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The lower bound on the number of n-variable balanced symmetric functions over finite fields GF ( p ) presented by Cusick et al. in T.W. Cusick, Y. Li, P. Sta?nica?, Balanced symmetric functions over GF(p), IEEE Trans. Inform. Theory 54 (3) (2008) 1304-1307] is improved in this paper. An equivalent characterization is also presented for the general case.

We introduce a class of real Jordan triple systems, called JH-triples, and show, via the Tits–Kantor–Koecher construction of Lie algebras, that they correspond to a class of Riemannian symmetric spaces including the Hermitian symmetric spaces and the symmetric R-spaces.

We first present a general property of nice polynomials, which leads to an important modification of the concept of equivalence classes of nice polynomials. We then establish several properties of nice symmetric and antisymmetric polynomials with an odd number of distinct roots. We give a complete description of all nice symmetric and antisymmetric polynomials with three distinct roots. We then apply our results to nice symmetric sextics and octics. As an important application, all the properties we have established have dramatically increased the speed of a computer search for examples, allowing us to discover the first examples of nice symmetric and antisymmetric polynomials with five distinct roots and the first two examples of nice polynomials with six distinct roots.

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.

We prove that Martin's conjecture holds for weight 3 blocks of symmetric group algebras—that if these blocks have Abelian defect group, then their projective (indecomposable) modules have a common radical length 7.

An Hermitian bounded symmetric domain in a complex vector space, given in its circled realization, is endowed with two natural symplectic forms: the flat form and the hyperbolic form. In a similar way, the ambient vector space is also endowed with two natural symplectic forms: the Fubini-Study form and the flat form. It has been shown in arXiv:math.DG/0603141 that there exists a diffeomorphism from the domain to the ambient vector space which puts in correspondence the above pair of forms. This phenomenon is called symplectic duality for Hermitian non compact symmetric spaces. In this article, we first give a different and simpler proof of this fact. Then, in order to measure the non uniqueness of this symplectic duality map, we determine the group of bisymplectomorphisms of a bounded symmetric domain, that is, the group of diffeomorphisms which preserve simultaneously the hyperbolic and the flat symplectic form. This group is the direct product of the compact Lie group of linear automorphisms with an infinite-dimensional Abelian group. This result appears as a kind of Schwarz lemma., Comment: 19 pages. Version 2: minor corrections

It is shown that central elements in G. Letzter’s quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter’s approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras.

In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree.

Independently Siemons and Wagner [I.J. Siemons, O. Wagner, private communication, 1986] and Stanley [R.P. Stanley, Positivity problems and conjectures in algebraic combinatorics, in: Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000, pp. 295–319] conjecture that if a partition λ dominates its conjugate partition λ ′ then an FS n -homomorphism ψ λ : 1 N S n ( S λ ) S n ↪ 1 N S n ( S λ ′ ) S n is injective. We demonstrate a method of adding columns to the partition λ that preserves the injectivity of ψ λ .

In this paper, we generalize the recent counting results about rotation symmetric Boolean functions to the rotation symmetric polynomials over finite fields GF(p). By using Mobius function, we obtain some formulas for more general n, the number of variables. Some known formula in Boolean case are simplified.

Symmetric property is a special property of Boolean functions, which has attracted much study on it. This chapter presents fast Walsh transforms of symmetric Boolean functions, correlation immunity of symmetric functions, construction of symmetric resilient Boolean functions, and some cryptographic properties of majority functions being a special class of symmetric Boolean functions. The study on the correlation immunity of majority functions shows that majority functions have good asymptotical behavior of correlation immunity, i.e., although they are not correlation immune, they have, however, asymptotical correlation immunity.

In this paper, we study shifted Schur functions $S_\mu^\star$, as well as a new family of shifted symmetric functions $\mathfrak{K}_\mu$ linked to Kostka numbers. We prove that both are polynomials in multi-rectangular coordinates, with nonnegative coefficients when written in terms of falling factorials. We then propose a conjectural generalization to the Jack setting. This conjecture is a lifting of Knop and Sahi's positivity result for usual Jack polynomials and resembles recent conjectures of Lassalle. We prove our conjecture for one-part partitions., Comment: 2nd version: minor modifications after referee comments

A symmetry on a group G is a mapping G ∋ x ↦ gx−1 g ∈ G, where g ∈ G. A subset A ⊆ G is symmetric if it is invariant under some symmetry, that is, A = gA−1g. The notion of symmetry has interesting relations to enumerative combinatorics. A coloring is symmetric if χ(gx−1g) = χ(x) for some g ∈ G. We discuss an approach how to compute the number of symmetric r-colorings for any finite group. Using this approach we derive the formula for the number of symmetric r-colorings of the dihedral group D3.

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces Gk/Kk, where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and Gk resp. Kk the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice \(\Lambda_{\mathfrak{a}}\) of the symmetric space in terms of the characters of the weight lattice \(\Lambda_{\mathfrak {t}}\) of the group. These algorithms can easily be implemented in a computer algebra package.

A number of generalized metric spaces have been defined or characterized in terms of g-functions. Symmetric g-functions are discussed by C. Good, D. Jennings and A. M. Mohamad. In this paper, some questions about symmetric g-functions are answered, particularly it is shown that every sym-wg-space is expandable.

Following the approach to pseudo-Riemannian symmetric spaces developed in [I. Kath, M. Olbrich, On the structure of pseudo-Riemannian symmetric spaces, arXiv:math.DG/0408249 , 2004] we exhibit examples of indefinite hyper-Kahler symmetric spaces with non-abelian holonomy. Moreover, we classify indecomposable hyper-Kahler symmetric spaces whose metric has signature ( 4 , 4 n ) . Such spaces exist if and only if n ∈ { 0 , 1 , 3 } .

In this paper we are interested in the higher-order derivatives of functions of the eigenvalues of symmetric matrices with respect to the matrix argument. We describe the formula for the k-th derivative of such functions in two general cases. The flrst case concerns the derivatives of the composition of an arbitrary (not necessarily symmetric) k-times difierentiable function with the eigenvalues of symmetric matrices at a symmetric matrix with distinct eigenvalues. The second case describes the derivatives of the composition of a k-times difierentiable separable symmetric function with the eigenvalues of symmetric matrices at an arbitrary symmetric matrix. We show that the formula signiflcantly simplifles when the separable symmetric function is k-times continuously difierentiable. As an application of the developed techniques, we re-derive the formula for the Hessian of a general spectral function at an arbitrary symmetric matrix. The new tools lead to a shorter, cleaner derivation than the original one in [16]. To make the exposition as self contained as possible, we have included the necessary background results and deflnitions. The proofs of the intermediate technical results are collected in the appendices.

We consider the Green polynomials at roots of unity. We obtain a recursive formula for the Green polynomials at roots of unity whose orders do not exceed some positive integer. The formula is described in a combinatorial manner. The coefficients of the recursive formula are realized by the cardinality of a set of permutations. The formula gives an interpretation of a combinatorial property on a family of graded modules for the symmetric group in terms of representation theory.

In this research announcement we present a new q q -analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q q -Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q q -Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker; (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev, and Lunts; (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville, and Vaughan and by Dollhopf, Goulden, and Greene; and (4) Stanley’s chromatic symmetric functions.

Stanley introduced expressions for the normalized characters of the symmetric group and stated some positivity conjectures for these expressions. Here, we give an affirmative partial answer to Stanley's positivity conjectures about the expressions using results on Kerov polynomials. In particular, we use new positivity results by Goulden and the present author. We shall see that the generating series $C(t)$ introduced by them is critical to our discussion., Comment: 20 pages, 2 figures, v2, minor revisions, fixed typos etc

A survey of general results about spectra of uniform algebras of symmetricholomorphic functions and algebras of symmetric analytic functions of bounded type onBanach spaces is given.

The problem of base changes for the classical symmetric functions has been solved a long time ago and has been incorporated into most computer software packages for symmetric functions. In this paper, we develop a simple recursive formula for the expansion of the augmented monomial symmetric functions into power sum symmetric functions. As corollaries, we present two algorithms that can be used to expressing the augmented monomial symmetric functions in terms of the power sum symmetric functions.

Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of "involution Schubert polynomials" can be viewed as a Billey-Jockusch-Stanley type formula for cohomology class representatives of $\mathrm{O}_n$- and $\mathrm{Sp}_{2n}$-orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type $B$., 38 pages; v2: some revisions and corrections, with an expanded introduction; v3, v4: added remarks, attribution, and acknowledgements; v5: revised introduction, updated references; v6: various revisions and corrections, removed geometric appendix, added index of notation, final version

We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$$n-1$, $B$$n$, $Ã$$n$, and the flag weak order on the wreath product ℤ$r$ ≀ $S$$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$$n-1$ case, in which case we obtain the classical Stanley symmetric function., On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$$n-1$, $B$$n$, $Ã$$n$, ainsi qu’une variante de l’ordre faible sur les produits en couronne ℤ$r$ ≀ $S$$n$ introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$$n-1$ où l’on obtient les séries de Stanley classiques.

To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function., A chaque sous-ensemble fini de $\mathbb{N}×\mathbb{N}$ (un diagramme), on peut associer une sous-variété d’une grassmannienne complexe et une représentation d’un groupe symétrique (un module de Specht). Liu a conjecturé que la classe de cohomologie de la variété d’un diagramme est représentée par la caractéristique de Frobenius du module de Specht correspondant. Nous donnons un contre-exemple à cette conjecture.Cependant, nous montrons que dans le cas de la variété du diagramme de permutation, la classe de cohomologie conjecturée par Liu est au moins un majorant de la classe juste $\tau$ , c’est-à-dire que $\sigma - \tau$ est une combinaison linéaire non-négative des classes de Schubert. Pour ce faire, nous considérons une dégénérescence des variétés de rang de Coskun qui contient la variété appropriée d’un diagramme comme une composante irréductible. Les variétés de rang sont des exemples de variétés de positroïde, dont les classes de cohomologie sont représentées par des fonctions symétriques de Stanley affines. En effet, nous montrons que la classe de cohomologie d’une variété de rang est représentée par une fonction symétrique de Stanley ordinaire.

We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigate the combinatorics of affine Schubert calculus for type $A$. We introduce Murnaghan-Nakayama elements and Dunkl elements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutative algebra generated by these operators is isomorphic to the cohomology of the affine flag variety. We show that the cohomology of the affine flag variety is product of the cohomology of an affine Grassmannian and a flag variety, which are generated by MN elements and Dunkl elements respectively. The Schubert classes in cohomology of the affine Grassmannian (resp. the flag variety) can be identified with affine Schur functions (resp. Schubert polynomials) in a quotient of the polynomial ring. Affine Schubert polynomials, polynomial representatives of the Schubert class in the cohomology of the affine flag variety, can be defined in the product of two quotient rings using the Bernstein-Gelfand-Gelfand operators interpreted as divided difference operators acting on the affine Fomin-Kirillov algebra. As for other applications, we obtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. We also define $k$-strong-ribbon tableaux from Murnaghan-Nakayama elements to provide a new formula of $k$-Schur functions. This formula gives the character table of the representation of the symmetric group whose Frobenius characteristic image is the $k$-Schur function., 21 pages, 1 figure. v2: fixed typos including the definition of c(D). v3. 29 pages, 1 figure, The definition of the affine Schubert polynomials, generalization of the nonnegativity conjecture for Fomin-Kirillov algebra, connection with representation theory are added. Due to these important changes, the title, abstract and introduction are modified. v4. Journal version

Precise uniqueness theorems are proved for the solutions of a convolution equation on Riemannian symmetric spaces of non-compact type of rank 1 and an application is given to lacunary series with respect to special functions.

We prove that the integral of n functions over a symmetric set L in R n , with additional properties, increases when the functions are replaced by their symmetric decreasing rearrangements. The result is known when L is a centrally symmetric convex set, and our result extends it to nonconvex sets. We deduce as consequences, inequalities for the average of a function whose level sets are of the same type as L, over measurable sets in R n . The average of such a function on E is maximized by the average over the symmetric set E * .

In this paper we prove the Dipper–James conjecture that the centre of the Iwahori–Hecke algebra of type A is the set of symmetric polynomials in the Jucys–Murphy operators.

We show that for every infinite-dimensional normed space E and every k > 3 there are extendible k-homogeneous polynomials which are not integral. As a consequence, we prove a symmetric version of a result of John.

We present an explicit description of the isomorphism between two models of finite factor representations of the infinite symmetric group: the tableau model in the space of functions on Young bitableaux and the dynamical model in the space of functions on pairs of Bernoulli sequences. The main tool used is the Fourier transform on the symmetric groups. We also start the investigation of the so-called tensor model of two-row representations of the symmetric groups, which plays an intermediate role between the tableau and dynamical models, and show its relations to both these models.

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of 2n−1 permutations that pairwise generate the symmetric group Sn. There is no set of 2n−1+1 permutations having this property. For all sufficiently large integers n with n≡2mod4, there exists a set of 2n−2 even permutations that pairwise generate the alternating group An. There is no set of 2n−2+1 permutations having this property.

We present a Grobner basis associated with the symmetric group of degree n , which is determined by a strong generating set of the symmetric group and is defined by means of a term ordering with the elimination property.

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1, and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula was conjectured for by Buch and Fulton in terms of factor sequences of Young tableaux; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in ``doubled'' versions, two for `double quiver polynomials', and the other two for their stable versions, the `double quiver functions', where setting half the variables equal to the other half specializes to the ordinary case., 75 pages, about 25 LaTeX-drawn figure-like objects; submitted. v2: references added, Corollary 1.16 corrected, Remark 6.25 added, Sections 7 and 8 reorganized around a direct proof that quiver constants are instances of Stanley coefficients. Introduction edited to accomodate these alterations

By the fundamental theorem of symmetric polynomials, if P ∈ ℚ[X1,…,Xn] is symmetric, then it can be written P = Q(σ1,…,σn), where σ1,…,σn are the elementary symmetric polynomials in n variables, and Q is in ℚ[S1,…,Sn]. We investigate the complexity properties of this construction in the straight-line program model, showing that the complexity of evaluation of Q depends only on n and on the complexity of evaluation of P. Similar results are given for the decomposition of a general polynomial in a basis of ℚ[X1,…,Xn] seen as a module over the ring of symmetric polynomials, as well as for the computation of the Reynolds operator.

In this paper we study the Calabi diastasis function of Hermitian symmetric spaces. This allows us to prove that if a complete Hermitian locally symmetric space ( M , g ) admits a Kahler immersion into a globally symmetric space ( S , G ) then it is globally symmetric and the immersion is injective. Moreover, if ( S , G ) is symmetric of a specified type (Euclidean, noncompact, compact), then ( M , g ) is of the same type. We also give a characterization of Hermitian globally symmetric spaces in terms of their diastasis function. Finally, we apply our analysis to study the balanced metrics, introduced by Donaldson, in the case of locally Hermitian symmetric spaces.

In this paper, we continue a long line of research which shows that many generating function identities for various permutation statistics arise from well known symmetric function identities by applying certain ring homomorphisms on the ring of symmetric functions. This idea was first introduced in a 1993 paper of Brenti who used it to show that the generating functions of permutations of the symmetric group Sn by descents and excedances could be derived in such a manner. In this paper, we define certain (q,t)-analogues of Brenti's homomorphism that lead to generating functions for statistics on m-tuples of permutations. Our results generalize previous work of generating functions for permutations statistics due to Carlitz, Stanley, and Fedou and Rawlings. We also introduce some new bases of symmetric functions which are necessary to extend Brenti's results on generating functions for permutations by excedances to m-tuples of permutations. Finally, we study the image of our homomorphisms on analogues of the q-basis of symmetric functions studied by Ram and King and Wybourne to describe the irreducible characters of the Hecke algebras of type A.

We define a new family of symmetric functions which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalise the (dual of the) $k$-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Conjecturally, affine Stanley symmetric functions should be related to the cohomology of the affine flag variety., Comment: 27 pages

We obtain explicit generators for the centre of the Ringel-Hall algebra of a cyclic quiver and define a canonical algebra monomorphism from Macdonald's ring of symmetric functions to the centre, which furthermore respects the comultiplication and the symmetric bilinear form.

In this paper, we introduce the notion of Minkowski symmetric Lie algebras to give an algebraic description of symmetric Berwald spaces. As an application, we study some geometric properties of these spaces. In particular, we generalize an elegant result of E. Cartan on the sectional curvature of Riemannian symmetric spaces to the flag curvature of symmetric Berwald spaces.

We calculate the homology and the cycles in tensor products of algebras of symmetric function over ℤ2

For probability measures on product spaces which are symmetric under permutations of coordinates, we study the rate of approximation by mixtures of product measures.

In (Adv. Math. 174(2) (2003) 236), a bijection between collections of reduced factorizations of elements of the symmetric group was described. Initially, this bijection was used to show the Schur positivity of the Stanley symmetric functions. Further investigations have revealed that our bijection has strong connections to other more familiar combinatorial algorithms. In this paper we will show how the Robinson–Schensted–Knuth correspondence can be decomposed into a sequence of applications of this bijection.

We use the Jack symmetric functions as a basis of the Fock space, and study the action of the Virasoro generators $L_n$. We calculate explicitly the matrix elements of $L_n$ with respect to the Jack-basis. A combinatorial procedure which produces these matrix elements is conjectured. As a limiting case of the formula, we obtain a Pieri-type formula which represents a product of a power sum and a Jack symmetric function as a sum of Jack symmetric functions. Also, a similar expansion was found for the case when we differentiate the Jack symmetric functions with respect to power sums. As an application of our Jack-basis representation, a new diagrammatic interpretation is presented, why the singular vectors of the Virasoro algebra are proportional to the Jack symmetric functions with rectangular diagrams. We also propose a natural normalization of the singular vectors in the Verma module, and determine the coefficients which appear after bosonization in front of the Jack symmetric functions., Comment: 23 pages, references added

Consider the algebra Qhhx1, x2, . . .ii of formal power series in countably many noncommuting variables over the rationals. The subalgebra Π(x1, x2, . . .) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as will as investigating their properties.

In this article we study the evaluation of symmetric functions on the alphabet of contents of a partition. Applying this notion of content evaluation to the computation of central characters of the symmetric group, we are led to the definition of a new basis of the algebra Λ of symmetric functions over Q (n) that we call the basis of class symmetric functions. By definition this basis provides an algebra isomorphism between Λ and the Farahat–Higman algebra FH governing for all n the products of conjugacy classes in the center Z n of the group algebra of the symmetric group S n . We thus obtain a calculus of all connexion coefficients of Z n inside Λ. As expected, taking the homogeneous components of maximal degree in class symmetric functions, we recover the symmetric functions introduced by Macdonald to describe top connexion coefficients. We also discuss the relation of class symmetric functions to the asymptotic of central characters and of the enumeration of standard skew young tableaux. Finally we sketch the extension of these results to Hecke algebras.

We present and study a new construction of singular symmetric functionals on Marcinkiewicz function spaces, which may be considered as a counterpart of an earlier construction by Dixmier of nonnormal traces on certain operator ideals. We also exhibit new examples of symmetric function spaces which are not subspaces of any Marcinkiewicz space and which admit nontrivial singular symmetric functionals. Bibliography: 9 titles.

In this paper we consider an algebraic generalization of symmetric spaces of noncompact type to a more general class of symmetric structures equipped with midpoints. These symmetric structures are shown to have close relationships to and even categorical equivalences with a variety of other algebraic structures: axiomatic midpoint spaces, uniquely 2-divisible twisted subgroups, transversal twisted subgroups of involutive groups, a special class of loops called B-loops, and gyrocommutative gyrogroups.

Correlation immunity of symmetric Boolean functions is studied in this paper. Lower bounds on the number of constructible correlation immune symmetric functions are given. Constructions for such new balanced functions are presented. These functions are also known as 1-resilient functions. In 1985, Chor et al. conjectured that the only 1-resilient symmetric functions are the exclusive-or of all n variables and its negation. This conjecture, however, was disproved by Gopalakrishnan, Hoffman and Stinson in 1993 by giving a class of infinite counterexamples, and they noted that it does not seem to extend any further in an obvious way. In this paper two more infinite classes of such examples are presented for n being even and being odd, respectively, and consequently one of the two open problems proposed by Gopalakrishnan et al., is addressed by constructing new symmetric resilient functions.

For symmetric spaces of measurable functions on the real half-line, we study the problem of existence of positive linear functionals monotone with respect to the Hardy-Littlewood semi-ordering, the so-called symmetric functionals. Two new wide classes of symmetric spaces are constructed which are distinct from Marcinkiewicz spaces and for which the set of symmetric functionals is non-empty. We consider a new construction of singular symmetric functionals based on the translation-invariance of Banach limits defined on the space of bounded sequences. We prove the existence of Banach limits invariant under the action of the Hardy operator and all dilation operators. This result is used to establish the stability of the new construction of singular symmetric functionals for an important class of generating sequences.