Showing papers on "Symmetric group published in 2016"

A second proof of the Shareshian--Wachs conjecture, by way of a new Hopf algebra

Abstract: This is a set of working notes which give a second proof of the Shareshian--Wachs conjecture, the first (and recent) proof being by Brosnan and Chow in November 2015. The conjecture relates some symmetric functions constructed combinatorially out of unit interval graphs (their $q$-chromatic quasisymmetric functions), and some symmetric functions constructed algebro-geometrically out of Tymoczko's representation of the symmetric group on the equivariant cohomology ring of a family of subvarieties of the complex flag variety, called regular semisimple Hessenberg varieties. Brosnan and Chow's proof is based in part on the idea of deforming the Hessenberg varieties. The proof given here, in contrast, is based on the idea of recursively decomposing Hessenberg varieties, using a new Hopf algebra as the organizing principle for this recursion. We hope that taken together, each approach will shed some light on the other, since there are still many outstanding questions regarding the objects under study.

Discrete Curvature and Abelian Groups

Abstract: We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of \curvature" in discrete spaces. An appealing feature of this discrete version of the so-called 2-calculus (of Bakry- Emery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specic graphs of interest { particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities ( a la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs { a result of independent interest.

An Orthogonal Basis for Functions over a Slice of the Boolean Hypercube

Abstract: We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young's orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form $\{(x_1,\ldots,x_n) \in \{0,1\}^n : \sum_i x_i = k\}$), which refines the eigenspaces of the Johnson association scheme; our basis is orthogonal with respect to any exchangeable measure. More concretely, our basis is an orthogonal basis for all multilinear polynomials $\mathbb{R}^n \to \mathbb{R}$ which are annihilated by the differential operator $\sum_i \partial/\partial x_i$. As an application of the last point of view, we show how to lift low-degree functions from a slice to the entire Boolean hypercube while maintaining properties such as expectation, variance and $L^2$-norm. As an application of our basis, we streamline Wimmer's proof of Friedgut's theorem for the slice. Friedgut's theorem, a fundamental result in the analysis of Boolean functions, states that a Boolean function on the Boolean hypercube with low total influence can be approximated by a Boolean junta (a function depending on a small number of coordinates). Wimmer generalized this result to slices of the Boolean hypercube, working mostly over the symmetric group, and utilizing properties of Young's orthogonal representation. Using our basis, we show how the entire argument can be carried out directly on the slice.

Representations of the Infinite Symmetric Group

Abstract: Representation theory of big groups is an important and quickly developing part of modern mathematics, giving rise to a variety of important applications in probability and mathematical physics. This book provides the first concise and self-contained introduction to the theory on the simplest yet very nontrivial example of the infinite symmetric group, focusing on its deep connections to probability, mathematical physics, and algebraic combinatorics. Following a discussion of the classical Thoma's theorem which describes the characters of the infinite symmetric group, the authors describe explicit constructions of an important class of representations, including both the irreducible and generalized ones. Complete with detailed proofs, as well as numerous examples and exercises which help to summarize recent developments in the field, this book will enable graduates to enhance their understanding of the topic, while also aiding lecturers and researchers in related areas.

Mod-ϕ Convergence: Normality Zones and Precise Deviations

Abstract: In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of random variables ($X_n$)$ _{n\in{\mathbb{N}}}$. The random variables considered can be lattice or non-lattice distributed, and single or multi-dimensional; and one obtains precise estimates of the fluctuations $\mathbb{P}[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay log($\mathbb{P}[X_{n} \in t_{n}B]$In the special setting of mod-Gaussian convergence, we shall see that our approach allows us to identify the scale at which the central limit theorem ceases to hold and we are able to quantify the "breaking of symmetry" at this critical scale thanks to the residue or limiting function occurring in mod-f convergence. In particular this provides us with a systematic way to characterise the normality zone, that is the zone in which the Gaussian approximation for the tails is still valid. Besides, the residue function measures the extent to which this approximation fails to hold at the edge of the normality zone. The first sections of the article are devoted to a proof of these abstract results. We then propose new examples covered by this theory and coming from various areas of mathematics: classical probability theory (multi-dimensional random walks, random point processes), number theory (statistics of additive arithmetic functions), combinatorics (statistics of random permutations), random matrix theory (characteristic polynomials of random matrices in compact Lie groups), graph theory (number of subgraphs in a random Erd˝os-Renyi graph), and non-commutative probability theory (asymptotics of random character values of symmetric groups). In particular, we complete our theory of precise deviations by a concrete method of cumulants and dependency graphs, which applies to many examples of sums of “weakly dependent” random variables. Although the latter methods can only be applied in the more restrictive setting of mod-Gaussian convergence, the large number as well as the variety of examples which are covered there hint at a universality class for second order fluctuations.

Discrete Ricci Curvature bounds for Bernoulli-Laplace and Random Transposition models

Abstract: We calculate a Ricci curvature lower bound for some classical examples of random walks, namely, a chain on a slice of the n-dimensional discrete cube (the so-called Bernoulli–Laplace model) and the random transposition shuffle of the symmetric group of permutations on n letters. Resume. Nous calculons une borne inferieure a la courbure de Ricci pour quelques exemples classiques de marches aleatoires. Notamment nous considerons une marche sur une tranche du cube discret (dite modele de Bernoulli–Laplace) et la marche sur le groupe symetrique de permutations par transpositions aleatoires.

Symmetric unimodal expansions of excedances in colored permutations

Abstract: We consider several generalizations of the classical γ -positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the γ -positivity for Eulerian polynomials for derangements of type B . More general expansion formulae are also given for Eulerian polynomials for r -colored derangements. Our results answer and generalize several recent open problems in the literature.

Four random permutations conjugated by an adversary generate Sn with high probability

Abstract: We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group Sn generate a transitive subgroup with probability for some independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of An or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 2015

Involution words II: braid relations and atomic structures

Abstract: Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori-Hecke algebra modules, and of orbit closures in flag varieties. Specifically, to any twisted involutions $x$, $y$ in a Coxeter group $W$ with automorphism $*$, we associate a set of involution words $\hat{\mathcal{R}}_*(x,y)$. This set is the disjoint union of the reduced words of a set of group elements $\mathcal{A}_*(x,y)$, which we call the atoms of $y$ relative to $x$. The atoms, in turn, are contained in a larger set $\mathcal{B}_*(x,y) \subset W$ with a similar definition, whose elements we refer to as Hecke atoms. Our main results concern some interesting properties of the sets $\hat{\mathcal{R}}_*(x,y)$ and $\mathcal{A}_*(x,y) \subset \mathcal{B}_*(x,y)$. For finite Coxeter groups we prove that $\mathcal{A}_*(1,y)$ consists of exactly the minimal-length elements $w \in W$ such that $w^* y \leq w$ in Bruhat order, and conjecture a more general property for arbitrary Coxeter groups. In type $A$, we describe a simple set of conditions characterizing the sets $\mathcal{A}_*(x,y)$ for all involutions $x,y \in S_n$, giving a common generalization of three recent theorems of Can, Joyce, and Wyser. We show that the atoms of a fixed involution in the symmetric group (relative to $x=1$) naturally form a graded poset, while the Hecke atoms surprisingly form an equivalence class under the "Chinese relation" studied by Cassaigne, Espie, et al. These facts allow us to recover a recent theorem of Hu and Zhang describing a set of "braid relations" spanning the involution words of any self-inverse permutation. We prove a generalization of this result giving an analogue of Matsumoto's theorem for involution words in arbitrary Coxeter groups.

The full automorphism group of the power (di)graph of a finite group

Abstract: We describe the full automorphism group of the power (di)graph of a finite group. As an application, we solve a conjecture proposed by Doostabadi, Erfanian and Jafarzadeh in 2013.

Divisibility and laws in finite simple groups

Abstract: We provide new bounds for the divisibility function of the free group \({\mathbf F}_2\) and construct short laws for the symmetric groups \({{\mathrm{Sym}}}(n)\). The construction is random and relies on the classification of the finite simple groups. We also give bounds on the length of laws for finite simple groups of Lie type.

Geometric Representations of the Braid Groups

Abstract: We show that the morphisms from the braid group with n strands in the mapping class group of a surface with a possible non empty boundary, assuming that its genus is smaller or equal to n/2 are either cyclic morphisms (their images are cyclic groups), or transvections of monodromy morphisms (up to multiplication by an element in the centralizer of the image, the image of a standard generator of the braid group is a Dehn twist, and the images of two consecutive standard generators are two Dehn twists along two curves intersecting in one point). As a corollary, we determine the endomorphisms, the injective endomorphisms, the automorphisms and the outer automorphism group of the following groups: the braid group with n strands where n is greater than or equal to 6, and the mapping class group of any surface of genus greater or equal than 2. For each statement involving the mapping class group, we study both cases: when the boundary is fixed pointwise, and when each boundary component is fixed setwise. We will also describe the set of morphisms between two different braid groups whose number of strands differ by at most one, and the set of all morphisms between mapping class groups of surfaces (possibly with boundary) whose genus (greater than or equal to 2) differ by at most one.

Sigma theory and twisted conjugacy-II: Houghton groups and pure symmetric automorphism groups

Abstract: We say that $x,y\in \Gamma$ are in the same $\phi$-twisted conjugacy class and write $x\sim_\phi y$ if there exists an element $\gamma\in \Gamma$ such that $y=\gamma x\phi(\gamma^{-1})$. This is an equivalence relation on $\Gamma$ called the $\phi$-twisted conjugacy. Let $R(\phi)$ denote the number of $\phi$-twisted conjugacy classes in $\Gamma$. If $R(\phi)$ is infinite for all $\phi\in Aut(\Gamma)$, we say that $\Gamma$ has the $R_\infty$-property. The purpose of this note is to show that the symmetric group $S_\infty$, the Houghton groups and the pure symmetric automorphism groups have the $R_\infty$-property. We show, also, that the Richard Thompson group $T$ has the $R_\infty$-property. We obtain a general result establishing the $R_\infty$-property of finite direct product of finitely generated groups.

Quantum groups with partial commutation relations

Abstract: We define new noncommutative spheres with partial commutation relations for the coordinates. We investigate the quantum groups acting maximally on them, which yields new quantum versions of the orthogonal group: They are partially commutative in a way such that they do not interpolate between the classical and the free quantum versions of the orthogonal group. Likewise we define non-interpolating, partially commutative quantum versions of the symmetric group recovering Bichon's quantum automorphism groups of graphs. They fit with the mixture of classical and free independence as recently defined by Speicher and Wysoczanski (rediscovering $\Lambda$-freeness of Mlotkowski), due to some weakened version of a de Finetti theorem.

A Tableau Approach to the Representation Theory of 0-Hecke Algebras

Abstract: A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob-Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to types B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of types B and D and quasisymmetric functions and noncommutative symmetric functions of types B and D. Other applications are also provided.

Quantum Hurwitz numbers and Macdonald polynomials

Abstract: Parametric families in the center Z(C[Sn]) of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements. Their eigenvalues provide coefficients in the double Schur function expansion of 2D Toda τ-functions of hypergeometric type. Expressing these in the basis of products of power sum symmetric functions, the coefficients may be interpreted geometrically as parametric families of quantum Hurwitz numbers, enumerating weighted branched coverings of the Riemann sphere. Combinatorially, they give quantum weighted sums over paths in the Cayley graph of Sn generated by transpositions. Dual pairs of bases for the algebra of symmetric functions with respect to the scalar product in which the Macdonald polynomials are orthogonal provide both the geometrical and combinatorial significance of these quantum weighted enumerative invariants.

Flag‐Transitive Point‐Primitive Automorphism Groups of Nonsymmetric 2−(v,k,2) Designs

Abstract: In this paper, we show that if \({\mathcal {D}}\) is a non-trivial non-symmetric 2-(v, k, 3) design admitting a flag-transitive point-primitive automorphism group G, then G must be an affine or almost simple group.

Modular generalized Springer correspondence I: the general linear group

Abstract: We define a generalized Springer correspondence for the group GL(n) over any field. We also determine the cuspidal pairs, and compute the correspondence explicitly. Finally we define a stratification of the category of equivariant perverse sheaves on the nilpotent cone of GL(n) satisfying the 'recollement' properties, and with subquotients equivalent to categories of representations of a product of symmetric groups.

$$G(\ell ,k,d)$$G(ℓ,k,d)-modules via groupoids

Abstract: In this note, we describe a seemingly new approach to the complex representation theory of the wreath product $$G\wr S_d$$G?Sd, where G is a finite abelian group. The approach is motivated by an appropriate version of Schur---Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of $$G\wr S_d$$G?Sd. This directly implies a classification of simple modules. As an application, we get a Gelfand model for $$G\wr S_d$$G?Sd from the classical involutive Gelfand model for the symmetric group. We describe the Schur---Weyl duality which motivates our approach and relate it to various Schur---Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type $$G(\ell ,k,d)$$G(l,k,d).

Permutations and the combinatorics of gauge invariants for general $N$

Abstract: Group algebras of permutations have proved highly useful in solving a number of problems in large N gauge theories. I review the use of permutations in classifying gauge invariants in one-matrix and multi-matrix models and computing their correlators. These methods are also applicable to tensor models and have revealed a link between tensor models and the counting of branched covers. The key idea is to parametrize $U(N)$ gauge invariants using permutations, subject to equivalences. Correlators are related to group theoretic properties of these equivalence classes. Fourier transformation on symmetric groups by means of representation theory offers nice bases of functions on these equivalence classes. This has applications in AdS/CFT in identifying CFT duals of giant gravitons and their perturbations. It has also lead to general results on quiver gauge theory correlators, uncovering links to two dimensional topological field theory and the combinatorics of trace monoids.

Some Other Algebraic Properties Of Folded Hypercubes.

Abstract: We construct explicity the automorphism group of the folded hypercube $FQ_n$ of dimension $n>3$, as a semidirect product of $N$ by $M$, where $N$ is isomorphic to the Abelian group $Z_2^n$, and $M$ is isomorphic to $Sym(n+1)$, the symmetric group of degree $n+1$, then we will show that the folded hypercube $FQ_n$ is a symmetric graph.

Normalizers of Primitive Permutation Groups

Abstract: Let $G$ be a transitive normal subgroup of a permutation group $A$ of finite degree $n$. The factor group $A/G$ can be considered as a certain Galois group and one would like to bound its size. One of the results of the paper is that $|A/G| < n$ if $G$ is primitive unless $n = 3^{4}$, $5^4$, $3^8$, $5^8$, or $3^{16}$. This bound is sharp when $n$ is prime. In fact, when $G$ is primitive, $|\mathrm{Out}(G)| < n$ unless $G$ is a member of a given infinite sequence of primitive groups and $n$ is different from the previously listed integers. Many other results of this flavor are established not only for permutation groups but also for linear groups and Galois groups.