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Using twists of fields of charged free fermions, we revise the generalized vertex operator presentation of Hall–Littlewood polynomials, propose a new version of the deformed boson-fermion correspondence, and construct new examples of tau-functions of the KP hierarchy in the one-parameter deformation of the ring of symmetric functions Λ[t].

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric functions are defined as weighted sums over cylindric reverse plane partitions or - alternatively - in terms of sets of affine permutations. We relate their combinatorial definition to an algebraic construction in terms of the principal Heisenberg subalgebra of the affine Lie algebra $\mathfrak{\widehat{sl}}_n$ and a specialised cyclotomic Hecke algebra. Using Schur-Weyl duality we show that the new cylindric symmetric functions arise as matrix elements of Lie algebra elements in the subspace of symmetric tensors of a particular level-0 module which can be identified with the small quantum cohomology ring of the $k$-fold product of projective space. The analogous construction in the subspace of alternating tensors gives the known set of cylindric Schur functions which are related to the small quantum cohomology ring of Grassmannians. We prove that cylindric Schur functions form a subcoalgebra in $\Lambda$ whose structure constants are the 3-point genus 0 Gromov-Witten invariants. We show that the new families of cylindric functions obtained from the subspace of symmetric tensors also share the structure constants of a symmetric Frobenius algebra, which we define in terms of tensor multiplicities of the generalised symmetric group $G(n,1,k)$., Comment: 63 pages, 5 figures (v3: version accepted for publication in Algebraic Combinatorics)

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized ( max ⁡ , + ) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.

In [33] , the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products. In addition, we identify some applications and instances where special cases of these products have occurred elsewhere in the mathematical literature.

The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.

The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group W0. The set of Weyl characters sλ forms a basis of the center and Lusztig showed in [11] that these characters act as translations on the Kazhdan–Lusztig basis element $$ {C}_{w_0} $$ where w0 is the longest element of W0, that is, we have $$ {C}_{w_0}{\mathrm{s}}_{\uplambda}={C}_{w_0{t}_{\uplambda}} $$ . As a consequence, the coefficients that appear when decomposing $$ {C}_{w_0{t}_{\uplambda}}{\mathrm{s}}_{\tau } $$ in the Kazhdan–Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group W0. The aim of this paper is to explain how admissible subsets and Littelmann paths, which are models to compute such multiplicities, naturally appear when working out this decomposition.

We prove a representation theorem for copulas that are (simultaneously) symmetric and radially symmetric. We use this representation theorem to propose a method to construct an n-ary symmetric function that is radially symmetric, starting from an ( n − 1 ) -dimensional copula and an n-ary auxiliary function. We study the necessary and sufficient conditions on this auxiliary function that guarantee our construction method to result in a symmetric and radially symmetric n-dimensional copula. We examine several options for defining the auxiliary function in the trivariate case, inspired by the nesting of copulas, the lifting of copulas and product-like extensions. For each choice of auxiliary function, we provide several examples for different families of copulas.

We study graded symmetric algebras, which are the symmetric monoids in the monoidal category of vector spaces graded by a group. We show that a finite dimensional graded semisimple algebra is graded symmetric. The center of a symmetric algebra is not necessarily symmetric, but we prove that the center of a finite dimensional graded division algebra is symmetric, provided that the order of the grading group is not divisible by the characteristic of the base field.

We present a proof of the compositional shuffle conjecture by Haglund, Morse and Zabrocki [Canad. J. Math., 64 (2012), 822–844], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra by Haglund, Haiman, Loehr, Remmel, and Ulyanov [Duke Math. J., 126 (2005), 195–232]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space V ∗ V_* whose degree zero part is the ring of symmetric functions Sym ⁡ [ X ] \operatorname {Sym}[X] over Q ( q , t ) \mathbb {Q}(q,t) . We then extend these operators to an action of an algebra A ~ \tilde {\mathbb A} acting on this space, and interpret the right generalization of the ∇ \nabla using an involution of the algebra which is antilinear with respect to the conjugation ( q , t ) ↦ ( q − 1 , t − 1 ) (q,t)\mapsto (q^{-1},t^{-1}) .

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.

This work examines the free and parametric vibrations of constrained cyclically symmetric ring structures and compares the restrictions from different modeling assumptions. A complete model and two simplified models are developed by introducing different restrictions on deflections. The eigenvalues are formulated by direct perturbation and mode superposition. The results verify that they are comparable on the condition that the stiffness ratio of the rotating support to ring bending is relatively small. The eigenvalue splitting, parametric instability and their relationships are determined in closed form. The simplified models can correctly predict the splitting behavior before or after the loci veering. Two types of models obtain the same rules governing the instability, but the simplified models cannot accurately approach the time-evolution of the instability. Hence, it needs to be cautious to use the simplified models to estimate the instability behaviors for a cyclically symmetric ring structure with larger stiffness ratios especially within the loci veering range. Main results are verified by numerical calculations.

The advanced theory of the first and higher symmetric Frechet differentials and K-sub-differentials is constructed including the mean value theorem and the Taylor formula. We give simple sufficient conditions for symmetric K-subdifferentiability and consider some applications to Fourier series and variational functionals.

In this paper, we study several properties of $m$-complex symmetric operators. In particular, we prove that if $T\in{\cal L(H)}$ is an $m$-complex symmetric operator and $N$ is a nilpotent operator of order $n>2$ with $TN=NT$, then $T+N$ is a $(2n+m-2)$-complex symmetric operator. Moreover, we investigate the decomposability of $T+A$ and $TA$ where $T$ is an $m$-complex symmetric operator and $A$ is an algebraic operator. Finally, we provide various spectral relations of such operators. As some applications of these results, we discuss Weyl type theorems for such operators.

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.

Entangling properties of a mixed two-qubit system can be described by local homogeneous unitary invariant polynomials in the elements of the density matrix. The structure of the corresponding ring of invariant polynomials for a special subclass of states, the so-called mixed X-states, is established. It is shown that for the X-states there is an injective ring homomorphism of the quotient ring of SU(2)×SU(2)-invariant polynomials modulo its syzygy ideal to the SO(2) × SO(2)-invariant ring freely generated by five homogeneous polynomials of degrees 1, 1, 1, 2, 2.

An algorithm is presented for decomposing a symmetric tensor into a sum of rank-1 symmetric tensors. For a given tensor, by using apolarity, catalecticant matrices and the condition that the mapping matrices are commutative, the rank of the tensor can be obtained by iteration. Then we can find the generating polynomials under a selected basis set. The decomposition can be constructed by the solutions of generating polynomials under the condition that the solutions are all distinct which can be guaranteed by the commutative property of the matrices. Numerical examples demonstrate the efficiency and accuracy of the proposed method.

A symmetric top is considered, which is a particular case of a mechanical top that is usually described by the canonical Poisson structure on T*SE (3). This structure is invariant under the right action of the rotation group SO(3), but the Hamiltonian of the symmetric top is invariant only under the right action of the subgroup S 1, which corresponds to the rotation of the symmetric top around its axis of symmetry. This Poisson structure is obtained as the reduction T* SE (3) / S 1. A Hamiltonian and motion equations are proposed that describe a wide class of interaction models of the symmetric top with an axially symmetric external field.

Let V be a finite dimensional complex vector space, and for each positive integer n let denote the vector space whose elements are the complex valued symmetric n-linear functions on (n-copies). If , that is, f is a linear functional on V, then we define the symmetric n-linear function by for all . It is well known that the elements , known as polar elements, span . A basis for consisting entirely of polar elements is a polar basis. We present a surprisingly simple recursive algorithm for the generation of polar bases. Explicit stand alone formulas for polar bases are also presented. One such explicit basis is so natural that it should probably be called the canonical polar basis for . We introduce the notion of subspace filter and show that each subspace filter of order n provides polar bases for all of the spaces simultaneously, and we extend this result to the entire symmetric algebra .

The common exponential functions whose exponents are the scalar products 〈λ,x〉, where x is a real variable and λ is an integer, admit two generalizations to any higher dimension, the symmetric and the antisymmetric ones [KLIMYK, A.—PATERA, J.: (Anti)symmetric multivariate exponential functions and corresponding Fourier transforms, J. Phys. A: Math. Theor. 40 (2007), 10473–10489]. Restriction in the paper to the three variables only allows us to work out many specific properties of the symmetric and antisymmetric functions useful in applications. Such are (i) the orthogonalities, both the continuous one and the discrete one on the 3D lattice of any density; (ii) corresponding discrete and continuous Fourier transforms; (iii) generating functions for the related polynomials in three variables, and others. Rapidly increasing precision of the interpolation with increasing density of the 3D lattice is shown in an example.

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.

Unlike in complex analysis, in all hypercomplex function theories a hypercomplex variable is not monogenic (regular) and there exists a problem to define analogues of positive and negative powers of the complex variable. R. Fueter firstly introduces a system of symmetric regular quaternion polynomials as analogues of positive powers of a complex variable and proves the Taylor theorem in his theory. In Clifford analysis an analogical idea is used. The Fueter symmetric polynomials are both left- and right-regular, the symmetric polynomials in Clifford analysis are also both left- and right-monogenic. In this paper we construct only left-regular quaternion polynomials and show that the theory of regular quaternion functions of a vector valued quaternion variable can be developed using these polynomials.

The paper contains a description of symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\ell_{1}$-sum of the space $\mathbb{C}^{2}.$ We show that the specrum of such algebra does not coincide of point evaluation functionals and described characters of the algebra as functions of exponential type with plane zeros.

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.

This paper considers the condition when a rational function $F(s)$ may be represented as the function of the argument $s+(1/s)$ . If this condition is satisfied then $F(s)$ is the ratio of recursive (symmetric) polynomials. This paper investigates the network properties of such rational functions and their realization. Then the symmetric polynomials are applied for synthesis of symmetric band-pass filters. Substituting $p=s+(1/s)$ into symmetric band-pass filter transfer function one obtains its low-pass generating filter. The slew rate and overshoot of generating filter step-response is closely connected with the step-response duration of symmetric band-pass filter. The choice of generating filter becomes an additional factor of symmetric band-pass filter design. As the generating filters the paper proposes using Lommel polynomial filters which have easy control of overshoot and slew rate. An example of six order symmetric band-pass filter is given.

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.

We introduce an enriched analogue of Lam and Pylyavskyy's theory of set-valued $P$-partitions. An an application, we construct a $K$-theoretic version of Stembridge's Hopf algebra of peak quasisymmetric functions. We show that the symmetric part of this algebra is generated by Ikeda and Naruse's shifted stable Grothendieck polynomials. We give the first proof that the natural skew analogues of these power series are also symmetric. A central tool in our constructions is a "$K$-theoretic" Hopf algebra of labeled posets, which may be of independent interest. Our results also lead to some new explicit formulas for the involution $\omega$ on the ring of symmetric functions., Comment: 51 pages; v2: one conjecture upgraded to a theorem, another new conjecture added, some minor edits; v3: minor edits, fixed typos, added references

It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei–Thoma theorem. In this paper, we study positive specializations of symmetric Grothendieck polynomials, K-theoretic deformations of Schur polynomials.

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis., arXiv admin note: text overlap with arXiv:1605.06672; remark 1: irreducible character basis of the rook algebra

Symmetric functions arise in many areas of mathematics including combinatorics, topology and algebraic geometry. Using ideals of symmetric functions, we tie these three branches together. This thesis generalizes work of Garsia and Procesi in 1992 that gave a quotient ring presentation for the cohomology ring of Springer varieties. Let R be the polynomial ring Z[x1, . . . , xn]. We present two different ideals in R. Both are parametrized by a Hessenberg function h, namely a nondecreasing function that satisfies h(i) ≥ i for all i. The first ideal, which we call Ih, is generated by modified elementary symmetric functions. The ideal Ih generalizes the work of Tanisaki who gave a combinatorial description of the ideal used in Garsia and Procesi’s quotient ring. Like the Tanisaki ideal, the generating set for Ih is redundant. We give a minimal generating set for this ideal. The second ideal, which we call Jh, is generated by modified complete symmetric functions. The generators of this ideal form a Grobner basis, which is a useful property. Using the Grobner basis for Jh, we identify a basis for the quotient R/Jh. We introduce a partial ordering on the Hessenberg functions, and in turn we discover nice nesting properties in both families of ideals. When h > h′, we have Ih ⊂ Ih′ and Jh ⊂ Jh′ . We prove that Ih equals Jh when h is maximal. Since Ih is the ideal generated by the elementary symmetric functions when h is maximal, the generating set for Jh forms a Grobner basis for the elementary symmetric functions. Moreover, the quotient R/Jh gives another description of the cohomology ring of the full flag variety. The generators of the ring R/Jh are in bijective correspondence with the Betti numbers of certain Hessenberg varieties. These varieties are a two-parameter

The problem of computing geometric measure of quantum entanglement for symmetric pure states can be regarded as the problem of finding the largest unitary symmetric eigenvalue (US-eigenvalue) for symmetric complex tensors, which can be taken as a multilinear optimization problem in complex number field. In this paper, we convert the problem of computing the geometric measure of entanglement for symmetric pure states to a real polynomial optimization problem. Then we use Jacobian semidefinite relaxation method to solve it. Some numerical examples are presented.

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.

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.