Permutation Group Sn Research Articles

Bootstrapping the mathcal{N} = 1 Wess-Zumino models in three dimensions

Using numerical bootstrap method, we determine the critical exponents of the minimal three-dimensional mathcal{N} = 1 Wess-Zumino models with cubic superpotetential mathcal{W}sim {d}_{ijk}{Phi}^i{Phi}^j{Phi}^k . The tensor dijk is taken to be the invariant tensor of either permutation group SN, special unitary group SU(N), or a series of groups called F4 family of Lie groups. Due to the equation of motion, at the Wess-Zumino fixed point, the operator dijkΦjΦk is a (super)descendant of Φi. We observe such super-multiplet recombination in numerical bootstrap, which allows us to determine the scaling dimension of the super-field ∆Φ.

Converting immanants on skew-symmetric matrices

Let Qn denote the space of all n×n skew-symmetric matrices over the complex field C and T:Qn→Qn be a map satisfying the condition dχ′(T(A)+zT(B))=dχ(A+zB) for all matrices A,B∈Qn and all constants z∈C. Here χ and χ′ are irreducible characters of the permutation group Sn and dχ(C) denotes the immanant of the matrix C associated with the character χ. Let Pn be the set of permutations with no cycles of odd length in the decomposition into the product of independent cycles. The main goal of this paper is twofold. The first one is to show that there are no maps T satisfying the above conditions if n≥6 is even and χ and χ′ are not proportional on Pn. The second is to characterize such maps T if χ and χ′ are proportional on Pn. In particular, we prove that T is bijective and linear. Observe that the general problem of the characterization for bijective linear maps on Qn, that convert one immanant into another, remained an open problem with the exception for determinant and permanent. Our results include all known characterizations for immanant preserving and converting maps and provide the complete solution of this problem.

A new metric for statistical properties of long time behaviors

Let (X,T) be a topological dynamical system with metric d. We define a new function F‾(x,y)=limsupn→+∞infσ∈Sn⁡1n∑k=1nd(Tkx,Tσ(k)y) by using permutation group Sn. It's shown F(x,y)=limn→+∞⁡infσ∈Sn⁡1n∑k=1nd(Tkx,Tσ(k)y) exists when x,y∈X are generic points. Applying this function, we prove (X,T) is uniquely ergodic if and only if F‾(x,y)=0 for any x,y∈X. The characterizations of ergodic measures and physical measures by F‾(x,y) are given. We introduce the notion of weak mean equicontinuity and prove that (X,T) is weak mean equicontinuous if and only if the time averages f⁎(x)=limn→+∞⁡1n∑k=1nf(Tkx) are continuous for all f∈C(X).

Symmetric group and the Axelrod model for dissemination of cultures

We review the Axelrod’s model for dissemination of cultures from an analytic point of view. We define ⟨s(t)⟩ to quantify possible culture configurations at time t in a society. Typical initial culture configurations of this model are characterised. Equation of motion in terms of ⟨s(t)⟩ is derived. We study the dynamic of this Axelrod system toward to its culture configurations equilibrium. Generically, we observe that this model undergoes three phases of development. We give a quantitative explanation about the latter. We characterize the culture configurations space at equilibrium point where 〈s(teq)〉=1. This space is called monoculture space. Understanding this space is equivalent to restrict to the space of culture configurations from one individual in the model. This individual cultural space is identified to the space VN⊗F up to isomorphisms. In this model, N and F are respectively the dimension and the number of attributes to define an individual state. VN is a N-dimensional vector space. Action of the permutation group SN on the space VN⊗F is considered. Under this action, the observable ⟨s(t)⟩ is an invariant of the Axelrod system. We explore this symmetry and classify the different inequivalent classes of culture configurations composing the monoculture space. To achieve this, we consider the case N ≥ F. We propose techniques from group representation theory to perform this classification. The inequivalent classes of culture configurations are indexed by the Dickau diagrams which are associated to the Bell number BF. A concrete example with F=4 and N ≥ 4 is considered for a full illustration of our analysis.

The automorphism group of [formula omitted] and [formula omitted

In this paper we show that the automorphism groups of M0,ntrop and M‾0,ntrop are isomorphic to the permutation group Sn for n≥5, while the automorphism groups of M0,4trop and M‾0,4trop are isomorphic to the permutation group S3.

The Full Classification of Orthogonal Easy Quantum Groups

We study easy quantum groups, a combinatorial class of orthogonal quantum groups introduced by Banica–Speicher in 2009. We show that there is a countable descending chain of easy quantum groups interpolating between Bichon’s free wreath product with the permutation group Sn and a semi-direct product of a permutation action of Sn on a free product. This reveals a series of new commutation relations interpolating between a free product construction and the tensor product. Furthermore, we prove a dichotomy result saying that every hyperoctahedral easy quantum group is either part of our new interpolating series of quantum groups or belongs to a class of semi-direct product quantum groups recently studied by the authors. This completes the classification of easy quantum groups. We also study combinatorial and operator algebraic aspects of the new interpolating series.

Existence and examples of quantum isometry groups for a class of compact metric spaces

We formulate a definition of isometric action of a compact quantum group (CQG) on a compact metric space, generalizing Banica's definition for finite metric spaces. For metric spaces (X,d) which can be isometrically embedded in some Euclidean space, we prove the existence of a universal object in the category of the compact quantum groups acting isometrically on (X,d). In fact, our existence theorem applies to a larger class, namely for any compact metric space (X,d) which admits a one-to-one continuous map f:X→Rn for some n such that d0(f(x),f(y))=ϕ(d(x,y)) (where d0 is the Euclidean metric) for some homeomorphism ϕ of R+.As concrete examples, we obtain Wang's quantum permutation group Sn+ and also the free wreath product of Z2 by Sn+ as the quantum isometry groups for certain compact connected metric spaces constructed by taking topological joins of intervals in [13].

Full ranking as local descriptor for visual recognition: A comparison of distance metrics on sn

In this paper we propose to use the full ranking of a set of pixels as a local descriptor. In contrast to existing methods which use only partial ranking information, the full ranking encodes the complete comparative information among the pixels, while retaining invariance to monotonic photometric transformations. The descriptor is used within the bag-of-visual-words paradigm for visual recognition. We demonstrate that the choice of distance metric for assigning the descriptors to visual words is crucial to the performance, and provide an extensive evaluation of eight distance metrics for the permutation group Sn on four widely used face verification and texture classification benchmarks. The results demonstrate that (1) full ranking of pixels encodes more information than partial ranking, consistently leading to better performance; (2) full ranking descriptor can be trivially made rotation invariant; (3) the proposed descriptor applies to both image intensities and filter responses, and is capable of producing state-of-the-art performance.

On the representation theory of partition (easy) quantum groups

Abstract Compact matrix quantum groups are strongly determined by their intertwiner spaces, due to a result by S. L. Woronowicz. In the case of easy quantum groups (also called partition quantum groups), the intertwiner spaces are given by the combinatorics of partitions, see the initial work of T. Banica and R. Speicher. The philosophy is that all quantum algebraic properties of these objects should be visible in their combinatorial data. We show that this is the case for their fusion rules (i.e. for their representation theory). As a byproduct, we obtain a unified approach to the fusion rules of the quantum permutation group S N + ${S_{N}^{+}}$ , the free orthogonal quantum group O N + ${O_{N}^{+}}$ as well as the hyperoctahedral quantum group H N + ${H_{N}^{+}}$ . We then extend our work to unitary easy quantum groups and link it with a “freeness conjecture” of T. Banica and R. Vergnioux.

Bruhat Order in Full Symmetric Toda System

In this paper we discuss some geometrical and topological properties of the full symmetric Toda system. We show by a direct inspection that the phase transition diagram for the full symmetric Toda system in dimensions n = 3, 4 coincides with the Hasse diagram of the Bruhat order of symmetric groups S 3 and S 4. The method we use is based on the existence of a vast collection of invariant subvarieties of the Toda flow in orthogonal groups. We show how one can extend it to the case of general n. The resulting theorem identifies the set of singular points of dim = n Toda flow with the elements of the permutation group S n , so that points will be connected by a trajectory, if and only if the corresponding elements are Bruhat comparable. We also show that the dimension of the submanifolds, spanned by the trajectories connecting two singular points, is equal to the length of the corresponding segment in the Hasse diagram. This is equivalent to the fact that the full symmetric Toda system is in fact a Morse–Smale system.

Interpolation via symmetric exponential functions

Complex valued functions on the Euclidean space ℝn, symmetric or antisymmetric with respect to the permutation group Sn, are often dealt with in various branches of physics, such as quantum theory or theory of integrable systems. One often needs to approximate such functions with series consisting of various special functions which satisfy nice properties. Questions of uniform convergence of such approximations are crucial for applications. In this article a family of special functions called the symmetric exponential functions are used for such approximation and the uniform convergence of their sums is considered.

Alternating multivariate trigonometric functions and corresponding Fourier transforms

We define and study multivariate sine and cosine functions, symmetric with respect to the alternating group An, which is a subgroup of the permutation (symmetric) group Sn. These functions are eigenfunctions of the Laplace operator. They determine Fourier-type transforms. There exist three types of such transforms: expansions into corresponding sine-Fourier and cosine-Fourier series, integral sine-Fourier and cosine-Fourier transforms, and multivariate finite sine and cosine transforms. In all these transforms, alternating multivariate sine and cosine functions are used as a kernel.

Polydiagonal compactification of configuration spaces

A smooth compactification X〈n〉 of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group Sn manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of De Concini–Procesi, and it has a strata-compatible surjection onto the Fulton–MacPherson compactification. The degenerate configurations added in the compactification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson. In characteristic 0, isotropy subgroups of the action of Sn on X〈n〉 are abelian, thus X〈n〉 may be a step toward an explicit resolution of singularities of the symmetric products X/Sn.

Fractional superLie algebras and groups

The nth root of a Lie algebra and its dual (that is the fractional supergroup) based on the permutation group Sn invariant forms is formulated in the Hopf algebra formalism. Detailed discussion of S3-graded sl(2) algebras is performed.

Symmetrization of wave functions of 4- and 5-atom molecules

Symmetrized wave functions of 4- and 5-atom molecules with two, three, or four identical atoms are constructed in general form. The symmetrization of the angular and radial parts of the vibrational-rotational wave function is examined in detail. The representation matrix and the Clebsch-Gordan coefficients of the permutation group Sn (n = 3, 4) are adapted for the use in molecular spectroscopy. This makes it easier to draw analogies with the theory of effective Hamiltonian. The use of symmetrized wave functions is shown to significantly alleviate the problem of calculating the vibrational-rotational energy levels from the potential energy surface for such molecules as CH4, CH3D, CH3Cl, etc.

Multiparticle SUSY quantum mechanics and representations of the permutation group

The method of multidimensional SUSY quantum mechanics is applied to the investigation of supersymmetrical N -particle systems on a line for the case of separable centre-of-mass motion. New decomposition of the super-Hamiltonian into block-diagonal form with elementary matrix components is constructed. Matrices of coefficients of these minimal blocks are shown to coincide with matrices of irreducible representations of the permutation group SN , which correspond to the Young tableaux (N -M ,1M ). The connections with known generalizations of N -particle Calogero and Sutherland models are established.

Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques.

When faced with a complex task, is it better to be systematic or to proceed by making random adjustments? We study aspects of this problem in the context of generating random elements of a finite group. For example, suppose we want to fill n empty spaces with zeros and ones such that the probability of configuration x = (x1, . . . , xn) is θ n−|x|(1− θ)|x|, with |x| the number of ones in x. A systematic scan approach works left to right, filling each successive place with a θ coin toss. A random scan approach picks places at random, and a given site may be hit many times before all sites are hit. The systematic approach takes order n steps and the random approach takes order 1 4n log n steps. Realistic versions of this toy problem arise in image analysis and Ising-like simulations, where one must generate a random array by a Monte Carlo Markov chain. Systematic updating and random updating are competing algorithms that are discussed in detail in Section 2. There are some successful analyses for random scan algorithms, but the intuitively appealing systematic scan algorithms have resisted analysis. Our main results show that the binary problem just described is exceptional; for the examples analyzed in this paper, systematic and random scans converge in about the same number of steps. Let W be a finite Coxeter group generated by simple reflections s1, s2, . . . , sn, where s2 i = id. For example, W may be the permutation group Sn+1 with si = (i, i + 1). The length function `(w) is the smallest k such that w = si1si2 · · · sik . Fix 0 < θ ≤ 1 and define a probability distribution on W by π(w) = θ −`(w) PW (θ−1) , where PW(θ −1) = ∑

Algebraic description of SN-invariant oscillator states

A group-theoretical description of N identical harmonic oscillators on a line is presented. It provides a scheme for labeling the energy eigenstates that are invariant under the permutation group SN and for obtaining the symmetric operators that transform these degenerate eigenfunctions among themselves. The symmetry algebra that these generators form is in general polynomial. The 2- and 3-particle cases are considered in detail. For the simple 2-body problem the invariance algebra is found to be the cubic SU(2) algebra: [J0,J±]=±J±, [J+,J−]=2J0−αJ03. In the 3-body case, the permutational invariant states are characterized with the help of the subgroup chain U(3)⊃U(2)⊃O(2). The labeling and step operators are obtained from determining an integrity basis for the S3 scalar in U(U(3)). Generating functions techniques are used to that end; an eight-dimensional basis is found whose elements span the symmetry algebra of the three identical oscillator problem. These constants of motion are seen to generate a nonlinear algebra whose representation on the symmetric states is provided.

Topological Structure of the Configuration Space and the Separation of Spin and Spatial Variables for N-Electron Systems

The relationship between the topological structure of the multidimensional configuration space and the structure of the symmetrical permutation group SN is analyzed. This relationship permits to separate the spin and spatial variables during the solution of the spinless stationary Schrödinger equation. It is possible to simplify the construction of wave functions for N-electron systems and to expand the class of trial functions in the variation method. The relation of the suggested approach to the construction of wave functions by traditional methods is demonstrated by the example of three-electron systems.

Correlated states and collective transition operators for multilevel atomic systems. I

The correlated states of an assembly of N multilevel atoms interacting with resonant radiation fields are determined. On the basis of the duality between the permutation group and the linear group, one can construct the internal states for an assembly of N atoms, with n levels each. The possible states are determined by various allowed Young diagrams, corresponding to various representations of the permutation group SN. The collective transition operators are obtained as the generators of the group SU(n).