Group Sn Research Articles

On a mod 3 property of ℓ-tuples of pairwise commuting permutations

Let Sn denote the symmetric group of permutations acting on n elements. We investigate the double sequence {Nℓ(n)} counting the number of ℓ tuples of elements of the symmetric group Sn, where the components commute, normalized by the order of Sn. Our focus lies on exploring log-concavity with respect to n: Nℓ(n)2-Nℓ(n-1)Nℓ(n+1)≥0.We establish that this depends on n(mod3) for sufficiently large ℓ. These numbers are studied by Bryan and Fulman as the nth orbifold characteristics, generalizing work by Macdonald and Hirzebruch–Hofer concerning the ordinary and string-theoretic Euler characteristics of symmetric products. Notably, N2(n) represents the partition numbers p(n), while N3(n) represents the number of non-equivalent n-sheeted coverings of a torus studied by Liskovets and Medynkh. The numbers also appear in algebra since |Sn|Nℓ(n)=HomZℓ,Sn.

Symmetric group gauge theories and simple gauge/string dualities

Abstract We study two-dimensional topological gauge theories with gauge group equal to the symmetric group Sn and their string theory duals. The simplest such theory is the topological quantum field theory of principal Sn fiber bundles. Its correlators are equal to Hurwitz numbers. The operator products in the gauge theory for each finite value of n are coded in one partial permutation algebra. We propose a generalization of the partial permutation algebra to the symmetric orbifold topological quantum field theory of any seed theory and show that the theory factorizes into marked partial permutation combinatorics and seed Frobenius algebra properties. Moreover, we exploit the established correspondence between Hurwitz theory and the stationary sector of Gromov–Witten theory on the sphere to prove an exact gauge/string duality. The relevant field theory is a grand canonical version of Hurwitz theory and its two-point functions are obtained by summing over all values of the instanton degree of the maps covering the sphere. We stress that one must look for a multiplicative basis on the boundary to match the bulk operator algebra of single string insertions. The relevant boundary observables are completed cycles.

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.

Constructing flag-transitive, point-primitive 2-designs from complete graphs

In this paper, we study 2-designs D=(P,BSn), where P can be viewed as the edge set of the complete graph Kn, and B can be identified as the edge set of a subgraph of Kn. We give a necessary condition for Sn to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group Sn. As an application, all pairs (D,G) are determined, where D is a 2-(v,k,λ) design with gcd⁡(v−1,k−1)=3 or 4, and G is flag-transitive with Soc(G)=An for n≥5. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-(v,k,λ) designs with gcd⁡(v−1,k−1)≤(v−1)1/2 and alternating socle An with v=(n2).

Probabilistic Galois theory in function fields

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial f=yn+∑i=0n−1ai(x)yi∈Fq[x][y] with i.i.d. coefficients ai taking values in the set {a(x)∈Fq[x]:deg⁡a≤d} with uniform probability, is irreducible with probability tending to 1−1qd as n→∞, where d and q are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group An. Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over Fq[x], then the Galois group of this polynomial is actually equal to the symmetric group Sn with probability tending to 1−1qd. We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and d→∞.

Excessive symmetry can preclude cutoff

In this paper we look at the families of random walks arising from FI-graphs. One may think of these objects as families of nested graphs, each equipped with a natural action by a symmetric group Sn, such that these actions are compatible and transitive. Families of graphs of this form were introduced by the authors in [9], while a systematic study of random walks on these families were considered in [10]. In the present work, we illustrate that these random walks never exhibit the so-called product condition, and therefore also never display total variation cutoff as defined by Aldous and Diaconis [1]. In particular, we provide a large family of algebro-combinatorially motivated examples of collections of Markov chains which satisfy some well-known algebraic heuristics for cutoff, while not actually having the property.

On the chirality of the pseudo-Hurwitz maps with alternating and symmetric groups

In a previous paper the authors with M. Conder proved that the alternating group An and the symmetric group Sn are pseudo-Hurwitz for any n>23, as well as are A18, A19, A21, A22 and A23, and also S10, S17, S19, S20, S21 and S22. In this paper, we provide geometric tools for constructing all pseudo-Hurwitz groups up to a “small” degree. This construction, applied up to degree 23, shows that there are no other degrees for which An and Sn are pseudo-Hurwitz.In the second part of this paper we show that there are both reflexible and chiral pseudo-Hurwitz maps with automorphism group An and Sn, for every n⩾36.

Infinite Families of Congruences Modulo 5 for the Number of Irreducible Characters of the Alternating Groups

In 1919, Ramanujan proved red three congruences for the partition function p(n) which denotes the number of partitions of n. The partition function p(n) can be understood as the number of irreducible characters of the symmetric group Sn . Recently, Nath and Sellers, and Xia established a number of congruences modulo 2, 3, and 5 for the numbers of spin characters of S ̂ n and A ̂ n , where S ̂ n and A ̂ n are the double covering group of Sn and the double covering group of the alternating group An , respectively. Motivated by their work, we establish infinite families of congruences modulo 5 for a(n), which denotes the number of irreducible characters of the alternating group An . In particular, we prove some strange congruences modulo 5 for a(n). For example, we prove that for k ≥ 0 , a ( 95 × 73 2 k + 1 24 ) ≡ − 3 k ( mod 5 ) .

Commutator subgroups and crystallographic quotients of virtual extensions of symmetric groups

The virtual braid group VBn, the virtual twin group VTn and the virtual triplet group VLn are extensions of the symmetric group Sn, which are motivated by the Alexander-Markov correspondence for virtual knot theories. The kernels of natural epimorphisms of these groups onto the symmetric group Sn are the pure virtual braid group VPn, the pure virtual twin group PVTn and the pure virtual triplet group PVLn, respectively. In this paper, we investigate commutator subgroups, pure subgroups and crystallographic quotients of these groups. We derive explicit finite presentations of the pure virtual triplet group PVLn, the commutator subgroup VTn′ of VTn and the commutator subgroup VLn′ of VLn. Our results complete the understanding of these groups, except that of VBn′, for which the existence of a finite presentation is not known for n≥4. We also prove that VLn/PVLn′ is a crystallographic group and give an explicit construction of infinitely many torsion elements in it.

Gauged permutation invariant matrix quantum mechanics: path integrals

We give a path integral construction of the quantum mechanical partition function for gauged finite groups. Our construction gives the quantization of a system of d, N × N matrices invariant under the adjoint action of the symmetric group SN. The approach is general to any discrete group. For a system of harmonic oscillators, i.e. for the non-interacting case, the partition function is given by the Molien-Weyl formula times the zero-point energy contribution. We further generalise the result to a system of non-square and complex matrices transforming under arbitrary representations of the gauge group.

Construction of the beta distributions using the random permutation divisors

A subset of cycles comprising a permutation σ in the symmetric group Sn, n ∈ N, is called a divisor of σ. Then the partial sums over divisors with sizes up to un, 0 ≤ u ≤ 1, of values of a nonnegative multiplicative function on a random permutation define a stochastic process with nondecreasing trajectories. When normalized the latter is just a random distribution function supported by the unit interval. We establish that its expectations under various weighted probability measures defined on the subsets of Sn are quasihypergeometric distribution functions. Their limits as n -> 1 cover the class of two-parameter beta distributions. It is shown that, under appropriate conditions, the convergence rate is of the negative power of n order.

Matrix factorizations of the discriminant of Sn

Consider the symmetric group Sn acting as a reflection group on the polynomial ring k[x1,…,xn] where k is a field, such that Char(k) does not divide n!. We use Higher Specht polynomials to construct matrix factorizations of the discriminant of this group action: these matrix factorizations are indexed by partitions of n and respect the decomposition of the coinvariant algebra into isotypical components. The maximal Cohen–Macaulay modules associated to these matrix factorizations give rise to a noncommutative resolution of the discriminant and they correspond to the nontrivial irreducible representations of Sn. All our constructions are implemented in Macaulay2 and we provide several examples. We also discuss extensions of these results to Young subgroups of Sn and indicate how to generalize them to the reflection groups G(m,1,n).

Theoretical guarantees for permutation-equivariant quantum neural networks

AbstractDespite the great promise of quantum machine learning models, there are several challenges one must overcome before unlocking their full potential. For instance, models based on quantum neural networks (QNNs) can suffer from excessive local minima and barren plateaus in their training landscapes. Recently, the nascent field of geometric quantum machine learning (GQML) has emerged as a potential solution to some of those issues. The key insight of GQML is that one should design architectures, such as equivariant QNNs, encoding the symmetries of the problem at hand. Here, we focus on problems with permutation symmetry (i.e., symmetry group Sn), and show how to build Sn-equivariant QNNs We provide an analytical study of their performance, proving that they do not suffer from barren plateaus, quickly reach overparametrization, and generalize well from small amounts of data. To verify our results, we perform numerical simulations for a graph state classification task. Our work provides theoretical guarantees for equivariant QNNs, thus indicating the power and potential of GQML.

Harish-Chandra bimodules for type A rational Cherednik algebras

We study Harish-Chandra bimodules for the rational Cherednik algebra associated to the symmetric group Sn. In particular, we show that for any parameter c∈C, the category of Harish-Chandra Hc-bimodules admits a fully faithful embedding into the category Oc, and describe the irreducibles in the image. We also construct a duality on the category of Harish-Chandra bimodules, and in fact we do this in a greater generality of quantizations of Nakajima quiver varieties. We use this duality, along with induction and restriction functors, to describe, as an abelian category, the smallest Serre subcategory of the category of Harish-Chandra bimodules containing the regular bimodule, as well as to explicitly describe the tensor products of its irreducible objects.

Hybrid Iodide Perovskites of Divalent Alkaline Earth and Lanthanide Elements.

Hybrid halide perovskites AMIIX3 (A = ammonium cation, MII = divalent cation, X = Cl, Br, I) have been extensively studied but have only previously been reported for the divalent carbon group elements Ge, Sn, and Pb. While they have displayed an impressive range of optoelectronic properties, the instability of GeII and SnII and the toxicity of Pb have stimulated significant interest in finding alternatives to these carbon group-based perovskites. Here, we describe the low-temperature solid-state synthesis of five new hybrid iodide perovskites centered around divalent alkaline earth and lanthanide elements, with the general formula AMIII3 (A = methylammonium, MA; MII = Sr, Sm, Eu, and A = formamidinium, FA; MII = Sr, Eu). Structural, calorimetric, optical, photoluminescence, and magnetic properties of these materials are reported.

On the difference graph of power graphs of finite groups

The power graph of a finite group G is the simple undirected graph with vertex set G whose two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is the simple undirected graph whose vertex set is the group G whose two vertices a and b are adjacent if there exists c ∈ G such that both a and b are powers of c. In this paper, we investigate the difference graph Ɗ(G) of a finite group G, which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We first characterize an arbitrary finite group G such that Ɗ(G) is a chordal graph, star graph, dominatable, threshold graph, and split graph. From this, we conclude that the latter four graph classes are equal for Ɗ(G). By applying these results, we classify the nilpotent groups G such that Ɗ(G) belong to the aforementioned five graph classes. This shows that all these graph classes are equal for Ɗ(G) when G is nilpotent. Then, we characterize the nilpotent groups whose difference graphs are cograph, bipartite, Eulerian, planar, and outerplanar. Finally, we consider the difference graph of non-nilpotent groups and determine the values of n such that the difference graphs of the symmetric group Sn and alternating group An are cograph, chordal, split, and threshold.

Ramanujan sums and rectangular power sums

For a fixed nonnegative integer u and positive integer n, we investigate the symmetric function∑d|n(cd(nd))updnd, where pn denotes the nth power sum symmetric function, and cd(r) is a Ramanujan sum, equal to the sum of the rth powers of all the primitive dth roots of unity. We establish the Schur positivity of these functions for u=0 and u=1, showing that, in each case, the associated representation of the symmetric group Sn decomposes into a sum of Foulkes representations, that is, representations induced from the irreducibles of the cyclic subgroup generated by the long cycle. We also conjecture Schur positivity for the case u=2.

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.

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.

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.