Symmetric Group Research Articles

An Alloy Engineering Strategy toward Helical Microstructures of Achiral π-Conjugated Molecules for Circularly Polarized Luminescence.

Helical assembly has been demonstrated to efficiently facilitate the circularly polarized luminescence (CPL) performances, but the synthesis of micro- and nanohelices from rigid achiral π-conjugated compounds remains challenging due to the absence of bilayer structures or complementary hydrogen-bonding interactions. Here, we develop an alloying strategy for the realization of helical microstructures from achiral anthracene/anthracene derivatives with x-/x-axis modification or anthracene/tetracene derivatives with x-/y-axis modification via solution coassembly. Interestingly, two anthracene derivatives bearing asymmetric phenyl/phenylethynyl groups and symmetric phenylethynyl groups can assemble into spiral microribbons with a fractal branching pattern. Using such an alloying strategy, color-tailorable ternary spiral microtubules/microribbons referring to high-efficiency energy transfer processes are achievable. Molecular dynamics simulations reveal that the Von Mises stress produced by symmetry differences of two components induces symmetry breaking of alloy structures associated with twisting. Additionally, the contents of the guest and H2O also play a vital role in the formation of intricate helical microstructures. Single binary and ternary spiral microribbons present considerable CPL properties with a dissymmetric factor ('glum') of more than 0.01. The present work provides new insights into the formation of helical microcrystals with complex topologies and new optoelectronic functions.

RoCK blocks for double covers of symmetric groups over a complete discrete valuation ring

RoCK blocks for double covers of symmetric groups over a complete discrete valuation ring

Hopf cocycles associated to pointed and copointed deformations over S 3

The algebras in the classification of finite dimensional pointed and copointed Hopf algebras over the symmetric group on three letters are all Hopf cocycle deformations of their associated graded objects. We show that these cocycles are generically pure, namely they are not co-homologous to exponentials of Hochschild 2-cocycles; apart from very specific examples. We provide a full description of these cocycles on a given basis.

Equality of skew Schur functions in noncommuting variables

AbstractThe question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li, and van Willigenburg introduced skew Schur functions in noncommuting variables, , where is a connected skew diagram with boxes and is a permutation in the symmetric group . In this paper, we combine these two and classify when two skew Schur functions in noncommuting variables are equal: such that if and only if is a nonsymmetric ribbon, is the antipodal rotation of , and is an explicit bijection between two set partitions determined by .

Spin- 1/2 Kagome Heisenberg Antiferromagnet: Machine Learning Discovery of the Spinon Pair-Density-Wave Ground State

The spin-1/2 kagome antiferromagnet (AFM) is one of the most studied models in frustrated magnetism since it is a promising candidate to host exotic spin-liquid states. However, despite numerous studies using both analytical and numerical approaches, the nature of the ground state and low-energy excitations in this system remains elusive. This challenge is related to the difficulty in determining the spin gap in various calculations. We present the results of our investigation of the kagome AFM using the recently developed group equivariant convolutional neural networks—a novel machine learning technique for studying strongly frustrated models. This approach, combined with variational Monte Carlo method, introduces significant improvement of the achievable results’ accuracy for frustrated spin systems in comparison with approaches based on other neural-network architectures. Contrary to the results obtained previously with various methods, which predicted Z2 or U(1) Dirac spin-liquid states, our results strongly indicate that the ground state of the kagome lattice antiferromagnet is a spinon pair density wave that does not break time-reversal symmetry or any of the lattice symmetries. The state appears due to the spinon Cooper pairing instability close to two Dirac points in the spinon energy spectrum, and it resembles the pair density wave state studied previously in the context of underdoped cuprate superconductors in connection with the pseudogap phase. This state has significantly lower energy than the lowest-energy states found by the SU(2) symmetric density matrix renormalization group calculations and other methods. Published by the American Physical Society 2025

Flow characteristics of supersonic oscillating jet impinging by Schlieren visualization

Flow characteristics of supersonic oscillating jets impinging upon a perpendicular flat plate were experimentally studied. The effects of nozzle-pressure-ratios (NPRs) and nozzle-to-wall distance variations were investigated by Schlieren visualization. The frequency of oscillation and deflection angle of the oscillating jet were compared with the subsonic oscillating jet using a smoke visualization technique. Key variables included NPRs ranging from 6 to 14 and two nozzle-to-wall distances (H = Dh and H = 2Dh). Flow characteristics were also studied by a proper orthogonal decomposition (POD) method. The results showed the independence of oscillation frequency from NPR while demonstrating significant variations in deflection angles and shock structures with increasing NPR. At higher NPR values, complex flow features such as D-shaped shock cells, secondary shock formations, and prominent expansion fans emerged and were affected by nozzle-to-wall separation. POD analysis categorized flow modes into symmetric and antisymmetric groups, showing multiscale interactions including Kelvin–Helmholtz instabilities, vortex streets, and Mach disk oscillations. In addition, the interplay between large-scale coherent structures and smaller turbulent features was emphasized across POD modes. Schlieren and POD analysis revealed periodic oscillation of the flow structures such as shock cells, stand-off shock wave, and wall jet boundary driven by unsteady flow motion.

Degree of Preoperative Bilateral Hearing Affects Patient-Reported Outcome in Primary Stapedotomy.

To investigate whether degree of asymmetric hearing impairment influences patient-reported outcome measures and objective hearing results in primary stapedotomy. Register study. Data from the Swedish Quality Register for Otosclerosis Surgery consisting of 90% of stapes operations performed in Sweden. The 984 patients eligible for inclusion were categorized on the basis of preoperative hearing impairment: unilateral, bilateral asymmetric, or bilateral symmetric. Pure-tone audiometry and patient-reported outcome measures were analyzed, and Glasgow benefit plots were constructed. Ordinal logistics regression analyses were performed to adjust for factors influencing PROMs associated with degree of asymmetric hearing. Over 90% of patients across all groups reported improved or much improved hearing ability post-surgery. Ninety-five percent of patients who rated their hearing as worse or much worse after surgery had an air-conductive gain of <20 dB PTA4. Individuals with unilateral hearing impairment were more likely to report lower satisfaction with hearing function and daily life activities after surgery compared with those with bilateral hearing impairment, especially bilateral symmetric hearing impairment. In terms of hearing function, the bilateral symmetric hearing impairment group showed a significant decrease in the log odds of reporting lower satisfaction with a coefficient of -0.71 (95% confidence interval, -1.13 to -0.33), whereas the bilateral asymmetric hearing impairment group showed a nonsignificant decrease with a coefficient of -0.14 (95% confidence interval, -0.41 to 0.14) compared with the unilateral hearing impairment group. Tinnitus was more frequent in those with unilateral hearing impairment. Those with preoperative unilateral hearing impairment were more likely to express lower satisfaction with the results, compared with patients with bilateral impairment. Our findings suggest that the degree of bilateral hearing impairment should be considered in preoperative counseling, to better align with patient expectations regarding the benefit of surgery. An estimated air-conductive gain of at least 20 dB PTA4 was favorable for patient satisfaction.

Metrics on Permutations With the Same Descent Set

A permutation in a finite symmetric group on a set of ordered elements has a descent at the i-th index if the permutation value at the i-th index is greater than the permutation value that follows. The descent set of a permutation is the set of all indices where the permutation has a descent. Each finite symmetric group can be partitioned by descent sets. In this paper we study the Hamming metric and the L-infinity metric on the sets of permutations that share the same descent set for all nonempty descent sets to determine the maximum possible value that these metrics can achieve when restricted to these subsets.

Applications and Related Concepts of Orbit-Stabilizer Theorem

As the essential part of abstract algebra, group theory takes a critical place on the field of pure mathematics. Approaching the study of it from the aspect of group action is a great starting point. This paper will firstly expound several related concepts and theorems, such as stabilizer, orbit, fixator, conjugacy class, center of group and centralizer, and the Lagrange’s Theorem. Based on that, the paper investigates three essential theorems themselves and their proofs, which are Orbit-Stabilizer Theorem, Class Equation, and Burnside’s Lemma. This paper specifically provides wonderful insights about the applications of the Orbit-Stabilizer Theorem. The theorem is directly utilized in the derivations of Burnside’s Lemma and the Class Equation, also in the deduction of the general formula of the order of symmetric group . There are always some connections between the concepts in these theorems and examples with the orbit and stabilizer, thus Orbit-Stabilizer Theorem can perform brilliantly. This paper provides wonderful insights that effectively connect the Orbit-Stabilizer Theorem with other parts of Group Theory, also brings some new ideas to solving problems and prove theorems.

The lattice of submonoids of the uniform block permutations containing the symmetric group

Abstract We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $$\mathscr {J}$$ J -classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.

EXCEPTIONAL GROUPS OF ORDER $p^6$ FOR PRIMES $p\geq 5$

Abstract The minimal faithful permutation degree $\mu (G)$ of a finite group G is the least integer n such that G is isomorphic to a subgroup of the symmetric group $S_n$ . If G has a normal subgroup N such that $\mu (G/N)&gt; \mu (G)$ , then G is exceptional. We prove that the proportion of exceptional groups of order $p^6$ for primes $p \geq 5$ is asymptotically zero. We identify $(11p+107)/2$ such groups and conjecture that there are no others.

An empirical relation between lepton masses from the symmetric permutation group

The automorphism group of the spinor space of the standard model introduces the permutations of S3 in addition to the gauge groups in a unified theory of the elementary particle interactions. The [Formula: see text] symmetry will be the basis of a theoretical explanation of the inclusion of a fermion term, which transforms under the spin-[Formula: see text] representation of the symmetric permutation group, in the Koide relation for charged leptons. This field is identified with the sterile neutrino, and the mass is compatible with experimental bounds.

Revisiting the Marcus–de Oliveira Conjecture

The Marcus–de Oliveira determinantal conjecture claims that the determinant of the sum of two normal matrices A and B with the prescribed spectra σ(A)={a1,⋯,an} and σ(B)={b1,⋯,bn}, respectively, is contained in the convex hull of the points zσ=∏i=1n(ai+bσ(i)) for σ∈Sn, the symmetric group of degree n. The conjecture was independently proposed by Marvin Marcus in 1973 and de Oliveira in 1982, inspired by a result obtained by Miroslav Fiedler in 1971. We survey the main achievements relating to this open problem in matrix analysis. Some related results and questions that it has raised are also briefly reviewed. This overview aims to bring the attention of researchers to this problem and to stimulate the development of original approaches and techniques in the area. Ideally, this work may inspire further progress towards the solution of this long-standing conjecture.

Low-spin ground state of the giant single-molecule magnets {Mn70} and {Mn84}

The single-molecule magnets {Mn70} and {Mn84} are characterized by a 14-site unit cell with S=2 spin sites arranged in a circular geometry. Experimentally, these systems exhibit a magnetic ground state with a notably low total spin Stot=5–7. Up to now, this low-spin ground state has been up difficult to describe theoretically due to the complexity of the quantum Heisenberg model for such a large system. In this paper, we fill this gap and demonstrate that the ground state of {Mn70} and {Mn84} is in fact governed by a small, finite Stot in quantitative agreement with the experiment. We employ accurate, large-scale SU(2)-symmetric density-matrix renormalization group calculations for a quantum Heisenberg model with previously published exchange parameters obtained by density-functional theory. We do not find a low-spin state for the same parameters and S=1 and thus propose that frustrated systems with S≥2 are inherently prone to weak ferromagnetic interactions. This could account for the prevalence of similar low-spin Mn-based single-molecule magnets. Finally, we compute the full magnetization curve and find wide plateaus at 10/14, 11/14, 12/14, and 13/14 of the saturation, which can be traced back to nearly independent three-site clusters with broken intercluster bonds. Published by the American Physical Society 2025

Eigenvalue systems for integer orthogonal bases of multi-matrix invariants at finite N

Multi-matrix invariants, and in particular the scalar multi-trace operators of N = 4 SYM with U(N) gauge symmetry, can be described using permutation centraliser algebras (PCA), which are generalisations of the symmetric group algebras and independent of N. Free-field two-point functions define an N-dependent inner product on the PCA, and bases of operators have been constructed which are orthogonal at finite N. Two such bases are well-known, the restricted Schur and covariant bases, and both definitions involve representation-theoretic quantities such as Young diagram labels, multiplicity labels, branching and Clebsch-Gordan coefficients for symmetric groups. The explicit computation of these coefficients grows rapidly in complexity as the operator length increases. We develop a new method for explicitly constructing all the operators with specified Young diagram labels, based on an N-independent integer eigensystem formulated in the PCA. The eigensystem construction naturally leads to orthogonal basis elements which are integer linear combinations of the multi-trace operators, and the N-dependence of their norms are simple known dimension factors. We provide examples and give computer codes in SageMath which efficiently implement the construction for operators of classical dimension up to 14. While the restricted Schur basis relies on the Artin-Wedderburn decomposition of symmetric group algebras, the covariant basis relies on a variant which we refer to as the Kronecker decomposition. Analogous decompositions exist for any finite group algebra and the eigenvalue construction of integer orthogonal bases extends to the group algebra of any finite group with rational characters.

Monitoring and Three-dimensional Numerical Analysis of Rock and Soil Deformation during Construction

In order to solve the problem of large monitoring errors in settlement deformation in construction engineering, the author proposes rock and soil deformation monitoring and three-dimensional numerical analysis under construction. The author proposes a settlement and deformation monitoring method for pile foundation construction in green building engineering. By setting settlement deformation monitoring points and setting the frequency of pile foundation contour monitoring, a contour line tracking edge monitoring model is constructed to obtain monitoring results, and longitudinal contour correction is used to correct monitoring errors, thus achieving settlement deformation monitoring. The experimental results show that compared with the traditional symmetrical settlement deformation monitoring group and the traditional compression observation settlement deformation monitoring group, the monitoring error obtained by the contour line tracking settlement deformation monitoring group designed by the author is controlled below 1.1. The author's monitoring method has better results, higher accuracy, and smaller errors, and has practical application value.

Representations of the Symmetric Group are Decomposable in Polynomial Time

Abstract We introduce an algorithm to decompose matrix representations of the symmetric group over the reals into irreducible representations, which as a by-product also computes the multiplicities of the irreducible representations. The algorithm applied to a d-dimensional representation of $$S_n$$ S n is shown to have a complexity of $${\mathcal {O}}(n^2 d^3)$$ O ( n 2 d 3 ) operations for determining which irreducible representations are present and their corresponding multiplicities and a further $${\mathcal {O}}(n d^4)$$ O ( n d 4 ) operations to fully decompose representations with non-trivial multiplicities. These complexity bounds are pessimistic and in a practical implementation using floating point arithmetic and exploiting sparsity we observe better complexity. We demonstrate this algorithm on the problem of computing multiplicities of two tensor products of irreducible representations (the Kronecker coefficients problem) as well as higher order tensor products. For hook and hook-like irreducible representations the algorithm has polynomial complexity as n increases. We also demonstrate an application to constructing a basis of homogeneous polynomials so that applying a permutation of variables induces an irreducible representation.

Full Galois groups of polynomials with slowly growing coefficients

AbstractChoose a polynomial uniformly at random from the set of all monic polynomials of degree with integer coefficients in the box . The main result of the paper asserts that if grows to infinity, then the Galois group of is the full symmetric group, asymptotically almost surely, as . When grows rapidly to infinity, say , this theorem follows from a result of Gallagher. When is bounded, the analog of the theorem is open, while the state‐of‐the‐art is that the Galois group is large in the sense that it contains the alternating group (if , it is conditional on the Extended Riemann Hypothesis). Hence the most interesting case of the theorem is when grows slowly to infinity. Our method works for more general independent coefficients.

Neutrino masses in left-right asymmetric model

Abstract This work presents a comprehensive investigation into the construction of a neutrino mass model utilizing the Γ(3) modular group, which exhibits an isomorphism with the A 4 symmetric group. The study focuses on developing a left-right asymmetric model by incorporating modular symmetry. An advantageous aspect of employing modular symmetry in the present model is the elimination of need for additional particles known as ”flavons” to break the flavor symmetry. To effectively implement the extended inverse seesaw mechanism, we have introduced one fermion singlet for each generation.

Divisibility of character values of the symmetric group by prime powers

Divisibility of character values of the symmetric group by prime powers