New convolutions for complete and elementary symmetric functions

In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.     

A generalization of the symmetry between complete and elementary symmetric functions

A generalization for the symmetry between complete symmetric functions and elementary symmetric functions is given. As corollaries we derive the inverse of a triangular Toeplitz matrix and the expression of the Toeplitz-Hessenberg determinant. A very large variety of identities involving integer partitions and multinomial coefficients can be generated using this generalization. The partitioned binomial theorem and a new formula for the partition function p(n) are obtained in this way.     

A convolution for complete and elementary symmetric functions

In this paper we give a convolution identity for complete and elementary symmetric functions. This result can be used to prove and discover some combinatorial identities involving r-Stirling numbers, r-Whitney numbers and q-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde’s convolution identity.     

Generalizations of two identities of Guo and Yang

In this paper, we provide generalizations of two identities of Guo and Yang [2] for the q-binomial coe?cients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coe?cients are obtained as very special cases of these results. A new relationship between restricted partitions and restricted partitions into parts of two kinds is derived in this context.     

An analytic formula for Macdonald polynomials

We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula. To cite this article: M. Lassalle, M. Schlosser, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Reduction and transformation formulae for bivariate basic hypergeometric series

The main object of this paper is to establish several bivariate basic hypergeometric series identities by means of elementary series manipulation. Some of them can be applied to yield transformation and reduction formulae for q-Kampé de Fériet functions.     

Schur function identities, their t-analogs, and k-Schur irreducibility

We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act trivially on the k-Schur functions, thus leading to a concept of irreducibility for these functions.     

Complexity and structure of near-minimal contact circuits for elementary symmetric functions

The paper studies the problem of the synthesis of contact circuits for elementary symmetric functions. The structure of minimal contact circuits realizing elementary symmetric functions is established and the estimates of the complexity of the obtained circuits, which are accurate to within an additive constant, are determined. It is proved that, for substantially large n, the complexity of an elementary symmetric function of n variables with the working number w satisfies the relation L(s _n ^w ) = (2w + 1)n ? B _w , whereB _w is a nonnegative constant.     

Young-Fibonacci insertion, tableauhedron and Kostka numbers

This work is first concerned with some properties of the Young-Fibonacci insertion algorithm and its relation with Fomin's growth diagrams. It also investigates a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebras associated to the Young-Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are conveniently interpreted as saturated chains in the Young-Fibonacci lattice. Using our conventions, we give a simpler proof of a property of Killpatrick's evacuation algorithm for Fibonacci tableaux. It also appears that this evacuation is no longer needed in making Roby's and Fomin's constructions coincide. We provide the set of Young-Fibonacci tableaux of size n with a structure of graded poset called tableauhedron, induced by the weak order of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients of the transition matrix from the analogue of complete symmetric functions to analogue of the Schur functions in Okada's algebra associated to the Young-Fibonacci lattice. We prove a similar result relating usual Kostka numbers with four partial orders on Young tableaux, studied by Melnikov and Taskin.     

Extended Bressoud-Wei and Koike skew Schur function identities

The Jacobi-Trudi identity expresses a skew Schur function as a determinant of complete symmetric functions. Bressoud and Wei extend this idea, introducing an integer parameter t?−1 and showing that signed sums of skew Schur functions of a certain shape are expressible once again as a determinant of complete symmetric functions. Koike provides a Jacobi-Trudi-style definition of universal rational characters of the general linear group and gives their expansion as a signed sum of products of Schur functions in two distinct sets of variables. Here we extend Bressoud and Wei's formula by including an additional parameter and extending the result to the case of all integer t. Then we introduce this parameter idea to the Koike formula, extending it in the same way. We prove our results algebraically using Laplace determinantal expansions.     

Quantum inverse scattering method for the q-boson model and symmetric functions

The purpose of this paper is to show that the quantum inverse scattering method for the so-called q-boson model has a nice interpretation in terms of the algebra of symmetric functions. In particular, in the case of the phase model (corresponding to q = 0) the creation operator coincides (modulo a scalar factor) with the operator of multiplication by the generating function of complete homogeneous symmetric functions, and the wave functions are expressed via the Schur functions s_λ(x). The general case of the q-boson model is related in a similar way to the Hall-Littlewood symmetric functions P_λ(x;q²).     

Descents, inversions, and major indices in permutation groups

A multivariate generating function involving the descent, major index, and inversion statistic first given by Ira Gessel is generalized to other permutation groups. We provide generating functions for variants of these three statistics for the Weyl groups of type B and D, wreath product groups, and multiples of permutations. All of our ideas are combinatorial in nature and exploit fundamental relationships between the elementary and homogeneous symmetric functions.     

On the relative extrema of a linear combination of elementary symmetric functions

We determine where a linear combination of elementary symmetric functions attains as maximum and minimum over a certain convex set in Rⁿ . We also show that an inequality for elementary symmetric functions proposed by S. Pierce is true.     

Cocharacters of Bilinear Mappings and Graded Matrices

Let M _k (F) be the algebra of k ×k matrices over a field F of characteristic 0. If G is any group, we endow M _k (F) with the elementary grading induced by the k-tuple (1,...,1,g) where g?∈?G, g ²?≠?1. Then the graded identities of M _k (F) depending only on variables of homogeneous degree g and g ^???1 are obtained by a natural translation of the identities of bilinear mappings (see Bahturin and Drensky, Linear Algebra Appl 369:95–112, 2003). Here we study such identities by means of the representation theory of the symmetric group. We act with two copies of the symmetric group on a space of multilinear graded polynomials of homogeneous degree g and g ^???1 and we find an explicit decomposition of the corresponding graded cocharacter into irreducibles.     

The Hecke group algebra of a Coxeter group and its representation theory

Let W be a finite Coxeter group. We define its Hecke-group algebra by gluing together appropriately its group algebra and its 0-Hecke algebra. We describe in detail this algebra (dimension, several bases, conjectural presentation, combinatorial construction of simple and indecomposable projective modules, Cartan map) and give several alternative equivalent definitions (as symmetry preserving operator algebra, as poset algebra, as commutant algebra, …).In type A, the Hecke-group algebra can be described as the algebra generated simultaneously by the elementary transpositions and the elementary sorting operators acting on permutations. It turns out to be closely related to the monoid algebras of respectively nondecreasing functions and nondecreasing parking functions, the representation theory of which we describe as well.This defines three towers of algebras, and we give explicitly the Grothendieck algebras and coalgebras given respectively by their induction products and their restriction coproducts. This yields some new interpretations of the classical bases of quasi-symmetric and noncommutative symmetric functions as well as some new bases.     

Congruences for the number of spin characters of the double covers of the symmetric and alternating groups

Let p be an odd prime. The bar partitions with sign and p-bar-core partitions with sign respectively label the spin characters and p-defect zero spin characters of the double cover of the symmetric group, and by restriction, those of the alternating group. The generating functions for these objects have been determined by J. Olsson. We study these functions from an arithmetic perspective, using classical analytic tools and elementary generating function manipulation to obtain many Ramanujan-like congruences.     

Bounds for symmetric elliptic integrals

Lower and upper bounds for the four standard incomplete symmetric elliptic integrals are obtained. The bounding functions are expressed in terms of the elementary transcendental functions. Sharp bounds for the ratio of the complete elliptic integrals of the second kind and the first kind are also derived. These results can be used to obtain bounds for the product of these integrals. It is shown that an iterative numerical algorithm for computing the ratios and products of complete integrals has the second order of convergence.     

Rings of invariants for modular representations of elementary abelian <Emphasis Type="Italic">p</Emphasis>-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.     

Algebraic and multiple integral identities

In this paper we develop some identities involving symmetric products in an abstract algebra which was formerly introduced by Rimark Ree to investigate the shuffle product and relations with skew symmetric (Lie) products. His motivation was partially the characterization of homogeneous Lie polynomials in noncommuting variables, while our motivation is derived from problems in systems theory. The main link in these applications is the need for identities involving multiple integrals of functions of many variables. The relation between these identities and some of the abstract identities developed here is also worked out and some of the applications to systems theory reviewed.