Theory Of Symmetric Group Research Articles

Permutation invariant tensor models and partition algebras

Matrix models with continuous symmetry are powerful tools for studying quantum gravity and holography. Tensor models have also found applications in holographic quantum gravity. Matrix models with discrete permutation symmetry have been shown to satisfy large N factorisation properties relevant to holography, while also having applications to the statistical analysis of ensembles of real-world matrices. Here, we develop 3-index tensor models in dimension D with a discrete symmetry of permutations in the symmetric group S_{D} . We construct the most general permutation invariant Gaussian tensor model using the representation theory of symmetric groups and associated partition algebras. We define a representation basis for the 3-index tensors, where the two-point function is diagonalised. Inverting the change of basis gives an explicit formula for the two-point function in the tensor basis for general D .

Biharmonic conjectures on hypersurfaces in a space form

We apply the Murnaghan-Nakayama rule in the representation theory of symmetric groups to develop new techniques for studying biharmonic hypersurfaces in a space form. As applications of the new techniques, we settle the well-known Chen’s conjecture on biharmonic hypersurfaces in R 6 \Bbb R^6 and BMO conjecture on biharmonic hypersurfaces in S 6 \mathbb S^6 .

A unified FFT-based approach to maximum assignment problems related to transitive finite group actions

This paper studies the cross-correlation of two real-valued functions whose common domain is a set on which a finite group acts transitively. Such a group action generalizes, e.g., circular time shifts for discrete periodic time signals or spatial translations in the context of image processing. Cross-correlations can be reformulated as convolutions in group algebras and are thus transformable into the spectral domain via Fast Fourier Transforms. The resulting general discrete cross-correlation theorem will serve as the basis for a unified FFT-based approach to maximum assignment problems related to transitive finite group actions. Our main focus is on those assignment problems arising from the actions of symmetric groups on the cosets of Young subgroups. The “simplest” nontrivial representatives of the latter class of problems are the NP-hard symmetric and asymmetric maximum quadratic assignment problem. We present a systematic spectral approach in terms of the representation theory of symmetric groups to solve such assignment problems. This generalizes and algorithmically improves Kondor's spectral branch-and-bound approach (SODA, 2010) to the exact solution of the asymmetric maximum quadratic assignment problem.

Primitive elements of the Hopf algebras of tableaux

The character theory of symmetric groups, and the theory of symmetric functions, both make use of the combinatorics of Young tableaux, such as the Robinson–Schensted algorithm, Schützenberger’s “jeu de taquin”, and evacuation. In 1995 Poirier and the second author introduced some algebraic structures, different from the plactic monoid, which induce some products and coproducts of tableaux, with homomorphisms. Their starting point are the two dual Hopf algebras of permutations, introduced by the authors in 1995. In 2006 Aguiar and Sottile studied in more detail the Hopf algebra of permutations: among other things, they introduce a new basis, by Möbius inversion in the poset of weak order, that allows them to describe the primitive elements of the Hopf algebra of permutations. In the present Note, by a similar method, we determine the primitive elements of the Poirier–Reutenauer algebra of tableaux, using a partial order on tableaux defined by Taskin.

Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory

The Kronecker, plethysm and Sylow branching coefficients describe the decomposition of representations of symmetric groups obtained by tensor products and induction. Understanding these decompositions has been hailed as one of the definitive open problems in algebraic combinatorics and has profound and deep connections with representation theory, symplectic geometry, complexity theory, quantum information theory, and local-global conjectures in representation theory of finite groups. The overarching theme of the Mini-Workshop has been the use of hidden, richer representation theoretic structures to prove and disprove conjectures concerning these coefficients. These structures arise from the modular and local-global representation theory of symmetric groups, graded representation theory of Hecke and Cherednik algebras, and categorical Lie theory.

Differential operators and the symmetric groups

In this paper, we study the action of the rational quantum Calogero-Moser system on polynomials. In this vein, we study polynomials ring over the complex field ℂ as a module over a ring of differential operators by elaborating its irreducible submodules. we endowed the polynomial ring ℂ[x 1 , …, x n ] with a differential structure by using directly the action of the Weyl algebra associated with the ring of symmetric polynomial ℂ[x 1 , …, x n ] Sn after a localization. Then we study the polynomials representation of the ring of invariant differential operators under the symmetric group. We use the representation theory of symmetric groups to exhibit the generators of its simple components.

Representation theory of symmetric groups and the strong Lefschetz property

We investigate the structure and properties of an Artinian monomial complete intersection quotient [Formula: see text]. We construct explicit homogeneous bases of [Formula: see text] that are compatible with the [Formula: see text]-module structure for [Formula: see text], all exponents [Formula: see text] and all homogeneous degrees [Formula: see text]. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the [Formula: see text]-module decomposition of homogeneous subspaces.

A Symmetric Group Method for Controllability Characterization of Bilinear Systems on the Special Euclidean Group

Bilinear systems form models of wide-ranging applications in diverse areas of engineering and natural sciences. Investigating fundamental properties of such systems has been a prosperous subject of interest and remains essential toward the advancement of systems science and engineering. In this paper, we introduce an algebraic framework utilizing the theory of symmetric group to characterize controllability of bilinear systems evolving on special orthogonal and Euclidean groups. Our development is based on the most notable Lie algebra rank condition and offers an alternative to controllability analysis. The main idea of the developed approach lies in identifying the mapping of Lie brackets of vector fields governing the system dynamics to permutation multiplications on a symmetric group. Then, by leveraging the actions of the resulting permutations on a finite set, controllability and controllable submanifolds for bilinear systems evolving on the special Euclidean group can be explicitly characterized.

Laplacian of a Graph Covering and Its Applications

Let G be a finite graph and \(\overline{G}\) be any graph covering over G. By applying the representation theory of symmetric groups, the Laplacian characteristic polynomial and the normalized Laplacian characteristic polynomial of \(\overline{G}\) are investigated. As applications, adopting the algebra method the Kirchhoff index, the multiplicative degree-Kirchhoff index and the complexity of any connected covering over a connected graph are derived.

Symmetric group representations and [formula omitted

We discuss implications of the following statement about representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation and every nonnegative integer appears infinitely often as a Littlewood–Richardson coefficient and as a Kronecker coefficient.

Jordan trialgebras and post-Jordan algebras

We compute minimal sets of generators for the Sn-modules (n≤4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp. tridendriform) algebra. These identities define Jordan trialgebras and post-Jordan algebras: Jordan analogues of the Lie trialgebras and post-Lie algebras introduced by Dotsenko et al., Pei et al., Vallette & Loday. We include an extensive review of analogous structures existing in the literature, and their interrelations, in order to identify the gaps filled by our two new varieties of algebras. We use computer algebra (linear algebra over finite fields, representation theory of symmetric groups), to verify in both cases that every polynomial identity of arity ≤6 is a consequence of those of arity ≤4. We conjecture that in both cases the next independent identities have arity 8, imitating the Glennie identities for Jordan algebras. We formulate our results as a commutative square of operad morphisms, which leads to the conjecture that the squares in a much more general class are also commutative.

Stability properties of the plethysm: A combinatorial approach

An important family of structural constants in the theory of symmetric functions and in the representation theory of symmetric groups and general linear groups are the plethysm coefficients. In 1950, Foulkes observed that they have some stability properties: certain sequences of plethysm coefficients are eventually constant. Such stability properties were proven by Brion with geometric techniques, and by Thibon and Carré by means of vertex operators.In this paper we present a new approach to prove such stability properties. Our proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients related to the complete homogeneous basis of symmetric functions. We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.

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

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.

Assembling homology classes in automorphism groups of free groups

The observation that a graph of rank n can be assembled from graphs of smaller rank k with s leaves by pairing the leaves together leads to a process for assembling homology classes for Out (F_n) and Aut (F_n) from classes for groups \Gamma_{k,s} , where the \Gamma_{k,s} generalize Out (F_k)=\Gamma_{k,0} and Aut (F_k)=\Gamma_{k,1} . The symmetric group \mathfrak G_s acts on H_*(\Gamma_{k,s}) by permuting leaves, and for trivial rational coefficients we compute the \mathfrak G_s -module structure on H_*(\Gamma_{k,s}) completely for k \leq 2 . Assembling these classes then produces all the known nontrivial rational homology classes for Aut (F_n) and Out (F_n) with the possible exception of classes for n=7 recently discovered by L. Bartholdi. It also produces an enormous number of candidates for other nontrivial classes, some old and some new, but we limit the number of these which can be nontrivial using the representation theory of symmetric groups. We gain new insight into some of the most promising candidates by finding small subgroups of Aut (F_n) and Out (F_n) which support them and by finding geometric representations for the candidate classes as maps of closed manifolds into the moduli space of graphs. Finally, our results have implications for the homology of the Lie algebra of symplectic derivations.

Generalized Stieltjes Transforms of Compactly-Supported Probability Distributions: Further Examples

For two families of beta distributions, we show that the generalized Stieltjes transforms of their elements may be written as elementary functions (powers and fractions) of the Stieltjes transform of the Wigner distribution. In particular, we retrieve the examples given by the author in a previous paper and relating generalized Stieltjes transforms of special beta distributions to powers of (ordinary) Stieltjes ones. We also provide further examples of similar relations which are motivated by the representation theory of symmetric groups. Remarkably, the power of the Stieltjes transform of the symmetric Bernoulli distribution is a generalized Stietljes transform of a probability distribution if and only if the power is greater than one. As to the free Poisson distribution, it corresponds to the product of two independent Beta distributions in $[0,1]$ while another example of Beta distributions in $[-1,1]$ is found and is related with the Shrinkage process. We close the exposition by considering the generalized Stieltjes transform of a linear functional related with Humbert polynomials and generalizing the symmetric Beta distribution.

Representation theory: a combinatorial viewpoint

This book discusses the representation theory of symmetric groups, the theory of symmetric functions and the polynomial representation theory of general linear groups. The first chapter provides a detailed account of necessary representation-theoretic background. An important highlight of this book is an innovative treatment of the Robinson–Schensted–Knuth correspondence and its dual by extending Viennot's geometric ideas. Another unique feature is an exposition of the relationship between these correspondences, the representation theory of symmetric groups and alternating groups and the theory of symmetric functions. Schur algebras are introduced very naturally as algebras of distributions on general linear groups. The treatment of Schur–Weyl duality reveals the directness and simplicity of Schur's original treatment of the subject. In addition, each exercise is assigned a difficulty level to test readers' learning. Solutions and hints to most of the exercises are provided at the end.

Law of Large Numbers for infinite random matrices over a finite field

Asymptotic representation theory of general linear groups $$\hbox {GL}(n,F_\mathfrak {q})$$ over a finite field leads to studying probability measures $$\rho $$ on the group $$\mathbb {U}$$ of all infinite uni-uppertriangular matrices over $$F_\mathfrak {q}$$ , with the condition that $$\rho $$ is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures $$\rho $$ ) was conjectured by Kerov in connection with nonnegative specializations of Hall–Littlewood symmetric functions. Vershik and Kerov also conjectured the following Law of Large Numbers. Consider an infinite random matrix drawn from an ergodic measure coming from the Kerov’s conjectural classification and its $$n \times n$$ submatrix formed by the first rows and columns. The sizes of Jordan blocks of the submatrix can be interpreted as a (random) partition of $$n$$ , or, equivalently, as a (random) Young diagram $$\lambda (n)$$ with $$n$$ boxes. Then, as $$n\rightarrow \infty $$ , the rows and columns of $$\lambda (n)$$ have almost sure limiting frequencies corresponding to parameters of this ergodic measure. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson–Schensted–Knuth (RSK) insertion algorithm which samples random Young diagrams $$\lambda (n)$$ coming from ergodic measures. The probability weights of these Young diagrams are expressed in terms of Hall–Littlewood symmetric functions. Our insertion algorithm is a modified and extended version of a recent construction by Borodin and Petrov (2013). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (SIAM J. Algebr. Discret. Math. 7(1):116–124, 1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).

Wreath determinants for group–subgroup pairs

The aim of the present paper is to generalize the notion of the group determinants for finite groups. For a finite group G and its subgroup H, one may define a rectangular matrix of size #H×#G by X=(xhg−1)h∈H,g∈G, where {xg|g∈G} are indeterminates indexed by the elements in G. Then, we define an invariant Θ(G,H) for a given pair (G,H) by the k-wreath determinant of the matrix X, where k is the index of H in G. The k-wreath determinant of an n by kn matrix is a relative invariant of the left action by the general linear group of order n and of the right action by the wreath product of two symmetric groups of order k and n. Since the definition of Θ(G,H) is ordering-sensitive, the representation theory of symmetric groups is naturally involved. When G is abelian, if we specialize the indeterminates to powers of another variable q suitably, then Θ(G,H) factors into the product of a power of q and polynomials of the form 1−qr for various positive integers r. We also give examples for non-abelian group–subgroup pairs.

Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)

Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a newapproach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope. Les coefficients du pléthysme sont des constantes de structure importantes de la théorie des fonctions symétriques, ainsi que de la théorie de la représentation des groupes symétriques et des groupes généraux linéaires. En 1950, Foulkes a observé pour ces coefficients de phénomènes de stabilité: certaines suites de coefficients du pléthysme sont stationnaires. De telles propriétés ont été démontrées par Brion, au moyen de techniques géométriques, et par Thibon et Carré, au moyen d’opérateurs vertex. Dans ce travail, nous présentons une nouvelle approche, purement combinatoire, pour démontrer des propriétés de stabilité de ce type. Nous décomposons les coefficients du pléthysme comme somme alternées de coefficients de pléthysme d’un autre type (liés à la base des fonctions symétriques sommes complètes), qui comptent les points entiers dans des polytopes. Nous démontrons la stabilité des suites de ces coefficients en exhibant des bijections entres les ensembles de points entiers des polytopes correspondants.

On the exterior algebra method applied to restricted set addition

In 1994 Dias da Silva and Hamidoune solved a long-standing open problem of Erdős and Heilbronn using the structure of cyclic spaces for derivatives on Grassmannians and the representation theory of symmetric groups. They proved that for any subset A of the p-element group Z/pZ (where p is a prime), at least min{p,m|A|−m2+1} different elements of the group can be written as the sum of m different elements of A. In this note we present an easily accessible simplified version of their proof for the case m=2, and explain how the method can be applied to obtain the corresponding inverse theorem.