Symmetric Group Research Articles

Mineralization and thickness of the condylar cortex in skeletal remains of children’s mandibles: A preliminary study

ObjectiveTo explore the relationship between the volumetric bone mineral density (vBMD), the thickness of the condylar cortex (Tcortex) and the hemimandibular volumes (Vhemimandible) of symmetrical and asymmetrical mandibles of children. DesignThe data collection consisted of 92 archeological skeletal remains of children’s mandibles between 1 and 12 years old. The mandibles were digitalized with a computed tomography (CT) scan, and three dimensional models were obtained. Vhemimandible was calculated using the optimal symmetry plane. The volumes were used to calculate the asymmetry index (AI). Mandibles with an AI of ≥ 3% (N = 9) and a sample of the most symmetrical mandibles (N = 9) were selected for this research. Three groups were created: a symmetrical, an asymmetrical and a pooled group. Micro-CT was used to measure the vBMD and Tcortex in four volumes of interest. The AI was calculated for these parameters as well. ResultsSignificant correlations were found between the vBMD and the Tcortex in the pooled group (P < .01) and between the AI of the vBMD and the AI of the Tcortex in the pooled (P < .01) and symmetrical group (P < .05). No significant correlations were found between the vBMD and the Vhemimandible and between the respective AIs. Between the Tcortex and the Vhemimandible a significant correlation was found in the pooled and asymmetrical group. ConclusionThere is a relationship between the vBMD and the Tcortex. The correlations between the Tcortex and the Vhemimandible are insufficient to draw firm conclusions. A relationship between the vBMD and Vhemimandible was not confirmed in this study.

Lie groups in the symmetric group: Reducing Ulam's problem to the simple case

Ulam asked whether all Lie groups can be represented faithfully on a countable set. We establish a reduction of Ulam's problem to the case of simple Lie groups.In particular, we solve the problem for all solvable Lie groups and more generally Lie groups with a linear Levi component. It follows that every amenable locally compact second countable group acts faithfully on a countable set.

Composing group action to permutation representation

A mapping that satisfies two specific axioms provides a common notion of group action. A homomorphism translating from a group to a symmetric group of a certain set can also be used to describe group action. Therefore, any example of the group actions can be stated based on the second equivalent definition, such as the regular action, natural matrix action, coset action, and Z^2 acting on R^2, etc. It is necessary to examine the concepts of the orbit and stabilizer of a group in order to reveal the orbit-stabilizer theorem. After the preparatory work, the orbit-stabilizer theorem can be proved by defining a mapping from the orbit to the stabilizer and then checking that the mapping is well-defined and bijective. To derive Burnsides lemma, it needs to introduce the set of fixed points which is related to the concept of the stabilizer. Through the orbit-stabilizer theorem along with the fact that a set is a disjoint union of orbits, Burnside's lemma can be confirmed. Moreover, it is natural to compose a group action with a linear representation, and then a representation would be obtained, which is permutation representation. Further, one must calculate the character of the permutation representation, the dimension of the fixed subspace, and the dimension of CX^G. Then it can show Burnsides lemma in another way by permutation representation.

On the determinant of representations of generalized symmetric groups ℤ r ≀Sn

In this paper, we study the determinant of irreducible representations of the generalized symmetric groups Z r ≀ S n . We give an explicit formula to compute the determinant of an irreducible representation of Z r ≀ S n . Recently, several authors have characterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results, given an integer n, r an odd prime and ζ a nontrivial multiplicative character of Z r ≀ S n with n < r, we obtain an explicit formula to compute N ζ ( n ) , the number of irreducible representations of Z r ≀ S n whose determinant is ζ.

Computing character tables of symmetric groups using higher order permutation representations

In this paper, we will introduce the topic of representation theory by focusing on the techniques of computing character tables of finite groups, in particular, the symmetric group Sn . First, we will introduce some basic concepts of representation theory of finite groups over ℂ. Then we will compute the character tables for S 3 and S 4 using the classical methods introduced in Section 2. Also, we will construct a series of representations for Sn and compute their characters. Finally, we will compute the character tables for S 5 and S 6 using our method introduced in Section 3.

Deep Reinforcement Learning Based Resource Allocation and Trajectory Planning in Integrated Sensing and Communications UAV Network

In this paper, multi-UAVs serve as mobile aerial ISAC platforms to sense and communicate with on-ground target users. To optimize the communication and sensing performance, we formulate a joint user association, UAV trajectory planning and power allocation problem to maximize the minimum weighted spectral efficiency among UAVs. This paper exploits the centralized and the decentralized deep reinforcement learning (DRL) solutions to solve the sequential decision-making problem. On one hand, we first introduce the centralized soft actor-critic (SAC) algorithm. Then, we explore the equivalent transformation of the optimization objective based on symmetric group, propose the random and the adaptive data augmentation schemes to design the replay memory buffer of SAC, and accordingly propose SAC algorithms assisted by data augmentation to tackle the transformed problem. On the other hand, the multi-agent soft actor-critic (MASAC), a decentralized solution, is also introduced to solve this sequential decision-making problem. The experiment results reveal the effectiveness of the centralized and the decentralized solutions in considered scenarios. Specifically, the SAC assisted by the adaptive scheme significantly outperforms other centralized solutions in the training speed and the weighted spectral efficiency. Meanwhile, the decentralized MASAC algorithm behaves best in the early training speed.

Synthesis of Cardanil Methyl Ether via Etherification Reaction Cardanol Isolated from Cashew Nut Shells (Anacardium occidentale L.) with Variations of Molar Methyl Iodide

The synthesis of the Cardanyl methyl ether is carried out through an etherization reaction between Cardanyl seed skin extract (CNSL) isolated from the methyl iodide. Cardanol is isolated with the addition of acetone, Ca(OH)2, NH4OH 25%, n-hexane: ethyl acetate (98:2), washing with NaOH 2.5%, HCl 5%, aquadest, and Na2SO4 anhydrate. The yield of the obtained cardanol is 74.39 g (74.39% of the initial weight of the CNSL) and has the following characteristics: acid count 1.2863 mgKOH/g, iodine count 215.6535 g I2/100 g, viscosity 38.7 cP, density 0.8266 g/mL, pH = 6.14. The FT-IR methyl ether Cardanyl spectrum shows an absorption band at 1162.9-1259.8 cm-1 shows a C-O stretching group at a wave number of 1043.6-1148.0 cm-1 showing the formation of an asymmetric and symmetrical C- O-C stretching band of the ether compound, the Spectra of the FT - IR methyl ether Cardanyl shows an absorptive band at 1162-1259-125,8 cm-1, showing a C-O stretching Group, at the wave number of 1043.6-1148.0 cm-1 indicating the formation, as a result, of an Asymmetric and Symmetric Stretching Group of the ethereal Compound. The Cardanyl methyl ether was synthesized by adding K2CO3, the solvent acetone and the iodide-methyl, with a reflux time of 8 hours. As for the parameters carried out in this study, the molar variation of the Cardanyl is the Methyl Iodide (1:1.5, 1:3, and 1:4.5) and yields 3.555 g; 12.084 g; 11.472 g of Cardanyl ether. Methyl ether Cardanyl was obtained in analysis with FT-IR and GC-MS spectroscopic photometers. The results of the analysis of Cardanyl methyl ethers with GC -MS were 63.28%, and the results of analysis of the Cardanyl ether with a variation of the molar ratio Cardanyl: Methyl iodide (1:1.5; 1:3; 1:4.5) with GC-MS obtaining 3.05%, 30.29%, and 26.58%. These results show the maximum percentage of methyl ethers Cardanyl at the reaction conditions of the variation molar ratios Cardanyl iodides (1:3), K2CO3 17.3 g, 50 mL of acetone, and 8 hours of reflux time.

A Barrier for Further Approximating Sorting by Transpositions.

The transposition distance problem is a classical problem in genome rearrangements, which seeks to determine the minimum number of transpositions needed to transform a linear chromosome into another represented by the permutations and , respectively. This article focuses on the equivalent problem of sorting by transpositions (SBT), where is the identity permutation . Specifically, we investigate palisades, a family of permutations that are "hard" to sort, as they require numerous transpositions above the celebrated lower bound devised by Bafna and Pevzner. By determining the transposition distance of palisades, we were able to provide the exact transposition diameter for 3-permutations (TD3), a special subset of the symmetric group Sn, essential for the study of approximate solutions for SBT using the simplification technique. The exact value for TD3 has remained unknown since Elias and Hartman showed an upper bound for it. Another consequence of determining the transposition distance of palisades is that, using as lower bound the one by Bafna and Pevzner, it is impossible to guarantee approximation ratios lower than 1.375 when approximating SBT. This finding has significant implications for the study of SBT, as this problem has been the subject of intense research efforts for the past 25 years.

The extendability of Cayley graphs generated by transpositions

A connected graph of order at least 2k+2 is k-extendable for a non-negative integer k if it contains a perfect matching and every matching of size k can be extended to a perfect matching. The extendability number of the graph is the maximum k such that the graph is k-extendable. In this paper, we prove that, for a Cayley graph Γ of a symmetric group with respect to a generating set of size m consisting of transpositions, the extendability number of Γ is m−1.

Representations on the cohomology of [formula omitted

The moduli space M‾0,n of n pointed stable curves of genus 0 admits an action of the symmetric group Sn by permuting the marked points. We provide a closed formula for the character of the Sn-action on the cohomology of M‾0,n. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of M‾0,n, equivariant with respect to the symmetric group action. Moreover we prove that H2k(M‾0,n) for k≤3 and H2k(M‾0,n)⊕H2k−2(M‾0,n) for any k are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the Sn-representation on the cohomology of the Fulton-MacPherson compactification P1[n] of the configuration space of n points on P1 and more generally on the cohomology of the moduli space M‾0,n(Pm−1,1) of stable maps.

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

On the lattice of pronormal subgroups of dicyclic, alternating and symmetric groups

Wachs permutations, Bruhat order and weak order

We study the partial orders induced on Wachs and signed Wachs permutations by the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.

A modular idealizer chain and unrefinability of partitions with repeated parts

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n − 2 terms in the chain. Moreover, they proved that the (n − 1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(pn), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over ℤm whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m − 1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As evidence of this, we establish a correspondence between the two constructions in the case m = 2.

The partition algebra and the plethysm coefficients I: Stability and Foulkes' conjecture

We propose a new approach to study plethysm coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. This provides an explanation of the stability properties of plethysm and Kronecker coefficients in a simple and uniform fashion for the first time. We prove the strengthened Foulkes' conjecture for stable plethysm coefficients in an elementary fashion.

On groups with chordal power graph, including a classification in the case of finite simple groups

AbstractWe prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by (Electron J Combin 28(3):14, 2021). Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic conditions. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups, and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.

On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups

On Algebras of Double Cosets of Symmetric Groups with Respect to Young Subgroups

Computations Stiefel-Whitney classes of real representations for non-prime power groups

In the current study, the researchers conduct computation of the Stiefel- Whitney classes of real representations of non-prime power groups such as Mathieu groups, symmetric groups, alternating groups and Janko groups. Stiefel- Whitney classes are conducted by using polytope convex hull with vector in n-dimention our computation using the HAP (Homological Algebra Programming.

Cyclic Degradation Parameters of Flow-Controlled Geomaterial for Pile Group Subjected to Torsional Loads

The pile groups support large foundation structures which often experience cyclic torsional loads. Several theoretical and experimental investigations were carried out on pile groups under axial and lateral cyclic loads; however, the study of the response of cyclic torsional loads on pile foundation groups remains elusive. This paper proposes a numerical scheme to capture the behavior of the symmetrical pile group in the flow-controlled geomaterial under cyclic torsional loading. Based on the numerical scheme, three-dimensional finite element analysis was performed to capture the nonlinear response of the pile group using a computational program. The results from the numerical analysis have been compared with the experimental observations. The peak twist and peak shear stress logarithmically decrease and then become asymptotic after a number of cycles of loading for a pile group (2,2) in a flow-controlled geomaterial. A set of cyclic degradation parameters of flow-controlled geomaterial is identified for the pile group and presented as a function of dilation.

Balance constants for Coxeter groups

The 1 / 3 1/3 - 2 / 3 2/3 Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant of any non-total order is at least 1 / 3 1/3 . By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets C C of any Coxeter group. Remarkably, we conjecture that the lower bound of 1 / 3 1/3 still applies in any finite Weyl group, with new and interesting equality cases appearing. We generalize several of the main results towards the 1 / 3 1/3 - 2 / 3 2/3 Conjecture to this new setting: we prove our conjecture when C C is a weak order interval below a fully commutative element in any acyclic Coxeter group (a generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the 1 / 3 1/3 - 2 / 3 2/3 Conjecture, and therefore on which methods are likely to be successful in resolving it.

Symmetric group fixed quotients of polynomial rings

Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p.