Symmetric Group Research Articles

A rank augmentation theorem for rank three string C-group representations of the symmetric groups

A rank augmentation theorem for rank three string C-group representations of the symmetric groups

An inductive approach to representations of general linear groups over compact discrete valuation rings

In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called PSH-algebras. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bialgebra that contains a large PSH-algebra that extends Zelevinsky's algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.

Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

AbstractWe calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group . We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.

Hilbert's 13th problem for algebraic groups

The algebraic form of Hilbert's 13th problem asks for the resolvent degree \operatorname{RD}(n) of the general polynomial f(x) = x^n + a_1 x^{n-1} + \cdots + a_n of degree n , where a_1, \ldots, a_n are independent variables. The resolvent degree is the minimal integer d such that every root of f(x) can be obtained in a finite number of steps, starting with \mathbb{C}(a_1, \ldots, a_n) and adjoining algebraic functions in \!{}\leqslant\nobreak d variables at each step. Recently Farb and Wolfson defined the resolvent degree \operatorname{RD}_k(G) for every finite group G and any base field k of characteristic 0 . In this setting \operatorname{RD}(n) = \operatorname{RD}_{\mathbb{C}}(\operatorname{S}_n) , where \operatorname{S}_n denotes the symmetric group. In this paper we extend their definition of \operatorname{RD}_k(G) to an arbitrary algebraic {group} G over an arbitrary field k . We investigate the dependency of this quantity on k and show that \operatorname{RD}_k(G) \leqslant 5 for any field k and any connected group G . The question whether \operatorname{RD}_k(G) can be bigger than 1 for any field k and any algebraic group G over k (not necessarily connected) remains open.

MAXIMAL SUBSEMIGROUPS OF INFINITE SYMMETRIC GROUPS

AbstractBrazil et al. [‘Maximal subgroups of infinite symmetric groups’, Proc. Lond. Math. Soc. (3)68(1) (1994), 77–111] provided a new family of maximal subgroups of the symmetric group $G(X)$ defined on an infinite set X. It is easy to see that, in this case, $G(X)$ contains subsemigroups that are not groups, but nothing is known about nongroup maximal subsemigroups of $G(X)$ . We provide infinitely many examples of such semigroups.

Bijection Between Trees in Stanley Character Formula and Factorizations of a Cycle

Stanley and Féray gave a formula for the irreducible character of the symmetric group related to a multi-rectangular Young diagram. This formula shows that the character is a polynomial in the multi-rectangularcoordinates and gives an explicit combinatorial interpretation for its coefficients in terms of counting certain decorated maps (i.e., graphs drawn on surfaces). In the current paper we concentrate on the coefficients of thetop-degree monomials in the Stanley character polynomial, which corresponds to counting certain decorated plane trees. We give an explicit bijection between such trees and minimal factorizations of a cycle.

Exploration of Symmetric Groups: Cayley Tables, Subgroup Analysis, and Real-World Applications in Card Tricks

This research delves into the extensive analysis of symmetric groups of various orders and degrees. We explore different representations of permutations, construct a Cayley table for the symmetric group of degree four, and systematically identify all its subgroups using Lagrange's theorem. An intriguing discovery unfolds as we demonstrate that the converse of the theorem countered by Sylow's theorem does not hold universally. Furthermore, we apply the concepts of permutations and product of disjoint cycles to address a real-world problem – the Card Trick game. This study amalgamates theoretical exploration with practical applications, showcasing the versatility and depth of symmetric group theory.

Tropical matrix groups and Boolean matrix groups

In this paper, we give a complete description of the maximal subgroups of the semigroup of all [Formula: see text] tropical matrices under multiplication up to isomorphism. Using the description, we prove that every subgroup of the semigroup of all [Formula: see text] Boolean matrices under multiplication is embedded in the symmetric group [Formula: see text].

The Application of Frieze Groups and Crystallographic Groups in Generating Batak Toba Ornament Motifs Using a Matlab Graphical User Interface

Gorga is a carving or sculpture typically found on the exterior of a Toba Batak traditional house. The Batak people use fractal (geometric) dimensions in Batak gorga carvings. In mathematics, repetitive and symmetrical patterns in planes that result from transformations are included in the plane symmetry groups. Ethnomathematics is a cultural approach to the concept of mathematics. A frieze group can be defined as a symmetrical group which arises from a unidirectional translation and subsequently generates a linear pattern that recurs exclusively in a single direction. There are seven different pattern types in the frieze groups. Meanwhile, crystallographic patterns are flat two-dimensional patterns that form a lattice. There are 17 crystallographic types of patterns with five different types of unit lattices. The purpose of this study is to generate motifs for Batak ornaments based on frieze groups and crystallographic groups using a Matlab Graphical User Interface (GUI). A total of 119 new motifs were generated based on seven types of patterns in the frieze groups, namely, F1,F2,F3,F4,F5,F6, and F7. Meanwhile, in the crystallographic groups, 153 new motifs were generated based on nine types of patterns, namely, p1,p2,pm,pg,cm,pmg,pmm,cmm, and pgg. To keep with trends, the new motifs generated can be used in everyday life as decorations or business symbols that are characteristic of the Toba Batak region.

Genuinely ramified maps and monodromy

For any genuinely ramified morphism f:Y⟶X between irreducible smooth projective curves we prove that (Y×XY)∖Δ‾ is connected, where Δ⊂Y×XY is the diagonal. Using this result the following are proved:(1)If f is further Morse then the Galois closure is the symmetric group Sd, where d=degree(f).(2)The Galois group of the general projection, to a line, of any smooth curve X⊂Pn of degree d, which is not contained in a hyperplane and contains a non-flex point, is Sd.

An Embedding of the Skein Action on Set Partitions into the Skein Action on Matchings.

Rhoades defined a skein action of the symmetric group on noncrossing set partitions which generalized an action of the symmetric group on matchings. The $\mathfrak{S}_n$-action on matchings is made possible via the Ptolemy relation, while the action on set partitions is defined in terms of a set of skein relations that generalize the Ptolemy relation. The skein action on noncrossing set partitions has seen applications to coinvariant theory and coordinate rings of partial flag varieties. In this paper, we will show how Rhoades' $\mathfrak{S}_n$-module can be embedded into the $\mathfrak{S}_n$-module generated by matchings, thereby explaining how Rhoades' generalized skein relations all arise from the Ptolemy relation.

Duplex Hecke algebras of type B

As a sequel to [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.], in this article we first introduce a so-called duplex Hecke algebras of type [Formula: see text] which is a [Formula: see text]-algebra associated with the Weyl group [Formula: see text] of type [Formula: see text], and symmetric groups [Formula: see text] for [Formula: see text], satisfying some Hecke relations (see Definition 3.1). This notion originates from the degenerate duplex Hecke algebra arising from the course of study of a kind of Schur–Weyl duality of Levi-type (see [B. Shu and Y. Yao, On enhanced reductive groups (I): Enhanced Schur algebras and the dualities related to degenerate duplex Hecke algebras, with an appendix by B. Liu, submitted (2023)]), extending the duplex Hecke algebra of type [Formula: see text] arising from the related [Formula: see text]-Schur–Weyl duality of Levi-type (see [C. Xue and A. Zhang, Doubled Hecke algebras and related quantum Schur duality, preprint (2021), arXiv:2108.07587[math.RT], accepted for publication in Algebra Colloq.]). A duplex Hecke algebra of type [Formula: see text] admits natural representations on certain tensor spaces. We then establish a Levi-type [Formula: see text]-Schur–Weyl duality of type [Formula: see text], which reveals the double centralizer property between such duplex Hecke algebras and [Formula: see text]quantum groups studied by Bao and Wang in [H. Bao and W. Wang, A new approach to Kazhdan–Lusztig theory of type B via quantum symmetric pairs, Astérisque 402 (2018)].

Some Results of Representation of the Character Table of S_6

n this essay we use standard Young tableaux to compute each equivalent irreducible representation of the Symmetric group by determining the matrices for adjacent transpositions (12) , (23) , (34), (45) and (56) of .

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

Abstract We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$ , there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$ , then $A^2 \supseteq A_n$ . This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$ . Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

Hypercontractivity on the symmetric group

Abstract The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$ , one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on $S_n$ , which are functions whose $2$ -norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small q-norms, for $q>2$ . As applications, we show the following: 1. An analog of the level-d inequality on the hypercube, asserting that the mass of a global function on low degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$ . 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice and stability versions of the Kruskal–Katona Theorem in some regimes of parameters.

Acentralizers of some Finite Groups

Let G be a finite group. The acentralizer of an automorphism α of G, is the subgroup of fixed points of α, i.e., CG(α) = {g ∈ G∣α(g) = g}. In this paper we determine the acentralizers of the dihedral group of order 2n, the dicyclic group of order 4n and the symmetric group on n letters. As a result we see that if n ≥ 3, then the number of acentralizers of the dihedral group and the dicyclic group of order 4n are equal. Also we determine the acentralizers of groups of orders pq and pqr, where p, q and r are distinct primes.

Generated new classes of BCK-algebras using permutation quasi-ordered sets

This work aims to develop new techniques for introducing new BCK-algebra classes via permutation in symmetric groups. permutation generalized BCK-algebras are introduced as a generalization of positive implicative permutation BCK-algebras. It shows how to create a permutation generalized BCK-algebra from a permutation quasi-ordered set. The concept of permutation generalized ideals of permutation generalized BCK-algebras is introduced, followed by examining the relationships between such ideals and congruence. Permutation generalized BCK-ideals are described in detail. It is established that a set generates a permutation generalized BCK-ideal.

The Hilbert series of the superspace coinvariant ring

Abstract Let $\Omega _n$ be the ring of polynomial-valued holomorphic differential forms on complex n-space, referred to in physics as the superspace ring of rank n. The symmetric group ${\mathfrak {S}}_n$ acts diagonally on $\Omega _n$ by permuting commuting and anticommuting generators simultaneously. We let $SI_n \subseteq \Omega _n$ be the ideal generated by ${\mathfrak {S}}_n$ -invariants with vanishing constant term and study the quotient $SR_n = \Omega _n / SI_n$ of superspace by this ideal. We calculate the doubly-graded Hilbert series of $SR_n$ and prove an ‘operator theorem’, which characterizes the harmonic space $SH_n \subseteq \Omega _n$ attached to $SR_n$ in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki.

On forbidden subgraphs of main supergraphs of groups

<p>In this study, we explore the main supergraph $ \mathcal{S}(G) $ of a finite group $ G $, defined as an undirected, simple graph with a vertex set $ G $ in which two distinct vertices, $ a $ and $ b $, are adjacent in $ \mathcal{S}(G) $ if the order of one is a divisor of the order of the other. This is denoted as either $ o(a)\mid o(b) $ or $ o(b)\mid o(a) $, where $ o(\cdot) $ is the order of an element. We classify finite groups for which the main supergraph is either a split graph or a threshold graph. Additionally, we characterize finite groups whose main supergraph is a cograph. Our classification extends to finite groups $ G $ with $ \mathcal{S}(G) $, a cograph that includes when $ G $ is a direct product of two non-trivial groups, as well as when $ G $ is either a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, or a sporadic simple group.</p>

Relationship between central corneal thickness and inter‐eye disease severity in primary angle closure glaucoma

Aims/Purpose: To determine whether subjects with asymmetric primary angle closure glaucoma (PACG) demonstrate greater central corneal thickness (CCT) asymmetry compared to those with symmetrical disease severity.Methods: The following clinical data were retrospectively extracted from the case records of 281 PACG subjects: presenting intraocular pressure (IOP), visual field mean deviation (MD) at presentation, CCT, anterior chamber depth (ACD), and axial length (AL). The eye with the worse visual field MD was considered the worse eye. Asymmetric PACG was defined as the inter‐eye visual field MD difference of greater than 5 dB; and symmetrical PACG disease as an inter‐eye difference of less than 5 dB. Subjects with previous acute primary angle closure were excluded.Results: Based on the definitions, a total of 160 out of 281 PACG subjects (56.9%) presented with asymmetrical disease and 121 (43.1%) with symmetrical disease. Subjects with asymmetric disease had a significantly thinner CCT in the worse eye (529.2 ± 35.2 μm) compared to the fellow eye (535.3 ± 37.2 μm) (p < 0.001), and higher presenting IOP (28.8 ± 12.0 vs 20.1 ± 8.4 mmHg, p < 0.001), but no inter‐eye differences were observed for measurements of AL (p = 0.71) and ACD (p = 0.51) respectively. For patients with symmetric disease, no significant inter‐eye differences were observed for CCT (531.7 ± 31.2 vs 532.9 ± 32.4 μm, p = 0.38), AL (p = 0.30) and ACD (p = 0.31), but the worse eye had significantly higher IOP (22.7 ± 7.9 vs 20.6 ± 6.9 mmHg, p < 0.001). In the asymmetric disease group, 30.6% of subjects had a thinner CCT (≥ 10 μm) in the worse eye compared to 16% of subjects in the symmetric group. There was no correlation between inter‐eye CCT difference and inter‐eye difference of visual field MD (p = 0.17).Conclusions: In PACG subjects with asymmetric disease at presentation, eyes with more severe disease have a thinner CCT and higher presenting IOP. Whether CCT is an independent risk factor for PACG through mechanisms beyond its influence on IOP measurements remains to be established.