Elementary Symmetric Functions Research Articles

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an n -variate polynomial f ( x ) over a field \({\mathbb {F}} \) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in {\mathrm{GL}}(n,{\mathbb {F}}) \text{ and } {\mathbf {b}} \in {\mathbb {F}}^n\rbrace \) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d} \) – the (1, 1)-th entry of a product of d generic 3 × 3 matrices – is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)} \) as shown by Ben-or and Cleve [12]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d}) \) before this work. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: (1) Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . (2) Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in {\mathbb {N}}} \) . (3) Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . (4) Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n) \) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [6]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [30], Forbes, Shpilka, and Saptharishi [27]. The proof of the third result adapts the algebraic independence based technique of Agrawal, Saha, Saptharishi, and Saxena [5], Beecken, Mittmann, and Saxena [11] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator based argument in Shpilka and Volkovich [90] yields the fourth result. The SV generator plays an important role in all the four results.

Representation theoretic interpretation and interpolation properties of inhomogeneous spin q-Whittaker polynomials

We establish new properties of inhomogeneous spin q-Whittaker polynomials, which are symmetric polynomials generalizing t=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=0$$\\end{document} Macdonald polynomials. We show that these polynomials are defined in terms of a vertex model, whose weights come not from an R-matrix, as is often the case, but from other intertwining operators of Uq′(sl^2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U'_q({\\widehat{\\mathfrak {sl}}}_2)$$\\end{document}-modules. Using this construction, we are able to prove a Cauchy-type identity for inhomogeneous spin q-Whittaker polynomials in full generality. Moreover, we are able to characterize spin q-Whittaker polynomials in terms of vanishing at certain points, and we find interpolation analogues of q-Whittaker and elementary symmetric polynomials.

On fully nonlinear Loewner-Nirenberg problem of Ricci curvature

We prove the existence of a smooth complete conformal metric with prescribed kth elementary symmetric function of negative Ricci curvature under certain condition on general domain in Euclidean space. We then formulate this problem for more general equations.

A Molev-Sagan type formula for double Schubert polynomials

We give a Molev-Sagan type formula for computing the product Su(x;y)Sv(x;z) of two double Schubert polynomials in different sets of coefficient variables where the descents of u and v satisfy certain conditions that encompass Molev and Sagan's original case and conjecture positivity in the general case. Additionally, we provide a Pieri formula for multiplying an arbitrary double Schubert polynomial Su(x;y) by a factorial elementary symmetric polynomial Ep,k(x;z). Both formulas remain positive in terms of the negative roots when we set y=z, so in particular this gives a new equivariant Littlewood-Richardson rule for the Grassmannian, and more generally a positive formula for multiplying a factorial Schur polynomial sλ(x1,…,xm;y) by a double Schubert polynomial Sv(x1,…,xp;y) such that m≥p. An additional new result we present is a combinatorial proof of a conjecture of Kirillov of nonnegativity of the coefficients of skew Schubert polynomials, and we conjecture a weight-preserving bijection between a modification of certain diagrams used in our formulas and RC-graphs/pipe dreams arising in formulas for double Schubert polynomials.

Delta and Theta Operator Expansions

Abstract We give an elementary symmetric function expansion for the expressions $M\Delta _{m_\gamma e_1}\Pi e_\lambda ^{\ast }$ and $M\Delta _{m_\gamma e_1}\Pi s_\lambda ^{\ast }$ when $t=1$ in terms of what we call $\gamma $ -parking functions and lattice $\gamma $ -parking functions. Here, $\Delta _F$ and $\Pi $ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$ . Our main results, in turn, give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta _{Fe_1} \Theta _{G} J$ whenever F is expanded in terms of monomials, G is expanded in terms of the elementary basis, and J is expanded in terms of the modified elementary basis $\{\Pi e_\lambda ^\ast \}_\lambda $ . Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an e-positivity conjecture for when t is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.

Formula omitted]-Stirling numbers in type [formula omitted

Stirling numbers of the first and second kind, which respectively count permutations in the symmetric group and partitions of a set, have found extensive application in combinatorics, geometry, and algebra. We study analogues and q-analogues of these numbers corresponding to the Coxeter group of type B. In particular, we show how they are related to complete homogeneous and elementary symmetric polynomials; demonstrate how they q-count signed partitions and permutations; compute their ordinary, exponential, and q-exponential generating functions; and prove various identities about them. Ordered analogues of the q-Stirling numbers of the second kind have recently appeared in conjectures of Zabrocki and of Swanson–Wallach concerning the Hilbert series of certain super coinvariant algebras. We provide conjectural bases for these algebras and show that they have the correct Hilbert series.

Symmetric Polynomials in Free Associative Algebras—II

Let K⟨Xd⟩ be the free associative algebra of rank d≥2 over a field, K. In 1936, Wolf proved that the algebra of symmetric polynomials K⟨Xd⟩Sym(d) is infinitely generated. In 1984 Koryukin equipped the homogeneous component of degree n of K⟨Xd⟩ with the additional action of Sym(n) by permuting the positions of the variables. He proved finite generation with respect to this additional action for the algebra of invariants K⟨Xd⟩G of every reductive group, G. In the first part of the present paper, we established that, over a field of characteristic 0 or of characteristic p>d, the algebra K⟨Xd⟩Sym(d) with the action of Koryukin is generated by (noncommutative version of) the elementary symmetric polynomials. Now we prove that if the field, K, is of positive characteristic at most d then the algebra K⟨Xd⟩Sym(d), taking into account that Koryukin’s action is infinitely generated, describe a minimal generating set.

A sharp inequality for trace-free matrices with applications to hypersurfaces

We derive a sharp inequality relating the second and fourth elementary symmetric functions of the eigenvalues of a trace-free matrix and give two applications. First, we give a new proof of the classification of conformally flat hypersurfaces in spaceforms. Second, we construct a functional which characterizes rotational hypersurfaces and catenoids.

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree n is called hyperbolic if all of its n roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article is on families of hyperbolic polynomials which are determined through k linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a hyperbolic slice. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the local extreme points of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most k distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

Fast approximation to supershell partition functions: Explicit forms of the coefficients

In a previous work, we derived a formula for supershell partition functions, which are the cornerstone of the Super-Transition-Array approach to radiative-opacity calculations. The new expression takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients (denoted Γk) for which we obtained a recursion relation. In the present short paper, as a complement of the previous one, we give an explicit formula for the coefficients Γk. Another recursion relation is also provided, as well as an alternative expression involving Bell polynomials. The connections with cycle indexes of permutation groups and with elementary symmetric polynomials are outlined.

KLEIN-4 İNVARYANTLARI ÜZERİNE

Let K[X_4 ]=K[x_1,x_2,x_3,x_4 ] be the polynomial algebra with 4 algebraically independent commuting variables over a field K of characteristic zero. The symmetric group S_4 acts on K[X_4 ] naturally by the action of permutations exchanging the indices of variables with respect to the corresponding permutation. It is well known that the algebra K[X_4 ]^S_4 of all polynomials preserved under the action of S_4 is generated by 4 algebraically independent elements called the elementary symmetric polynomials. In this study, we consider the subalgebra G of S_4 generated by the transpositions (13) and (24) which is isomorphic to the Klein-4 group, and find a free generating set for the algebra K[X_4 ]^G of G-invariants.

Infinite Series Concerning Tails of Riemann Zeta Values

Infinite series involving Riemann’s zeta and Dirichlet’s lambda tails, and weighted by three harmonic-like elementary symmetric functions are examined. By means of integral representations of zeta tails together with the telescopic approach, twelve general summation theorems are established that express these series as coefficients of the bivariate beta function Beta(u,v). By further expanding Beta(u,v) into Laurent series in u and v, several explicit summation formulae are shown as consequences.

The first infinite family of orthogonal Steiner systems S(3,5,v)

The research on orthogonal Steiner systems S(t,k,v) was initiated in 1968. For (t,k)∈{(2,3),(3,4)}, this corresponds to orthogonal Steiner triple systems (STSs) and Steiner quadruple systems (SQSs), respectively. The existence problem of a pair of orthogonal STSs or SQSs was settled completely thirty years ago. However, for Steiner systems with t≥3 and k≥5, only two small examples of orthogonal pairs were known to exist before this work. An infinite family of orthogonal Steiner systems S(3,5,v) is constructed in this paper. In particular, the existence of a pair of orthogonal Steiner systems S(3,5,4m+1) is established for any even m≥2; additionally a pair of orthogonal G-designs G(4m+15,5,5,3) is displayed for any odd m≥3. The construction is based on the Steiner systems admitting 3-transitive automorphism groups supported by elementary symmetric polynomials. Moreover, 50 mutually orthogonal Steiner systems S(5,8,24) are shown to exist.

State Preparation of Antisymmetrized Geminal Power on a Quantum Computer without Number Projection

The antisymmetrized geminal power (AGP) is equivalent to the number projected Bardeen-Cooper-Schrieffer (PBCS) wave function. It is also an elementary symmetric polynomial (ESP) state. We generalize previous research on deterministically implementing the Dicke state to a state preparation algorithm for an ESP state, or equivalently AGP, on a quantum computer. Our method is deterministic and has polynomial cost, and it does not rely on number symmetry breaking and restoration. We also show that our circuit is equivalent to a disentangled unitary paired coupled cluster operator and a layer of unitary Jastrow operator acting on a single Slater determinant. The method presented herein highlights the ability of disentangled unitary coupled cluster to capture nontrivial entanglement properties that are hardly accessible with traditional Hartree-Fock based electronic structure methods.

Symmetric polynomials over finite fields

It is shown that two vectors with coordinates in the finite q-element field of characteristic p belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree pk,2pk,…,(q−1)pk, k=0,1,2,… has the same value on them. This separating set of polynomial invariants for the natural permutation representation of the symmetric group is not far from being minimal when q=p and the dimension is large compared to p. A relatively small separating set of multisymmetric polynomials over the field of q elements is derived.

Clifford-symmetric polynomials

Based on the NilHecke algebra , the odd NilHecke algebra developed by Ellis, Khovanov and Lauda, and on Kang, Kashiwara and Tsuchioka’s quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra as another super-algebraic analogue of . We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.

Optimal unlabeled set partitioning with application to risk-based quarantine policies

We consider the problem of partitioning a set of items into unlabeled subsets so as to optimize an additive objective, i.e., the objective function value of a partition is equal to the sum of the contribution of each subset. Under an arbitrary objective function, this family of problems is known to be an -complete combinatorial optimization problem. We study this problem under a broad family of objective functions characterized by elementary symmetric polynomials, which are “building blocks” to symmetric functions. By analyzing a continuous relaxation of the problem, we identify conditions that enable the use of a reformulation technique in which the set partitioning problem is cast as a more tractable network flow problem solvable in polynomial-time. We show that a number of results from the literature arise as special cases of our proposed framework, highlighting its generality. We demonstrate the usefulness of the developed methodology through a novel and timely application of quarantining heterogeneous populations in an optimal manner. Our case study on real COVID-19 data reveals significant benefits over conventional measures in terms of both spread mitigation and economic impact, underscoring the importance of data-driven policies.

Newton’s identities and positivity of trace class integral operators

We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton’s identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger’s uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.

Lattice Paths and Negatively Indexed Weight-Dependent Binomial Coefficients

In 1992, Loeb (Adv Math, 91:64–74, 1992) considered a natural extension of the binomial coefficients to negative entries and gave a combinatorial interpretation in terms of hybrid sets. He showed that many of the fundamental properties of binomial coefficients continue to hold in this extended setting. Recently, Formichella and Straub (Ann Comb, 23:725–748, 2019) showed that these results can be extended to the q-binomial coefficients with arbitrary integer values and extended the work of Loeb further by examining the arithmetic properties of the q-binomial coefficients. In this paper, we give an alternative combinatorial interpretation in terms of lattice paths and consider an extension of the more general weight-dependent binomial coefficients, first defined by Schlosser (Sém Lothar Combin, 81:24, 2020), to arbitrary integer values. Remarkably, many of the results of Loeb, Formichella and Straub continue to hold in the general weighted setting. We also examine important special cases of the weight-dependent binomial coefficients, including ordinary, q- and elliptic binomial coefficients as well as elementary and complete homogeneous symmetric functions.

Integral formulas for a foliated sub-Riemannian manifold

We apply the notion of foliation to a nonholonomic manifold, which was introduced for the geometric interpretation of constrained systems in mechanics. We prove a series of integral formulas for a foliated sub-Riemannian manifold, that is, a Riemannian manifold equipped with a distribution $${{\mathscr {D}}}$$ and a foliation $${{\mathscr {F}}}$$ whose tangent bundle is a subbundle of $${{\mathscr {D}}}$$ . Our integral formulas generalize some results for a foliated Riemannian manifold and involve the shape operators of $${\mathscr {F}}$$ with respect to normals in $${\mathscr {D}}$$ , the curvature tensor of induced connection on $${\mathscr {D}}$$ and arbitrary functions depending on elementary symmetric functions of eigenvalues of the shape operators. For a special choice of these functions, integral formulas with the Newton transformations of the shape operators of $${\mathscr {F}}$$ are obtained. Application to a foliated sub-Riemannian manifold with restrictions on the curvature and extrinsic geometry of $${\mathscr {F}}$$ and also to codimension-one foliations are given.