Showing papers on "Symmetric group published in 2022"

OUP accepted manuscript

Abstract: Fix a word $w$ in a free group $F$ on $r$ generators. A $w$-random permutation in the symmetric group $S_N$ is obtained by sampling $r$ independent uniformly random permutations $\sigma_{1},\ldots,\sigma_{r}\in S_{N}$ and evaluating $w\left(\sigma_{1},\ldots,\sigma_{r}\right)$. In [arXiv:1104.3991, arXiv:1202.3269] it was shown that the average number of fixed points in a $w$-random permutation is $1+\theta\left(N^{1-\pi\left(w\right)}\right)$, where $\pi\left(w\right)$ is the smallest rank of a subgroup $H\le F$ containing $w$ as a non-primitive element. We show that $\pi\left(w\right)$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge2$, the average number of $t$-cycles is $\frac{1}{t}+O\left(N^{-\pi\left(w\right)}\right)$. As an application, we prove that for every $s$, every $\varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have second eigenvalue at most $2\sqrt{2r-1}+\varepsilon$ asymptotically almost surely. An important ingredient in this work is a systematic study of not-necessarily connected Stallings core graphs.

Invariant theory for the free left-regular band and a q-analogue

Abstract: We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by D\'esarm\'enien and Wachs.

Intersecting principal Bruhat ideals and grades of simple modules

Abstract: We prove that the grades of simple modules indexed by boolean permutations, over the incidence algebra of the symmetric group with respect to the Bruhat order, are given by Lusztig's $\mathbf{a}$-function. Our arguments are combinatorial, and include a description of the intersection of two principal order ideals when at least one permutation is boolean. An important object in our work is a reduced word written as minimally many runs of consecutive integers, and one step of our argument shows that this minimal quantity is equal to the length of the second row in the permutation's shape under the Robinson-Schensted correspondence. We also prove that a simple module over the above-mentioned incidence algebra is perfect if and only if its index is the longest element of a parabolic subgroup.Mathematics Subject Classifications: 20F55, 06A07, 05E15

A Novel Method to Generate Key-Dependent S-Boxes with Identical Algebraic Properties

Abstract: The s-box plays the vital role of creating confusion between the ciphertext and secret key in any cryptosystem, and is the only nonlinear component in many block ciphers. Dynamic s-boxes, as compared to static, improve entropy of the system, hence leading to better resistance against linear and differential attacks. It was shown in [2] that while incorporating dynamic s-boxes in cryptosystems is sufficiently secure, they do not keep non-linearity invariant. This work provides an algorithmic scheme to generate key-dependent dynamic $n\times n$ clone s-boxes having the same algebraic properties namely bijection, nonlinearity, the strict avalanche criterion (SAC), the output bits independence criterion (BIC) as of the initial seed s-box. The method is based on group action of symmetric group $S_n$ and a subgroup $S_{2^n}$ respectively on columns and rows of Boolean functions ($GF(2^n)\to GF(2)$) of s-box. Invariance of the bijection, nonlinearity, SAC, and BIC for the generated clone copies is proved. As illustration, examples are provided for $n=8$ and $n=4$ along with comparison of the algebraic properties of the clone and initial seed s-box. The proposed method is an extension of [3,4,5,6] which involved group action of $S_8$ only on columns of Boolean functions ($GF(2^8)\to GF(2)$ ) of s-box. For $n=4$, we have used an initial $4\times 4$ s-box constructed by Carlisle Adams and Stafford Tavares [7] to generated $(4!)^2$ clone copies. For $n=8$, it can be seen [3,4,5,6] that the number of clone copies that can be constructed by permuting the columns is $8!$. For each column permutation, the proposed method enables to generate $8!$ clone copies by permuting the rows.

The cohomology class of the mod 4 braid group

Abstract: The mod 4 braid group, [Formula: see text], is defined to be the quotient of the braid group by the subgroup of the pure braid group generated by squares of all elements, [Formula: see text]. Kordek and Margalit proved [Formula: see text] is an extension of the symmetric group by [Formula: see text]. For [Formula: see text], we construct a 2-cocycle in the group cohomology of the symmetric group with twisted coefficients classifying [Formula: see text]. We show this cocycle is the mod 2 reduction of the 2-cocycle corresponding to the extension of the symmetric group by the abelianization of the pure braid group. We also construct the 2-cocycle corresponding to this second extension and show that it represents an order two element in the cohomology of the symmetric group. Furthermore, we give presentations for both extensions and a normal generating set for [Formula: see text].

Stack-sorting for Coxeter groups

Abstract: Given an essential semilattice congruence \(\equiv\) on the left weak order of a \linebreak Coxeter group \(W\), we define the Coxeter stack-sorting operator \({\bf S}_\equiv:W\to W\) by \({{\bf S}_\equiv(w)=w\big(\pi_\downarrow^\equiv(w)\big)^{-1}}\), where \(\pi_\downarrow^\equiv(w)\) is the unique minimal element of the congruence class of \(\equiv\) containing \(w\). When \(\equiv\) is the sylvester congruence on the symmetric group \(S_n\), the operator \({\bf S}_\equiv\) is West's stack-sorting map. When \(\equiv\) is the descent congruence on \(S_n\), the operator \({\bf S}_\equiv\) is the pop-stack-sorting map. We establish several general results about Coxeter stack-sorting operators, especially those acting on symmetric groups. For example, we prove that if \(\equiv\) is an essential lattice congruence on \(S_n\), then every permutation in the image of \({\bf S}_\equiv\) has at most \(\left\lfloor\frac{2(n-1)}{3}\right\rfloor\) right descents; we also show that this bound is tight. We then introduce analogues of permutree congruences in types \(B\) and \(\widetilde A\) and use them to isolate Coxeter stack-sorting operators \(\mathtt{s}_B\) and \(\widetilde{\mathtt{s}}\) that serve as canonical type-\(B\) and type-\(\widetilde A\) counterparts of West's stack-sorting map. We prove analogues of many known results about West's stack-sorting map for the new operators \(\mathtt{s}_B\) and \(\widetilde{\mathtt{s}}\). For example, in type \(\widetilde A\), we obtain an analogue of Zeilberger's classical formula for the number of \(2\)-stack-sortable permutations in \(S_n\).Mathematics Subject Classifications: 06A12, 06B10, 37E15, 05A05, 05E16

Towards combinatorial invariance for Kazhdan-Lusztig polynomials

Abstract: Kazhdan-Lusztig polynomials are important and mysterious objects in representation theory. Here we present a new formula for their computation for symmetric groups based on the Bruhat graph. Our approach suggests a solution to the combinatorial invariance conjecture for symmetric groups, a well-known conjecture formulated by Lusztig and Dyer in the 1980s.

Stable Centres of Iwahori-Hecke Algebras of Type A

Abstract: A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres $Z(\mathbb {Z}S_{n})$ of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to $\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }$ (where $\mathcal {R}$ is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

A "power" conjugate equation in the symmetric group

Abstract: First we consider the solutions of the general "cubic" equation a_{1}x^{r1}a_{2}x^{r2}a_{3}x^{r3}=1 (with r1,r2,r3 in {1,-1}) in the symmetric group S_{n}. In certain cases this equation can be rewritten as aya^{-1}=y^{2} or as aya^{-1}=y^{-2}, where a in S_{n} depends on the a_{i}'s and the new unknown permutation y in S_{n} is a product of x (or x^{-1}) and one of the permutations a_{i}^{1} and a_{i}^{-1}. Using combinatorial arguments and some basic number theoretical facts, we obtain results about the solutions of the so-called power conjugate equation aya^{-1}=y^{e} in S_{n}, where e is an integer exponent. Under certain conditions, the solutions are exactly the solutions of y^{e-1}=1 in the centralizer of a.

Schur–Weyl Duality for Twin Groups

Abstract: The twin group TWn on n strands is the group generated by $t_{1}, \dots , t_{n-1}$ with defining relations ${t_{i}^{2}}=1$ , titj = tjti if |i − j| > 1. We find a new instance of semisimple Schur–Weyl duality for tensor powers of a natural n-dimensional reflection representation of TWn, depending on a parameter q. At q = 1, the representation coincides with the natural permutation representation of the symmetric group, so the new Schur–Weyl duality may be regarded as a q-analogue of the one motivating the definition of the partition algebra.

The weak order on the hyperoctahedral group and the monomial basis for the Hopf algebra of signed permutations

Abstract: We give a combinatorial description for the weak order on the hyperoctahedral group. This characterization is then used to analyze the order-theoretic properties of the shifted products of hyperoctahedral groups. It is shown that each shifted product is a disjoint union of some intervals, which can be convex embedded into a hyperoctahedral group. As an application, we investigate the monomial basis for the Hopf algebra $\mathfrak{H}Sym$ of signed permutations, related to the fundamental basis via M\"obius inversion on the weak order on hyperoctahedral groups. It turns out that the image of a monomial basis element under the descent map from $\mathfrak{H}Sym$ to the algebra of type $B$ quasi-symmetric functions is either zero or a monomial quasi-symmetric function of type $B$.

Permutation Groups Generated by γ-Cycles

Abstract: A γ-cycle is a cycle of the form (i+1,i+2,⋯,i+m) in the symmetric group Sn. We study the subgroups of Sn generated by several sets of γ-cycles. Our mathematical development is strongly supported by computational experiments and proofs based on do-it-yourself programming with the logic-based language Maude.

A Note on Two-Generator 2-Group Covers of Cubic Symmetric Graphs of Order 2p

Abstract: Let [Formula: see text] be a prime. In this paper, a complete classification of edge-transitive [Formula: see text]-covers of a cubic symmetric graph of order [Formula: see text] is given for the case when [Formula: see text] is a two-generator 2-group whose derived subgroup is either isomorphic to [Formula: see text] or generated by at most two elements. As an application, it is shown that 11 is the smallest value of [Formula: see text] for which there exist infinitely many cubic semisymmetric graphs with order of the form [Formula: see text].

Invariant Theory for the free left-regular band and a q-analogue

Abstract: We examine from an invariant theory viewpoint the monoid algebras for two monoids having large symmetry groups. The first monoid is the free left-regular band on $n$ letters, defined on the set of all injective words, that is, the words with at most one occurrence of each letter. This monoid carries the action of the symmetric group. The second monoid is one of its $q$-analogues, considered by K. Brown, carrying an action of the finite general linear group. In both cases, we show that the invariant subalgebras are semisimple commutative algebras, and characterize them using Stirling and $q$-Stirling numbers. We then use results from the theory of random walks and random-to-top shuffling to decompose the entire monoid algebra into irreducibles, simultaneously as a module over the invariant ring and as a group representation. Our irreducible decompositions are described in terms of derangement symmetric functions introduced by D\'esarm\'enien and Wachs.

Splitting the square of homogeneous and elementary functions into their symmetric and anti-symmetric parts

Abstract: We consider the expansion of the square of a complete homogeneous function $h_\lambda$, or of an elementary symmetric function $e_\lambda$, in the basis of Schur functions. This square also decomposes into two plethysms, $s_2[h_\lambda]$ and $s_{11}[h_\lambda]$ (resp. $s_2[h_\lambda]$ and $s_{11}[h_\lambda]$), which are called its symmetric and anti-symmetric parts, respectively. We define a sign statistic on the set of tableaux that index the Schur functions appearing in the square of those symmetric functions. This sign statistic allows to determine to which plethysm each Schur function contributes. We use mainly combinatorial tools on tableaux (product on tableau and RSK) and basic manipulations on plethysm and symmetric functions.

Second Subgroup of General Linear Group in Dimention 2

Abstract: In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ Sn is said to be non-synchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number. When trying to check if some group is synchronizing, typically, we have only partial information about the graph but enough to say that it has an automorphism of some type, and the goal would be to have in hand a classification of the primitive groups containing permutations of that type. As an illustration of this, the key ingredient in some of the results in [2] was the observation that the primitive graph understudy has a 2-cycle automorphism and hence the automorphism group of the graph is the symmetric group. For many more examples of the importance of knowing the groups that contain permutations of a given type. This type of investigation is certainly very natural since it appears on the eve of group theory, with Jordan, Burnside, Marggraff, but the difficulty of the problem is well illustrated by the very slow progress throughout the twentieth century. Given the new tools available (chiefly the classification of finite simple groups), the topic seems to have new momentum. Let Sn denote the symmetric group on n points; a permutation g∈Sn is said to be imprimitive if there exists an imprimitive group contain in gg. An imprimitive group is said to beminimally imprimitiveifit contains no transitive proper subgroup. An imprimitive group G≤Sn is said to be maximally imprimitive if for all g Sn\G, the group〈g, G 〉 is primitive. The next result, whose proof isstraightforward, provides some alternative characterizations of imprimitive permutations. Now in this paper we discuse Presentation for imprimitive second Subgroup of general linear group in dimention 2 over the field of pk-elements .