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In this paper we define a canonical locally flat generalized conformal structure on a symmetric R-space of the rank greater than 1. We prove that the group of automorphisms of this structure coincides with the noncompact group of automorphisms of the symmetric space.

We study the construction and properties of positive linear functionals on symmetric spaces of measurable functions which are monotone with respect to submajorization. The construction of such functionals may be lifted to yield the existence of singular traces on certain non-commutative Marcinkiewicz spaces which generalize the notion of Dixmier trace.

In analogy with the symmetric group, we define the vexillary elements in the hyperoctahedral group to be those for which the Stanley 1 symmetric function is a single Schur Q-function. We show that the vexillary elements can be again determined by pattern avoidance conditions. These results can be extended to include the root systems of types A, B, C, and D. Finally, we give an algorithm for multiplication of Schur Q -functions with a superfied Schur function and a method for determining the shape of a vexillary signed permutation using jeu de taquin.

This purpose of this chapter is to introduce Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The presentation roughly follows three lectures I gave at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.

This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus

Many properties of symmetric functions are related to properties of the set of partitions, as a lattice.In this paper we give properties of the remarkable distributive sublattice associated with Pieri's formula, which expresses the multiplication of a Schur function by an elementary symmetric function, or the intersection of a Schubert cycle by a special one.

We present theSPpackage devoted to the manipulation of Schubert polynomials. These polynomials contain as a subfamily the Schur symmetric functions and allow to extend to non symmetric polynomials the classical combinatorial techniques of the theory of symmetric functions. They have many applications, ranging from multivariate interpolation to intersection theory in algebraic geometry.

We determine the character of the action of the symmetric group on the homology of the induced subposet of the lattice of partitions of the set { 1 , 2 , … , n } \{1,2,\ldots ,n\} obtained by restricting block sizes to the set { 1 , k , k + 1 , … } \{1,k,k+1,\ldots \} . A plethystic formula for the generating function of the Frobenius characteristic of the representation is given. We combine techniques from the theory of nonpure shellability, recently developed by Björner and Wachs, with symmetric function techniques, developed by Sundaram, for determining representations on the homology of subposets of the partition lattice.

We present representation theoretical interpretations of quasi-symmetric functions and noncommutative symmetric functions in terms of quantum linear groups and Hecke algebras at q=0. We obtain in this way a noncommutative realization of quasi-symmetric functions analogous to the plactic symmetric functions of Lascoux and Schutzenberger. The generic case leads to a notion of quantum Schur function.

We determine the number of ordered factorisations of an arbitrary permutation on n symbols into transpositions such that the factorisations have minimal length and such that the factors generate the full symmetric group on n symbols. Such factorisations of the identity permutation have been considered by Crescimanno and Taylor in connection with a class of topologically distinct holomorphic maps on the sphere. As with Macdonald's construction for symmetric functions that multiply as the classes of the class algebra, essential use is made of Lagrange inversion.

(1997). On mullineux' conjecture in the representation theory of symmetric groups. Communications in Algebra: Vol. 25, No. 6, pp. 1797-1803.

We continue our development of a new basis for the algebra of non-commutative symmetric functions. This basis is analogous to the Schur basis for the algebra of symmetric functions, and it shares many of its wonderful properties. For instance, in this article we describe non-commutative versions of the Littlewood-Richardson rule and the Murnaghan-Nakayama rule. A surprising relation develops among non-commutative Littlewood-Richardson coefficients, which has implications to the commutative case. Finally, we interpret these new coefficients geometrically as the number of integer points inside a certain polytope., 30 pages: we cleaned and fixed many details in the proofs. The interested reader may toggle \specialcomments in the TeX file to reveal 6 added pages of details and ideas (in red)

We give the power-sum symmetric functions in terms of the Q-functions with spin character coefficients.

A basis of symmetric functions, which we denote byqλ(X; q, t), was introduced in the work of Ram and King and Wybourne in order to describe the irreducible characters of the Hecke algebras of type A. In this work we give combinatorial descriptions of the expansions of the functionsqλ(X; q, t) in terms of the classical bases of symmetric functions and apply these results in determining the determinant of the character table of the Iwahori–Hecke algebras and in giving a generating function for certain permutation statistics.

In [5] Stanley associated to a (finite) graph G a symmetric function X G generalizing the chromatic polynomial of G . Using an involution on a special type of arrays constructed by Gessel and Viennot [1], we show that if G is the incomparability graph of a ( 3 + 1 )-free poset, then X G is a nonnegative linear combination of Schur functions. Since the elementary symmetric functions are nonnegative linear combinations of Schur functions, this result gives supportive evidence for a conjecture of Stanley and Stembridge ([5, Conjecture 5.1] or [6, Conjecture 5.5]).

This paper discusses the definitions and properties of two kinds of fundamental symmetric functions, which are based on AND-OR-NOT algebraic system and AND-Exclusive OR algebraic system, respectively. Based upon it, some mapping transformation methods between two kinds of expansion coefficients of an arbitrary symmetric function in the complete set of two fundamental symmetric functions.

Waring's formula for expressing power sum symmetric functions in terms of elementary symmetric functions is generalized to monomial symmetric functions with equal exponents by applying the orthogonality property of Ramanujan sums together with the Möbius function over the set partition lattice.

We present a general survey of some recent developments regarding the construction of compact quantum symmetric spaces and the analysis of their zonal spherical functions in terms of $q$-orthogonal polynomials. In particular, we define a one-parameter family of two-sided coideals in $\Uq(\gog\gol(n,\CC))$ and express the zonal spherical functions on the corresponding quantum projective spaces as Askey-Wilson polynomials containing two continuous and one discrete parameter.

We define noncommutative analogues of the characters of the symmetric group which are induced by transitive cyclic subgroups (cyclic characters). We investigate their properties by means of the formalism of noncommutative symmetric functions. The main result is a multiplication formula whose commutative projection gives a combinatorial formula for the resolution of the Kronecker product of two cyclic representations of the symmetric group. This formula can be interpreted as a multiplicative property of the major index of permutations.

We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schutzenberger, and provide simplified proofs of their basic properties, along with various generalizations thereof. Our techniques make use of an explicit combinatorial interpretation of these polynomials in terms of configurations of labelled pseudo-lines.

In this paper, we develop the combinatorial interpretations of the transition matrices between the bases of the Bn analogues of the symmetric functions. In order to provide a complete reference, we have included a summary of the transition matrices for symmetric functions as well.

The explicit formulas for fundamental solutions of the modified wave equations on certain symmetric spaces are found. These symmetric spaces have the following characteristic property: all multiplicities of their restricted roots are even. As a corollary in the odd-dimensional case one has that the Huygens' principle in Hadamard's sense for these equations is fulfilled. We consider also the heat and Laplace equations on such a symmetric space and give explicitly the corresponding fundamental solutions-heat kernel and Green's function. This continues our previous investigations [16] of the spherical functions on the same symmetric spaces based on the fact that the radial part of the Laplace-Beltrami operator on such a space is related to the algebraically integrable case of the generalised Calogero-Sutherland-Moser quantum system. In the last section of this paper we apply the methods of Heckman and Opdam to extend our results to some other symmetric spaces, in particular to complex and quaternian grassmannians.

The Kronecker product of two homogeneous symmetric polynomials P1 and P2 is defined by means by the Frobenius map by the formula . By using the combinatorial definitions for the transition matrices between the various bases of the space of homogeneous symmetric functions, one obtains a procedure for expanding this Kronecker product. In this paper, we employ this technique to study this expansion for the monomial symmetric functions.

Combinatorial B n B_{n} -analogues of Schubert polynomials and corresponding symmetric functions are constructed and studied. The development is based on an exponential solution of the type B B Yang-Baxter equation that involves the nilCoxeter algebra of the hyperoctahedral group.

Stanley [9] introduced the chromatic symmetric function ${\bf X}_G$ associated to a simple graph $G$ as a generalization of the chromatic polynomial of $G$. In this paper we present a novel technique to write ${\bf X}_G$ as a linear combination of chromatic symmetric functions of smaller graphs. We use this technique to give a sufficient condition for two graphs to have the same chromatic symmetric function. We then construct an infinite family of pairs of unicyclic graphs with the same chromatic symmetric function, answering the question posed by Martin, Morin, and Wagner [7] of whether such a pair exists. Finally, we approach the problem of whether it is possible to determine a tree from its chromatic symmetric function. Working towards an answer to this question, we give a classification theorem for single-centroid trees in terms of data closely related to its chromatic symmetric function.

In this paper, we give some results towards the conjecture that σ2t+1l-1,2t are the only nonlinear balanced elementary symmetric Boolean functions where t and l are positive integers. At first, a unified and simple proof of some earlier results is shown. Then a property of balanced elementary symmetric Boolean functions is presented. With this property, we prove that the conjecture is true for n=2m+2t-1 where m,t(m>t) are two non-negative integers, which verified the conjecture for a large infinite class of integer n. Published version

We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of . Our research leads to connections to many known objects/structures yet to be explained combinatorially.

We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.

We present an explicit formula of the Virasoro singular vectors in terms of Jack symmetric polynomials. The parametert in the Virasoro central chargec=13-6(t+1/t) is just identified with the deformation parameter α of Jack symmetric polynomialsJγ(α). As a by-product, we obtain an integral representation of Jack symmetric polynomials indexed by the rectangular Young diagrams.

We describe the man features of a package of Maple programs for manipulating symmetric polynomials and related structures. Among the highlights of the package are (1) a collection of procedures for converting between polynomial expressions involving several fundamental bases, and (2) a general mechanism that allows the user to easily add new bases to the existing collection. The latter facilitates computations involving numerous important families of symmetric functions, including Schur functions, Zonal polynomials, Jack symmetric functions, Hall-Littlewood functions, and the two-parameter symmetric functions of Macdonald.

Let [Formula: see text] with [Formula: see text] a prime and [Formula: see text] a symmetric group. We prove in this paper that if [Formula: see text], then [Formula: see text], where [Formula: see text] is the nilpotent ideal constructed in [Radicals of symmetric cellular algebras, Collog. Math. 133 (2013) 67–83]. Finally we give two remarks on algebras [Formula: see text] with [Formula: see text].

We show that symmetric functions introduced by Di Francesco and co-workers, to prove Kontsevich's formula and Witten's conjecture in the intersection theory of the moduli space of curves are, up to sign, SchurQ-functions.

For a finite graph G with d vertices we define a homogeneous symmetric function XG of degree d in the variables x1, x2, ... . If we set x1 = ... = xn= 1 and all other xi = 0, then we obtain χG(n), the chromatic polynomial of G evaluated at n. We consider the expansion of XG in terms of various symmetric function bases. The coefficients in these expansions are related to partitions of the vertices into stable subsets, the Mobius function of the lattice of contractions of G, and the structure of the acyclic orientations of G. The coefficients which arise when XG is expanded in terms of elementary symmetric functions are particularly interesting, and for certain graphs are related to the theory of Hecke algebras and Kazhdan-Lusztig polynomials.

The symmetric [Formula: see text]-analysis is used to construct a type of minimum-uncertainty [Formula: see text]-coherent states in the Fock representation space of the symmetric [Formula: see text]-oscillator ∗-algebra with [Formula: see text]. Then, its corresponding [Formula: see text]-Hermite polynomials are derived by using the [Formula: see text]-Bargmann–Fock realization of the symmetric [Formula: see text]-oscillator algebra.

A new class of symmetric functions called factorial Schur symmetric functions has recently been discovered in connection with a branch of mathematical physics. We align this theory more closely with the standard symmetric function theory, giving the factorial Schur function a tableau definition, introducing a shift operator and a new generating function with which we extend to factorial symmetric functions proofs of various determinantal identities for classical symmetric functions, and defining a new factorial symmetric function—the factorial elementary symmetric function.

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-atoms, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$., Comment: 23 pages

By properties of the Schur-convex function, Schur-concavity for a class of symmetric functions is simply proved uniform. 2000 Mathematics Subject Classification: Primary 26D15; 05E05; 26B25.

We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian polynomials, the one in symmetric function theory deals with a refinement of the chromatic symmetric functions of Stanley, and the one in algebraic geometry deals with Tymoczko’s representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some remarkable connections between these topics.

We define new families of noncommutative symmetric functions and quasi-symmetric functions depending on two matrices of parameters, and more generally on parameters associated with paths in a binary tree. Appropriate specializations of both matrices then give back the two-vector families of Hivert, Lascoux, and Thibon and the noncommutative Macdonald functions of Bergeron and Zabrocki., Nous définissons de nouvelles familles de fonctions symétriques non-commutatives et de fonctions quasi-symétriques, dépendant de deux matrices de paramètres, et plus généralement, de paramètres associés à des chemins dans un arbre binaire. Pour des spécialisations appropriées, on retrouve les familles à deux vecteurs de Hivert-Lascoux-Thibon et les fonctions de Macdonald non-commutatives de Bergeron-Zabrocki.

We study the problem of expanding the product of two Stanley symmetric functions $F_w·F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, and study the behavior of the expansion of $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability., Nous étudions le problème de développement du produit de deux fonctions symétriques de Stanley $F_w·F_u$ en fonctions symétriques de Stanley de façon naturelle. Notre méthode consiste à considérer une fonction symétrique de Stanley comme un polynôme du Schubert stabilisè $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, et à étudier le comportement de développement de $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ en polynômes de Schubert lorsque $n$ augmente. Nous prouvons que cette développement se stabilise et donc nous obtenons une développement naturelle pour le produit de deux fonctions symétriques de Stanley. Dans le cas où l'une des permutations est Grassmannienne, nous avons une meilleure compréhension de cette stabilité.

In this note, we prove certain inequalities for elementary symmetric funtions that are relevant to the study of partial differential equations associated with curvature problems.

Bent functions are the boolean functions having the maximal possible Hamming distance from the linear boolean functions. Bent functions were introduced and first studied by O. S. Rothaus in 1976.We prove that there are exactly four symmetric bent functions on every even number of variables. These functions are exactly the four symmetric quadratic polynomials of the given number of variables.

In unpublished work, Macdonald gave an indirect proof that the connexion coefficients for certain symmetric functions coincide with the connexion coefficients of the class algebra of the symmetric group. We give a direct proof of this result and demonstrate the use of these functions in a number of combinatorial questions associated with ordered factorisations of permutations into factors of specified cycle-type, including factorisations considered up to commutation in the symmetric group. Several related properties of the symmetric functions are given.