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Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial variation, but here we demonstrate how to extract explicit formulas from that machinery which may be applied to other positivity problems as well.

Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the well-known problem of finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannians. In this paper, we prove cylindric Schur positivity of cylindric skew Schur functions, as conjectured by McNamara. We also show that all the coefficients appearing in the expansion are the same as the 3-point Gromov-Witten invariants. We start by discussing the properties of affine Stanley symmetric functions for general affine permutations and 321-avoiding affine permutations, and we explain how these functions are related to cylindric skew Schur functions. In addition, we provide an effective algorithm to compute the expansion of cylindric skew Schur functions in terms of cylindric Schur functions, as well as the expansion of affine Stanley symmetric functions in terms of affine Schur functions.

We relate the combinatorial definitions of the type $A_n$ and type $C_n$ Stanley symmetric functions, via a combinatorially defined "double Stanley symmetric function," which gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives the type $C$ case at $(\mathbf{x},\mathbf{x})$. We induce a type $A$ bicrystal structure on the underlying combinatorial objects of this function which has previously been done in the type $A$ and type $C$ cases. Next we prove a few statements about the algebraic relationship of these three Stanley symmetric functions. We conclude with some conjectures about what happens when we generalize our constructions to type $C$.

We study a family of symmetric functions $\hat F_z$ indexed by involutions $z$ in the affine symmetric group. These power series are analogues of Lam's affine Stanley symmetric functions and generalizations of the involution Stanley symmetric functions introduced by Hamaker, Pawlowski, and the first author. Our main result is to prove a transition formula for $\hat F_z$ which can be used to define an affine involution analogue of the Lascoux-Sch\"utzenberger tree. Our proof of this formula relies on Lam and Shimozono's transition formula for affine Stanley symmetric functions and some new technical properties of the strong Bruhat order on affine permutations., Comment: 28 pages, 1 figure; v2: fixed typos, added reference; v3: added figure, extra discussion in Section 5, updated references; v4: minor corrections, final version

We consider Z 2 -contractions of classical Lie algebras and the behaviour of the symmetric invariants under these contractions. It is demonstrated on three different examples how the theory of symmetric polynomials works in this invariant-theoretic problem.

The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words, and are indexed by the involutions in the symmetric group. By construction each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands, and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila-Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials., 38 pages, 3 figures; v2: some typos corrected, updated references; v3: various corrections and a few new results, article now split into two parts with the second half its own submission; v4: updated references and fixed typos, final version

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer $n$ or have algebraic degree no more than 4. In this paper, for any even integer $n\ge 4$ , three classes of bent functions are presented respectively. Most notably, the proposed $n$ -variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to $n/2$ . Besides, we obtain bent idempotent functions with the maximal algebraic degree $n/2$ .

We consider the algebra of holomorphic functions on L∞ that are symmetric, i.e. that are invariant under composition of the variable with any measure-preserving bijection of [0, 1]. Its spectrum is identified with the collection of scalar sequences such that is bounded and turns to be separable. All this follows from our main result that the subalgebra of symmetric polynomials on L∞ has a natural algebraic basis.

In this paper we provide new families of balanced symmetric functions over any finite field. We also generalize a conjecture of Cusick, Li, and Stǎnicǎ about the non-balancedness of elementary symmetric Boolean functions to any finite field and prove part of our conjecture.

Subword complexes were defined by A.Knutson and E.Miller in 2004 for describing Gr\"obner degenerations of matrix Schubert varieties. The facets of such a complex are indexed by pipe dreams, or, equivalently, by the monomials in the corresponding Schubert polynomial. In 2017 S.Assaf and D.Searles defined a basis of slide polynomials, generalizing Stanley symmetric functions, and described a combinatorial rule for expanding Schubert polynomials in this basis. We describe a decomposition of subword complexes into strata called slide complexes, that correspond to slide polynomials. The slide complexes are shown to be homeomorphic to balls or spheres., Comment: 16 pages, 5 figures. v2: proof of Thm 4.4 simplified, typos corrected. v3: minor improvements, Sec.4.3 added. To appear in Sbornik: Mathematics

We introduce and study affine analogues of the fixed-point-free (FPF) involution Stanley symmetric functions of Hamaker, Marberg, and Pawlowski. Our methods use the theory of quasiparabolic sets introduced by Rains and Vazirani, and we prove that the subset of FPF-involutions is a quasiparabolic set for the affine symmetric group under conjugation. Using properties of quasiparabolic sets, we prove a transition formula for the affine FPF involution Stanley symmetric functions, analogous to Lascoux and Sch\"utzenberger's transition formula for Schubert polynomials. Our results suggest several conjectures and open problems., Comment: 23 pages

In this paper, we firstly compute the dual functions of elementary symmetric bent functions. Next, we derive a new secondary construction of bent functions (given with their dual functions) involving symmetric bent functions, leading to a generalization of the well-know Rothaus' construction.

Rotation symmetric Boolean functions have been extensively studied in the recent years because of their applications in cryptography. In this study, a novel method to construct p-variable 1-resilient rotation symmetric functions over GF(p) is proposed based on a Latin square with maximum cycle structure, which is not required to solve any equation system. And a lower bound on the number of p-variable 1-resilient rotation symmetric functions is given. At last, an equivalent characterization of p-variable 1-resilient rotation symmetric functions over GF(p) is demonstrated, as a direct corollary, the number of p-variable 1-resilient rotation symmetric functions is represented by all the solutions of the equation system. Copyright © 2017 John Wiley & Sons, Ltd.

It has been known since 1996 (work of Cai et al.) that the sequence of Hamming weights { w t ( f n ( x 1 , ź , x n ) ) } , where f n is a symmetric Boolean function of degree d in n variables, satisfies a linear recurrence with integer coefficients. In 2011, Castro and Medina used this result to show that a 2008 conjecture of Cusick, Li and Stźnicź about when an elementary symmetric Boolean function can be balanced is true for all sufficiently large n . Quite a few papers have been written about this conjecture, but it is still not completely settled. Recently, Guo, Gao and Zhao proved that the conjecture is equivalent to the statement that all balanced elementary symmetric Boolean functions are trivially balanced. This motivates the further study of the weights of symmetric Boolean functions f n . In this paper we prove various new results on the trivially balanced functions. We also determine a period (sometimes the minimal one) for the sequence of weights w t ( f n ) modulo any odd prime p , where f n is any symmetric function, and we prove some related results about the balanced symmetric functions.

Rotation symmetric Boolean functions have been used as components of different cryptosystems. In this paper, based on the knowledge of compositions of an integer, a new construction of balanced 2 p -variable rotation symmetric Boolean functions with optimal algebraic immunity is provided, where p is an odd prime. The nonlinearity of our new functions is significantly higher than all previously obtained balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity, and is higher than the best nonlinearity of even-variable rotation symmetric Boolean functions with optimal algebraic immunity in most cases. We also show that our new functions have high algebraic degree and a good behavior against fast algebraic attacks.

Via the chip-firing game, a class of Schur positive symmetric functions depending on four parameters is introduced for any labeled connected simple graph. Tableaux formulae are stated to expand such symmetric functions in terms of the complete symmetric functions. In the case of a simple path, the resulting symmetric functions reduce to the transformed Hall-Littlewood symmetric functions when the parameters are suitable specialized.

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.

This paper focuses on the balancedness of symmetric Boolean functions. We prove a conjecture presented by Canteaut and Videau, which states that the balanced symmetric Boolean functions of fixed algebraic degree are trivially balanced when the number of variables is large enough. Denoted by $\sigma _{n,d}$ , the $n$ -variable elementary symmetric Boolean function of degree $d$ . As an application of this result to elementary symmetric Boolean functions, we show that all the trivially balanced elementary symmetric Boolean functions are of the form $\sigma _{2^{t+1}l-1,2^{t}}$ , where $t$ and $l$ are any positive integers. It implies that Cusick et al. ’s conjecture, which claims that $ \sigma _{2^{t+1}l-1,2^{t}}$ is the only nonlinear balanced elementary symmetric Boolean functions, is equivalent to the conjecture that all the balanced elementary symmetric Boolean functions are trivially balanced.

We propose effective constructions of dual bases for the noncommutative symmetric and quasi-symmetric functions. To this end, we use an effective variation of Schutzenberger's factorization adapted to the diagonal pairing between a graded space and its dual.

In this paper we find an exact formula for the number of affine equivalence classes under permutations for binary polynomials degree $$d=6$$d=6 invariant under the cyclic group (also, called monomial rotation symmetric), for a prime number of variables; this extends previous work for $$2\le d\le 5$$2≤d≤5.

To each finite subset of $\mathbb{Z}^2$ (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 exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians., Comment: 15 pages

In an article of Pommerenke [10] he remarked that, for an -fold symmetric functions in the class , the well known lemma stated by Caratheodary for a one fold symmetric functions in still holds good. Exploiting this concept, we introduce certain new subclasses of the bi-univalent function class in which both and are -fold symmetric analytic with their derivatives in the class of analytic functions. Furthermore, for functions in each of the subclasses introduced in this paper, we obtain the coefficient bounds for and We remark here that the concept of -fold symmetric bi-univalent is not in the literature and the authors hope it will make the researchers interested in these type of investigations in the forseeable future. By the working procedure and the difficulty involved in these procedures, one can clearly conclude that there lies an unpredictability in finding the coefficients of a -fold symmetric bi-univalent functions.

Following the definition of q-Frobenius-Euler polynomials introduced in [3], we derive some new symmetric identities under sym (5), also termed symmetric group of degree five, which are derived from the fermionic p-adic q-integral over the p-adic numbers field.

The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg.

The notion of left-right (resp. right-left) symmetric biderivation of KU-algebras is introduced and some related properties are investigated.

An operator T on a complex Hilbert space H is called skew symmetric if T can be represented as a skew symmetric matrix relative to some orthonormal basis for H. In this paper, we study skew symmetric operators with eigenvalues. First, we provide an upper-triangular operator matrix representation for skew symmetric operators with nonzero eigenvalues. On the other hand, we give a description of certain skew symmetric triangular operators, which is based on the geometric relationship between eigenvectors.

We show that the classification of symmetric spaces can be achieved by K-theoretical methods. We focus on Hermitian symmetric spaces of non-compact type, and define K-theory for JB*-triples along the lines of C*-theory. K-groups have to be provided with further invariants in order to classify the spaces in question. Among these are cycles obtained from so called grids, intimately connected to root systems of an underlying Lie-algebra and thus reminiscent of the classical classification scheme.

The class algebra and the double coset algebra are two classical commutative subalgebras of the group algebra of the symmetric group. The connexion coefficients of these two algebraic structures are important numbers with significant applications. From a combinatorial point of view, they give the number of factorizations of a given permutation into the ordered product of permutations with specific cyclic properties and count in some cases the number of hypermaps and constellations on (locally) orientable surfaces. They are also of notable interest in the study of Schur and zonal polynomials as well as in the theory of the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore as shown by Hanlon, Stanley, Stembridge (1992), the respective generating series of these coefficients in the basis of power sum symmetric functions are equal to the mathematical expectation of the trace of ( X U Y U ? ) n where X and Y are given symmetric (respectively hermitian) matrices, U is a random real (respectively complex) valued square matrix of standard normal distribution and n a non negative integer.This paper is devoted to the explicit evaluation of these series in terms of monomial symmetric functions. Morales and Vassilieva (2009, 2011) and Vassilieva (2013) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type a , b , 1 n - a - b ] .

An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijection with reduced words for the longest element in $A_n = S_{n+1}$, which are known to be in bijection with standard tableaux of shape $(n, n-1, \ldots, 2, 1)$., Comment: 26 pages, 3 figures; v2: fixed several typos; v3: some minor corrections, a few added remarks, final version

We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbf{\Pi}$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from $\mathbf{\Pi}$ to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. We also describe analogues of $\mathbf{\Pi}$ for the other classical types. In these cases, the relevant objects are module coalgebras rather than bialgebras, but there are again natural morphisms to the quasi-symmetric functions, under which the image of a signed permutation is the corresponding Stanley symmetric function of type B, C, or D., Comment: 24 pages; v2: fixed several typos; v3: minor corrections, updated references, final version

Following the work of B. Kuelshammer, J. B. Olsson and G. R. Robinson on generalized blocks of the symmetric groups, we give a definition for the \ell-defect of characters of the symmetric group S_n, where \ell > 1 is an arbitrary integer. We prove that the \ell -defect is given by an analogue of the hook-length formula, and use it to prove, when n < \ell^2, an \ell-version of the McKay Conjecture in S_n ., To appear in Journal of Group Theory, 10 pages

The paper considers a class of symmetric Boolean functions called Reed-Muller type Fundamental Symmetric Functions, it reviews some of their properties and presents some new ones. The main contribution of the paper is a proof that the Reed-Muller transform of a symmetric Boolean function is also symmetric and that of a rotation symmetric Boolean function is also rotation symmetric. Since symmetric n-place Boolean functions may be given a compact representation with a value vector of n+1 elements and this holds also for its Reed-Muller spectrum, some methods are reviewed, to calculate the Reed-Muller spectrum of a symmetric Boolean function based on its compact value vector. Furthermore a method is presented to calculate the Reed-Muller spectrum of a rotation symmetric Boolean function from the compact value vector representation of the function.

The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representatives $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functions $\hat{F}^{\tt{FPF}}_z$, which are stable limits of the polynomials $\hat{\mathfrak{S}}^{\tt{FPF}}_z$, are Schur $P$-positive. To do so, we construct an analogue of the Lascoux-Sch\"utzenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest for $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ when $z$ is a fixed-point-free version of a Grassmannian permutation. We also classify the fixed-point-free involution Stanley symmetric functions that are single Schur $P$-functions, and show that the decomposition of $\hat{F}^{\tt{FPF}}_z$ into Schur $P$-functions is unitriangular with respect to dominance order on strict partitions. These results and proofs mirror previous work by the authors related to the orthogonal group action on the type A flag variety., Comment: 34 pages, 1 figure. This article was formerly the second half of arXiv:1701.02824; v2: revised introduction, expanded proofs and examples, added index of notation, minor corrections

The algebra $\mathcal{H}_{bs}(\ell_1)$ of symmetric analytic functions of bounded type is investigated. In particular, we study continuity of some homomorphisms of the algebra of symmetric polynomials on $\ell_p$ and composition operators of the algebra of symmetric analytic functions. The paper contains several open questions.

It is known that the set of rotation symmetric Boolean functions has many functions with various useful properties for cryptography. This study shows how to construct some families of rotation symmetric functions which are balanced or plateaued. The authors also consider vectorial Boolean functions [that is, maps from GF (2) n to GF (2) m ] which are k-rotation symmetric and they give two infinite families of such functions which are permutations with the maximum possible algebraic degree. The families of functions that they give provide a source, which can be searched for functions with other useful cryptographic properties.

Generalizing the notion of a vexillary permutation, we introduce a filtration of S_infinity by the number of Schur function terms in the Stanley symmetric function, with the kth filtration level called the k-vexillary permutations. We show that for each k, the k-vexillary permutations are characterized by avoiding a finite set of patterns. A key step is the construction of a Specht series, in the sense of James and Peel, for the Specht module associated to the diagram of a permutation. As a corollary, we prove a conjecture of Liu on diagram varieties for certain classes of permutation diagrams. We apply similar techniques to characterize multiplicity-free Stanley symmetric functions, as well as permutations whose diagram is equivalent to a forest in the sense of Liu., 29 pages; corrected typos in section 3, fixed arguments and improved results in section 6

The complete and elementary symmetric functions are special cases of Schur functions. It is well-known that the Schur functions can be expressed in terms of complete or elementary symmetric functions using two determinant formulas: Jacobi–Trudi identity and Nagelsbach–Kostka identity. In this paper, we study new connections between complete and elementary symmetric functions.

In this paper, we derive a relation of new kind between certain character values of symmetric groups in terms of the so-called maya diagrams. We also investigate a relation between our result and Bernstein's creation operators for Schur functions, and consider analogous relations for projective characters of symmetric groups through creation operators for Schur $$Q$$ Q -functions. We also consider analogous relations for characters of Brauer algebras and walled Brauer algebras.

We investigate properties of sets of functions comprising countably many elements An such that every function An is orthogonal to all polynomials of degrees less than n. We propose an effective method for solving Fredholm integral equations of the first kind whose kernels are generating functions for these sets of functions. We study integral equations used to solve some axially symmetric problems in physics. We prove that their kernels are generating functions that produce functions in the studied families and find these functions explicitly. This allows determining the elements of the matrices of systems of linear equations related to the integral equations for considering the physical problems.

Tight bounds on an elementary symmetric function are established under the assumption that the values of the elementary symmetric functions of lower orders are given. The explicit form of the inverse Hankel moment matrix leads to inequalities for moments and for elementary symmetric polynomials.

The Little map and the Edelman---Greene insertion algorithm, a generalization of the Robinson---Schensted correspondence, are both used for enumerating the reduced decompositions of an element of the symmetric group. We show the Little map factors through Edelman---Greene insertion and establish new results about each map as a consequence. In particular, we resolve some conjectures of Lam and Little.

International audience; Our main purpose is to provide for homogeneous (even or odd) symmetricantiassociative quasialgebras a structure theory analogous to that for homogeneoussymmetric associative superalgebras given in [5] and to present an inductive descriptionof these classes of algebras.