accuracy than non-equivariant QCNNs. The $S_{n}$-equivariant QCNN architecture shows significantly improved training and test performance than non-equivariant QCNN for classification of connected and non-connected graphs. When trained with sufficiently large number of data, the $S_{n}$-equivariant QCNN shows better average performance compared to $S_{n}$-equivariant QNN . These results contribute towards building powerful quantum machine learning architectures in permutation-symmetric systems.

Charmed roots and the Kroweras complement

AbstractAlthough both noncrossing partitions and nonnesting partitions are uniformly enumerated for Weyl groups, the exact relationship between these two sets of combinatorial objects remains frustratingly mysterious. In this paper, we give a precise combinatorial answer in the case of the symmetric group: for any standard Coxeter element, we construct an equivariant bijection between noncrossing partitions under the Kreweras complement and nonnesting partitions under a Coxeter‐theoretically natural cyclic action we call the Kroweras complement. Our equivariant bijection is the unique bijection that is both equivariant and support‐preserving, and is built using local rules depending on a new definition of charmed roots. Charmed roots are determined by the choice of Coxeter element — in the special case of the linear Coxeter element , we recover one of the standard bijections between noncrossing and nonnesting partitions.

Normalizer quotients of symmetric groups and inner holomorphs

We show that every finite group T is isomorphic to a normalizer quotient NSn(H)/H for some n and a subgroup H≤Sn. We show that this holds for all large enough n≥n0(T) and also with Sn replaced by An. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group G with Out(G)≅T (for any given finite group T) and the determination of the normalizer in Sym(G) of the inner holomorph InHol(G)≤Sym(G) for any centerless indecomposable finite group G, which may be of independent interest.

Hesitant Fuzzy Monotonic Dependent OWA Operator and Its Application in Symmetric Group Decision-Making

Some new techniques of aggregating hesitant fuzzy numbers (HFNs) by using monotonic dependent OWA (MDOWA) operators are investigated. By utilizing the score value of HFN, the concepts of hesitant fuzzy configuration vector and hesitant fuzzy hybrid configuration vector are proposed. Then, some methods of calculating variable weights related to the MDOWA operators under hesitant fuzzy environments are presented. Further, some operators, including hesitant fuzzy monotonic dependent OWA (HFMDOWA) operators and hesitant fuzzy hybrid monotonic dependent OWA (HFHMDOWA) operators, are developed, such as balanced HFMDOWA operators, rewarded HFMDOWA operators, balanced HFHMDOWA operators, rewarded HFHMDOWA operators, and so on. These developed operators are applied to multiple criteria group decision making (MCGDM), and a novel MCGDM algorithm is presented. By using the presented operators and algorithm, we can obtain symmetric decision-making results. Finally, an application example is provided to demonstrate the effectiveness of the developed MCGDM techniques.

Stable homology isomorphisms for the partition and Jones annular algebras

We show that the homology of the Jones annular algebras is isomorphic to that of the cyclic groups below a line of gradient 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{2}$$\\end{document}. We also show that the homology of the partition algebras is isomorphic to that of the symmetric groups below a line of gradient 1, strengthening a result of Boyd–Hepworth–Patzt. Both isomorphisms hold in a range exceeding the stability range of the algebras in question. Along the way, we prove the usual odd-strand and invertible parameter results for the Jones annular algebras.

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

Group Theory and Solutions of the Rubik’s Cube

The paper explores the application of group theory to solving the Rubik’s Cube. Group theory, a branch of abstract algebra, studies sets equipped with an operation that satisfies specific properties: closure, associativity, identity, and inverses. The Rubik’s Cube is treated as a group because its set of moves forms a structure that aligns with these properties. By representing the cube’s various configurations and moves through the mathematical constructs of groups, the paper analyzes how the elements transform during cube manipulations. Key concepts such as symmetric groups, homomorphisms, alternating groups, and disjoint cycle decomposition are used to break down the cube’s complexity. Additionally, the study demonstrates how group actions, particularly the parity of permutations and orientation sums, govern valid cube configurations. The paper also references methodologies from other studies to apply and refine these mathematical approaches for systematically solving the Rubik’s Cube. Through these methods, the research illustrates how the cube can be navigated using logical algorithms grounded in group theory principles

Expressing even permutations as commutators

We prove that an element of the symmetric group of degree n > 3 can be represented as the commutator of two permutations whose supports intersect at exactly p ≤ n points if and only if it is the product of p cycles of length 3. As an accompanying result, and also as a step of the main result’s proof, we give a lower bound for p to express an even permutation as the product of p cycles of length 3, in terms of its decomposition into disjoint cycles.

Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

The study of suborbital graphs is a key area in group theory for it provides a graphical representation of a group action on a set. Moreover, it helps in understanding the combinatorial structures of the action of a group on a set. In this paper, we construct suborbital graphs based on the group action of the direct product of the symmetric group on Cartesian product of three sets through computation of the ranks and subdegrees of the group action. Suborbital graphs are constructed by the use of Sims theorem. The properties of the suborbital graphs are analyzed. In the study it is proven that the rank of the group action of direct product of the symmetric group acting on the Cartesian product of three sets is 8 for all <i>n</i> ≥ 2 and the suborbits are length 1, (<i>n</i>-1), (<i>n</i>-1), (<i>n</i>-1), (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>2</sup>, (<i>n</i>-1)<sup>3</sup>. We show that the suborbits of the group action are self-paired. Furthermore, it is demostrated that each graph has a girth of 3 for all <i>n</i> > 2 and suborbital graphs of the group action are undirected. It is shown that graphs <I>Γ<sub>2</sub></I> and <I>Γ</I><sub>3</sub> are regular of degree <i>n</i>-1, graphs <I>Γ</I><sub>4</sub>, <I>Γ</I><sub>5</sub> and <I>Γ</I><sub>6</sub> of degree (<i>n</i>-1)<sup>2</sup> and graph <I>Γ</I><sub>7</sub> is regular of degree (<i>n</i>-1)<sup>3</sup>. The suborbital graphs <i>Γ<sub>i</sub></i>(<i>i=</i>1, 2,…, 6) are disconnected, with the number of connected components equal to <i>n</i><sup>2</sup> while suborbital graph <I>Γ</I><sub>7</sub><i> </i>is connected for all <i>n</i> > 2.

On the monogenity and Galois group of certain classes of polynomials

Abstract We say a monic polynomial g(x) ∈ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, …, θ n−1} is a basis for the ring ℤ K of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f ( x ) = x n + c ∑ i = 1 n ( a x ) n − i ∈ Z [ x ] and F ( x ) = x n + c ∑ i = 1 n a i − 1 x n − i ∈ Z [ x ] $\begin{array}{} \displaystyle f(x)=x^n+c\sum_{i=1}^{n}(ax)^{n-i} \in \mathbb{Z}[x] \,\text{and}\, F(x)=x^n+c\sum_{i=1}^{n}a^{i-1}x^{n-i} \in \mathbb{Z}[x] \end{array}$ be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn , the symmetric group on n letters.

Multiplicity-free induced characters of symmetric groups

Multiplicity-free induced characters of symmetric groups

Generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic 2

AbstractWe prove generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic 2, similar to Donkin’s results for decomposition numbers of representations of symmetric groups.

Asymptotics of commuting ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-tuples in symmetric groups and log-concavity

Denote by Nℓ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_{\\ell }(n)$$\\end{document} the number of ℓ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell $$\\end{document}-tuples of elements in the symmetric group Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n$$\\end{document} with commuting components, normalized by the order of Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n$$\\end{document}. In this paper, we prove asymptotic formulas for Nℓ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_\\ell (n)$$\\end{document}. In addition, general criteria for log-concavity are shown, which can be applied to Nℓ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N_\\ell (n)$$\\end{document} among other examples. Moreover, we obtain a Bessenrodt-Ono type theorem which gives an inequality of the form c(a)c(b)>c(a+b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$c(a)c(b) > c(a+b)$$\\end{document} for certain families of sequences c(n).

Novel Dissymmetric Formal Quinoidal Molecules End-Capped by Dicyanomethylene and Triphenylphosphonium.

A number of quinoidal molecules with symmetric end-capping groups, particularly dicyanomethylene units, have been synthesized for organic optoelectronic materials. In comparison, dissymmetric quinoidal molecules, characterized by end-capping with different groups, are less explored. In this paper, we present the unexpected formation of new formal quinoidal molecules, which are end-capped with both dicyanomethylene and triphenylphosphonium moieties. The structures of these dissymmetric quinoidal molecules were firmly verified by single crystal structural analyses. On the basis of the control experiments and DFT calculations, we proposed the reaction mechanism for the formation of these dissymmetric quinoidal molecules. The respective zwitterionic forms should make contributions to the ground state structures of these quinoidal molecules based on the analysis of their bond lengths and aromaticity and Mayer Bond Orbital (MBO) calculation. This agrees well with the fact that negative solvatochromism was observed for these quinoidal molecules. Although these new quinoidal molecules are non-emissive both in solutions and crystalline states, they become emissive with quantum yields up to 51.4 % after elevating the solvent viscosity or dispersing them in a PMMA matrix. Interestingly, their emissions can also be switched on upon binding with certain proteins, in particular with human serum albumin.

Investigations on the growth, spectral, optical, electrical, thermal and nonlinear optical properties of L-valine itaconic acid single crystal

A nonlinear optical single crystal of L-Valine Itaconic Acid (LVIA) was grown at room temperature from an aqueous solution by using the slow evaporation method. The grown crystal was undergone X-ray diffraction investigation, which confirm that the crystal belongs to orthorhombic system with the non-centro symmetric Pbca space group. This single crystal contains several functional groups, according to FTIR spectroscopic analysis there were identified. By analyzing the crystal's UV–visible spectrum, based on its optical transparency this crystal may use for optoelectronic device manufacturing. The emission happens at 553 nm wavelength, according to the photoluminescence spectrum. Thermo gravimetric and differential thermal analytical studies helps us to measure title crystal melting point and thermal stability. At various temperatures, the dielectric property of the crystal was found out. The Kurtz-powder method has been used to study the phenomenon of Second Harmonic Generation (SHG) in the crystal.

The excedance quotient of the Bruhat order, quasisymmetric varieties, and Temperley–Lieb algebras

AbstractLet be the ring of polynomials in variables and consider the ideal generated by quasisymmetric polynomials without constant term. It was shown by J. C. Aval, F. Bergeron, and N. Bergeron that the th Catalan number. In the present work, we explain this phenomenon by defining a set of permutations with the following properties: first, is a basis of the Temperley–Lieb algebra , and second, when considering as a collection of points in , the top‐degree homogeneous component of the vanishing ideal is . Our construction has a few byproducts that are independently noteworthy. We define an equivalence relation on the symmetric group using weak excedances and show that its equivalence classes are naturally indexed by noncrossing partitions. Each equivalence class is an interval in the Bruhat order between an element of and a 321‐avoiding permutation. Furthermore, the Bruhat order induces a well‐defined order on . Finally, we show that any section of the quotient gives an (often novel) basis for .

Mini-Worskhop: Artin Groups meet Triangulated Categories

Artin and Coxeter groups are naturally occurring generalisations of the braid and symmetric groups respectively. However, unlike for Coxeter groups, many basic group theoretic questions remain unanswered for general Artin groups – most notably the K(\pi,1) -conjecture for Artin groups remains open except for certain special families of Artin groups. Recently, Artin groups have also appeared as groups acting on triangulated categories, where the associated spaces of Bridgeland’s stability conditions provide new realisations of the corresponding K(\pi,1) spaces. The aim of the workshop is to bring together experts and early career researchers from two seemingly different areas of research: (i) geometric and combinatorial group theory and topology, and (ii) triangulated categories and stability conditions, to explore their intersection via the K(\pi,1) -conjecture.