Symmetric Group Research Articles

Identifying the optimal mandibular midsagittal plane for systematic asymmetry correction: application in virtual surgery planning.

Abstract Motivated by Stanley's generalization of the chromatic polynomial of a graph to the chromatic symmetric function, we introduce the characteristic polynomial of a representation of the symmetric group, or more generally, of a symmetric function. When the representation arises from geometry, the coefficients of its characteristic polynomial tend to form a log‐concave sequence. To illustrate, we investigate explicit examples, including the ‐fold products of the projective spaces, the GIT moduli spaces of points on and Hessenberg varieties. Our main focus lies on the cohomology of the moduli space of pointed rational curves, for which we prove asymptotic formulas of its characteristic polynomial and establish asymptotic log‐concavity.

Given a smooth, projective curve Y , a point y_{0}\in Y , a positive integer n , and a transitive subgroup G of the symmetric group S_{d} , we study smooth, proper families, parameterized by algebraic varieties, of pointed degree d covers of (Y,y_{0}) , (X,x_{0})\to (Y,y_{0}) , branched in n points of Y\setminus y_{0} , whose monodromy group equals G . We construct a Hurwitz space H , an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of (Y,y_{0}) of this type. We construct explicitly a family parameterized by H , whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry.

Abstract We compute the asymptotic probability that a random pair of Sylow 2-subgroups in S n S_{n} or A n A_{n} intersects trivially. This calculation complements recent work of Diaconis, Giannelli, Guralnick, Law, Navarro, Sambale, and Spink.

Carbon-based hole-transport-layer (HTL)-free CsPbI2Br solar cells well balance power conversion efficiency (PCE), stability, and cost, but suffer from defects including undercoordinated Pb2+ and mobile I- in CsPbI2Br, and undercoordinated Sn4+ and oxygen vacancies (VO) in the SnO2 electron transport layers. To address these issues, biphenyl oxyacid additives including [1, 1'-biphenyl]-4, 4'-diphosphonic acid (BDPA), [1, 1'-biphenyl]-4, 4'-dicarboxylic acid, and [1, 1'-biphenyl]-4, 4'-disulfonic acid are investigated. It is found that the para-positioned oxyacid double bonds can coordinate with uncoordinated Pb2+ to form stable Pb─O bonds, while hydroxyls can anchor mobile I- via H-bonding. The opposing oxyacid double bonds can bind with uncoordinated Sn4+ to form stable Sn─O bonds, thus inhibiting VO formation. Concurrently, the symmetric oxyacid groups bridge the SnO2 and CsPbI2Br layers via coordination, thus enabling the biphenyl structure to function as an electron transport channel. Moreover, the additives increase the CsPbI2Br grain dimensions alongside enhanced surface density and reduced roughness. BDPA exhibits superior passivation efficacy due to the reduced electronegativity of its central phosphorus atom, strengthening oxygen coordination capability. Consequently, the BDPA-optimized device delivers a leading PCE of 15.55%, ≈24% increment over 11.80% for the control device, as well as the improved operational stability and reduced current-voltage hysteresis.

When groups of inertial swimmers move together, hydrodynamic interactions play a key role in shaping their collective dynamics, including the cohesion of the group. To explore how hydrodynamic interactions influence group cohesion, we develop a three-dimensional, inviscid, far-field model of a swimmer, neglecting the vortical wake produced by swimmers in order to determine the role that potential flow interactions play on group dynamics. Focusing on symmetric triangular, diamond, and circular group arrangements, we investigate whether passive hydrodynamics alone can promote cohesive behavior, and what role symmetry of the group plays. Under the idealized conditions of our model, we find that far-field interactions alone significantly impact the cohesion of groups of swimmers. This is an important result because, contrary to common belief, it shows that interactions with a vortical wake do not solely determine the cohesion of groups of swimmers. While small symmetric (and even asymmetric) groups can be cohesive, larger groups typically are not, instead breaking apart into smaller, self-organized subgroups that are cohesive. Notably, we discover circular arrangements of swimmers that chase each other around a circle, resembling the milling behavior of natural fish schools; we call this hydrodynamic milling. Hydrodynamic milling is cohesive in the sense that it is a fixed point of a particular Poincaré map, but it is unstable, especially to asymmetric perturbations. Our findings suggest that while passive hydrodynamics alone cannot sustain large-scale cohesion indefinitely, controlling interactions between subgroups, or controlling the behavior of only the periphery of a large group, could potentially enable stable collective behavior with minimal active input.

Given a finite group G , the order graph of G, denoted by S(G), is a graph whose vertex set is G, and two distinct vertices a and b are adjacent if o(a) | o(b) or o(b) | o(a), where o(a), and o(b), are the orders of a and b in G, respectively. In this paper, by the order of an element, we give a characterization of the finite groups whose order graph is C4-free. As applications, we classify a few families of finite groups whose order graph is C4-free, such as nilpotent groups, dihedral groups and symmetric groups.

The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of U ( g l ( n ) ) U(\mathfrak {gl}(n)) which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for U ( g l ( n ) ) U(\mathfrak {gl}(n)) in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull’s identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello [Electron. J. Combin. 16 (2009), Research Paper 103, 43].

Computing Young's Natural Representations for Generalized Symmetric Groups

Bilateral sagittal split ramus osteotomy (BSSRO) is the gold-standard surgical treatment for mandibular prognathism and facial symmetry restoration in class III malocclusion. However, it remains unclear whether asymmetric and symmetric mandibles exhibit differential relapse patterns that impact both functional stability and aesthetic outcomes. This study aims to compare intraoperative displacement and postoperative skeletal relapse between asymmetric and symmetric mandibles following BSSRO. This retrospective cohort study analyzed class III malocclusion patients who underwent BSSRO between January 2018 and December 2023. Three-dimensional CBCT datasets were obtained at three time points: preoperative (T0), immediately postoperative (T1), and at least 6 months postoperative (T2). Translational and rotational changes in mandibular segments (distal and proximal) were quantitatively assessed during surgical correction (T0-T1) and relapse (T1-T2). Intraoperatively, the asymmetric group (AG) exhibited greater displacement toward the non-deviated side of the distal segment (P=0.001) and movement of the non-deviated proximal segment toward the deviated side (P=0.044) compared to the symmetric group (SG). Rotational analysis revealed increased clockwise yaw in AG's distal segment (P=0.009) and enhanced clockwise roll in the non-deviated proximal segment (P=0.021). Postoperatively, AG demonstrated significant relapse toward the deviated side in both proximal segments (non-deviated side: P<0.001; deviated side: P=0.027), counterclockwise yaw in the distal segment (P=0.020), and a counterclockwise rolling tendency in the non-deviated proximal segment (P=0.050). Asymmetric and symmetric mandibles exhibit distinct relapse patterns following BSSRO, involving both translational and rotational movements across all segments. These findings highlight the importance of individualized surgical planning and postoperative management in asymmetric cases. Further validation through prospective multicenter studies with long-term follow-up is recommended. This journal requires that authors assign a level of evidence to each article. For a full description of these Evidence-Based Medicine ratings, please refer to the Table of Contents or the online Instructions to Authors www.springer.com/00266 .

The automorphism groups of Cayley graphs on symmetric groups, Cay(G, S), where S is a complete set of transpositions have been determined. In a similar spirit, automorphism groups of Cayley graphs Cay(An , S) on alternating groups An , where S is a set of all 3-cycles have also been determined. It has, in addition, been shown that these graphs are not normal. In all these Cayley graphs, one observes that their corresponding Cayley sets are a union of conjugacy classes. In this paper, we determine in their generality, the automorphism groups of Cay(G, S), where G ∈ {An , Sn } and S is a conjugacy class type Cayley set. Further, we show that the family of these graphs form a Boolean algebra. It is first shown that Aut(Cay(G, S)), S ∉ {∅, G \\ {e}}, is primitive if and only if G = An . Using one of the results obtained by Praeger in 1990, we exploit further the other cases, thereby proving that, for n > 4 and n ≠ 6, Aut(Cay(An , S)) ≅ Hol(An ) ⋊ 2, with Hol(G) ∼= G ⋊ Aut(G), provided that S is preserved by the outer automorphism defined by the conjugation by an odd permutation. Finally, in the remaining case G = Sn , n > 4 and n ≠ 6, we show that Aut(Cay(Sn, S) ≅ (Hol(An ) ⋊ 2) ≀ S 2 for S ⊂ An \\ {e}, and that Aut(Cay(Sn , S)) ≅ Hol(Sn ) ⋊ 2 otherwise; provided that S does not contain Sn \\ An or S ≠ An \\ {e}, S ∉ {∅, Sn \\ {e}}.

For [Formula: see text] with [Formula: see text], let [Formula: see text] denote the maximum number of edge disjoint trees connecting [Formula: see text] in [Formula: see text]. For [Formula: see text], the generalized [Formula: see text]-connectivity [Formula: see text] of an [Formula: see text]-vertex connected graph [Formula: see text] is defined to be [Formula: see text]. The generalized [Formula: see text]-connectivity can serve for measuring the fault tolerance of an interconnection network. The bubble-sort graph [Formula: see text] for [Formula: see text] is a Cayley graph over the symmetric group of permutations on [Formula: see text] generated by transpositions from the set [Formula: see text]. In this paper, we show that for the bubble-sort graphs [Formula: see text] with [Formula: see text], [Formula: see text].

Greene's Theorem and ideals of the group algebra of a symmetric group

We study the geometry and topology of Δ-Springer varieties associated with two-row partitions. These varieties were introduced in recent work by Griffin–Levinson–Woo to give a geometric realization of a symmetric function appearing in the Delta conjecture by Haglund–Remmel–Wilson. We provide an explicit and combinatorial description of the irreducible components of the two-row Δ-Springer variety and compare it to the ordinary two-row Springer fiber as well as Kato’s exotic Springer fiber corresponding to a one-row bipartition. In addition to that, we extend the action of the symmetric group on the homology of the two-row Δ-Springer variety to an action of a degenerate affine Hecke algebra and relate this action to a 𝔤𝔩 2 -tensor space.

The quantification of disorder in data remains a fundamental challenge in science, as many phenomena yield short length datasets with order-disorder behavior, significant (un)correlated fluctuations, and indistinguishable characteristics even when arising from distinct natures, such as chaotic or stochastic processes. In this Letter, we propose a novel method to directly quantify disorder in data through recurrence microstate analysis, showing that maximizing this measure is essential for its optimal estimation. Our approach reveals that the disorder condition corresponds to the action of the symmetric group on recurrence space, producing classes of equiprobable recurrence microstates. By leveraging information entropy, we define a robust quantifier that reliably differentiates between chaotic, correlated, and uncorrelated stochastic signals even using just small time series. Additionally, it uncovers the characteristics of corrupting noise in dynamical systems. As an application, we show that disorder minima over time often align with well-known stage transitions of the Cenozoic era, indicating periods of dominant drivers in paleoclimatic data.

Permutations on a set, endowed with function composition, build a group called a symmetric group. In addition to their algebraic structure, symmetric groups have two metrics that are of particular interest to us here: the Cayley distance and the Kendall tau distance. In fact, the aim of this paper is to introduce the concept of distance in a general finite group based on them. The main tool that we use to this end is Cayley’s theorem, which states that any finite group is isomorphic to a subgroup of a certain symmetric group. We also discuss the advantages and disadvantage of these permutation-based distances compared to the conventional generator-based distances in finite groups. The reason why we are interested in distances on groups is that finite groups appear in symbolic representations of time series, most notably in the so-called ordinal representations, whose symbols are precisely permutations, usually called ordinal patterns in that context. The natural extension from groups to group-valued time series is also discussed, as well as how such metric tools can be applied in time series analysis. Both theory and applications are illustrated with examples and numerical simulations.

Ridaifen (RID) analogues are identified as nonpeptide, noncovalent inhibitors of the catalytic subunits of the human 20S proteasome. They demonstrated effectiveness against both multiple myeloma and solid tumors. Herein, the synthesis and biological evaluation of 20 novel RID analogs that exhibit inhibitory effects on the three catalytic subunits of the 20S proteasome are reported. All the compounds were tested on the National Cancer Institute (NCI) 60 cancerous cell lines. The compounds bear symmetric aminoalkoxy groups on rings B and C, different substituents on ring A, and the terminal side chain on the ethylene backbone was modified to methyl and cyclopentyl groups.Compound 43 was the most potent inhibitor for both CT-L and PGPH (IC50 = 0.22, 0.05 μM), which is threefold more potent for CT-L and tenfold more potent on PGPH than RID-F. Most of the analogs showed pan activity toward different cancer cell lines, and compound 20 was more potent than tamoxifen. Compound 20 showed submicromolar IC50 values for CT-L and PGPH activities, indicating that it mediates its cytotoxic activity via proteasomal inhibition. Selected compounds were tested against Ebolavirus, and compound 36 showed the highest antiviral activity, surpassing the EC50 of the reference compound favipiravir.

Abstract We realise the Bott-Samelson resolutions of type A Schubert varieties as quiver Grassmannians. In order to explicitly describe this isomorphism, we introduce the notion of a geometrically compatible decomposition for any permutation in the symmetric group $$S_n$$ S n . For smooth type A Schubert varieties, we identify a suitable dimension vector such that the corresponding quiver Grassmannian is isomorphic to the Schubert variety. To obtain these isomorphisms, we construct a special quiver with relations and investigate two classes of quiver Grassmannians for this quiver.

Using the matrix-resolvent method and a formula of the second-named author on the n -point function for a KP tau-function, we show that the tau-function of an arbitrary solution to the Toda lattice hierarchy is a KP tau-function. We then generalize this result to tau-functions for the extended Toda hierarchy (ETH) by developing the matrix-resolvent method for the ETH. As an example the partition function of Gromov-Witten invariants of the complex projective line is a KP tau-function, and an application on irreducible representations of the symmetric group is obtained.

Abstract Let $$G=G_{n}=GL(n)$$ be the $$n\times n$$ complex general linear group and embed $$G_{n-1}=GL(n-1)$$ in the top left hand corner of $$G$$ . The standard Borel subgroup of upper triangular matrices $$B_{n-1}$$ of $$G_{n-1}$$ acts on the flag variety $$\mathcal {B}_{n}$$ of $$G$$ with finitely many orbits. In this paper, we show that each $$B_{n-1}$$ -orbit is the intersection of orbits of two Borel subgroups of $$G$$ acting on $$\mathcal {B}_{n}$$ . This allows us to give a new combinatorial description of the $$B_{n-1}$$ -orbits on $$\mathcal {B}_{n}$$ by associating to each orbit a pair of Weyl group elements. The closure relations for the $$B_{n-1}$$ -orbits can then be understood in terms of the Bruhat order on the symmetric group, and the Richardson-Springer monoid action on the orbits can be understood in terms of a well-understood monoid action on the symmetric group. This approach makes the closure relation more transparent than in Magyar (J. Algebraic Combin 21:71–101, 2005) and the monoid action significantly more computable than in our papers (Colarusso and Evens, J. Algebra 596:128–154, 2022) and (Colarusso and Evens, J. Algebra 619:249–297, 2023), and also allows us to obtain new information about the orbits including a simple formula for the dimension of an orbit.