Homogeneous Symmetric Polynomials Research Articles

Compton Amplitude for Rotating Black Hole from QFT.

We construct a candidate tree-level gravitational Compton amplitude for a rotating Kerr black hole, for any quantum spin s=0,1/2,1,…,∞, from which we extract the corresponding classical amplitude to all orders in the spin vector S^{μ}. We use multiple insights from massive higher-spin quantum field theory, such as massive gauge invariance and improved behavior in the massless limit. A chiral-field approach is particularly helpful in ensuring correct degrees of freedom, and for writing down compact off-shell interactions for general spin. The simplicity of the interactions is echoed in the structure of the spin-s Compton amplitude, for which we use homogeneous symmetric polynomials of the spin variables. Where possible, we compare to the general-relativity results in the literature, available up to eighth order in spin.

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.

Norms on complex matrices induced by random vectors

AbstractWe introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.

Norms on complex matrices induced by complete homogeneous symmetric polynomials

We introduce a remarkable new family of norms on the space of $n \times n$ complex matrices. These norms arise from the combinatorial properties of symmetric functions, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. Our norms enjoy many desirable analytic and algebraic properties, such as an elegant determinantal interpretation and the ability to distinguish certain graphs that other matrix norms cannot. Furthermore, they give rise to new dimension-independent tracial inequalities. Their potential merits further investigation.

On an integral representation of the normalized trace of the k-th symmetric tensor power of matrices and some applications

Let A be an matrix and let be its k-th symmetric tensor power. We express the normalized trace of as an integral of the k-th powers of the numerical values of A over the unit sphere of with respect to the rotation-invariant probability measure. Equivalently, this expression in turn can be interpreted as an integral representation for the (normalized) complete symmetric polynomials over . As applications, we present a new proof for the MacMahon Master Theorem in enumerative combinatorics. Then, our next application deals with a generalization of the work of Cuttler et al. in [Cuttler A, Greene C, Skandera M. Inequalities for symmetric means. Eur J Comb. 2011;32(6):745–761] concerning the monotonicity of products of complete symmetric polynomials. Finally, we give a solution to an open problem that was raised by Rovena and Temereanca in [Roventa I, Temereanca LE. A note on the positivity of the even degree complete homogeneous symmetric polynomials. Mediterr J Math. 2019;16(1):1–16].

Weighted means of B-splines, positivity of divided differences, and complete homogeneous symmetric polynomials

We employ the fact that certain divided differences can be written as weighted means of B-splines and hence are positive. These divided differences include the complete homogeneous symmetric polynomials of even degree 2p, the positivity of which is a classical result by D.B. Hunter. We extend Hunter's result to complete homogeneous symmetric polynomials of fractional degree, which are defined via Jacobi's bialternant formula. We show in particular that these polynomials have positive real part for real degrees μ with |μ−2p|<1/2. We also prove results on linear combinations of the classical complete homogeneous symmetric polynomials and on linear combinations of products of such polynomials.

Polynomial solutions of algebraic difference equations and homogeneous symmetric polynomials

This article addresses the problem of computing an upper bound of the degree d of a polynomial solution P(x) of an algebraic difference equation of the form G(x)(P(x−τ1),…,P(x−τs))+G0(x)=0 when such P(x) with the coefficients in a field K of characteristic zero exists and where G is a non-linear s-variable polynomial with coefficients in K[x] and G0 is a polynomial with coefficients in K.It will be shown that if G is a quadratic polynomial with constant coefficients then one can construct a countable family of polynomials fl(u0) such that if there exists a (minimal) index l0 with fl0(u0) being a non-zero polynomial, then the degree d is one of its roots or d≤l0, or d<deg⁡(G0). Moreover, the existence of such l0 will be proven for K being the field of real numbers. These results are based on the properties of the modules generated by special families of homogeneous symmetric polynomials.A sufficient condition for the existence of a similar bound of the degree of a polynomial solution for an algebraic difference equation with G of arbitrary total degree and with variable coefficients will be proven as well.

An Interesting Family of Symmetric Polynomials

We discuss two natural extremal problems for homogeneous polynomials. These problems have simple solutions for polynomials in one or two variables but become interesting for polynomials in three or more variables. We introduce a family of homogeneous symmetric polynomials in three variables that solve one of these problems and have a number of other interesting properties. For example, their coefficients are integers that can be expressed as sums of binomial coefficients and possess a certain divisibility property. Furthermore, these polynomials are connected in a simple way to a family of polynomials arising as sharp examples in the study of proper polynomial mappings between balls in complex Euclidean space.

What Is Schur Positivity and How Common Is It?

This is a short note about Schur positivity. We introduce Schur polynomials and explain how they appear in the representation theory of the general linear group. We end with a new result of the author with F. Bergeron and V. Reiner that gives the probability that a homogeneous symmetric polynomial with positive coefficients is Schur positive.

A Note on the Positivity of the Even Degree Complete Homogeneous Symmetric Polynomials

This article deals with the positivity of a nice family of symmetric polynomials, namely complete homogeneous symmetric polynomials. We are able to give a positive answer to a question arising in Tao ( https://terrytao.wordpress.com/2017/08/06/schur-convexity-and-positive-definiteness-of-the-even-degree-complete-homogeneous-symmetric-polynomials/ , 2017). Our strategy follows two different ideas, one of them based on a Schur-convexity argument and the other one uses a method with divided differences. Several Newton’s type inequalities are also discussed.

New concavity and convexity results for symmetric polynomials and their ratios

ABSTRACT We prove ‘power’ generalizations of Marcus–Lopes style (including McLeod and Bullen) concavity inequalities for elementary symmetric polynomials, and similar generalizations to convexity inequalities of McLeod and Baston for complete homogeneous symmetric polynomials. We also present additional concavity results for elementary symmetric polynomials, of which the main result is a concavity theorem that yields a well-known log-convexity 1972 result of Muir for positive definite matrices as a corollary.

An Orthosymplectic Pieri Rule

The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundaram's Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule.

Resultant of an equivariant polynomial system with respect to the symmetric group

Given a system of n⩾2 homogeneous polynomials in n variables which is equivariant with respect to the symmetric group of n symbols, it is proved that its resultant can be decomposed into a product of several resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.

Pairs of Comparable Relations for Complete Homogeneous Symmetric Polynomial

&lt;p&gt;Complete homogeneous symmetric polynomial has connections with binomial coefficient, composition, elementary symmetric polynomial, exponential function, falling factorial, generating series, odd prime and Stirling numbers of the second kind by different summations. Surprisingly the relations in the context are comparable in pairs. &lt;/p&gt;

Enumerations for Compositions and Complete Homogeneous Symmetric Polynomial

&lt;p class="abstract"&gt;We count the number of occurrences of &lt;em&gt;t &lt;/em&gt;as the summands&lt;em&gt; &lt;/em&gt;(i) in the compositions of a positive integer &lt;em&gt;n&lt;/em&gt; into &lt;em&gt;r&lt;/em&gt; parts; and (ii) in all compositions of &lt;em&gt;n&lt;/em&gt;; and subsequently obtain other results involving compositions. The initial counting further helps to solve the enumeration problems for complete homogeneous symmetric polynomial.&lt;/p&gt;

Generic expansion of the Jastrow correlation factor in polynomials satisfying symmetry and cusp conditions.

Jastrow correlation factors play an important role in quantum Monte Carlo calculations. Together with an orbital based antisymmetric function, they allow the construction of highly accurate correlation wave functions. In this paper, a generic expansion of the Jastrow correlation function in terms of polynomials that satisfy both the electron exchange symmetry constraint and the cusp conditions is presented. In particular, an expansion of the three-body electron-electron-nucleus contribution in terms of cuspless homogeneous symmetric polynomials is proposed. The polynomials can be expressed in fairly arbitrary scaling function allowing a generic implementation of the Jastrow factor. It is demonstrated with a few examples that the new Jastrow factor achieves 85%-90% of the total correlation energy in a variational quantum Monte Carlo calculation and more than 90% of the diffusion Monte Carlo correlation energy.

Multivariate juggling probabilities

We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities. The normalization factor in one case can be explicitly written as a homogeneous symmetric polynomial. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in bounded time.

Symmetric Polynomials and Symmetric Mean Inequalities

We prove generalized arithmetic-geometric mean inequalities for quasi-means arising from symmetric polynomials. The inequalities are satisfied by all positive, homogeneous symmetric polynomials, as well as a certain family of non-homogeneous polynomials; this family allows us to prove the following combinatorial result for marked square grids.Suppose that the cells of a $n \times n$ checkerboard are each independently filled or empty, where the probability that a cell is filled depends only on its column. We prove that for any $0 \leq \ell \leq n$, the probability that each column has at most $\ell$ filled sites is less than or equal to the probability that each row has at most $\ell$ filled sites.

Transformation formulas for complete symmetric polynomials and identities for generalized binomial coefficients and [formula omitted]-Stirling numbers

We obtain explicit formulas that express the complete homogeneous symmetric polynomials of the sequence of partial sums sk of a sequence xk as polynomials in the xk, and conversely, the complete homogeneous symmetric polynomials of the xk as polynomials in the sk. Taking particular sequences xk, we obtain combinatorial identities for several types of generalized binomial coefficients, including Gaussian coefficients, Stirling numbers of the second kind, q-Stirling numbers of the second kind, and Legendre–Stirling numbers. We also introduce some new families of generalized binomial coefficients and Stirling numbers.

Cyclotomic Polynomials, Symmetric Polynomials, and a Generalization of Euler's Totient Function

SummaryWe introduce a generalization of Euler's totient function that, when applied to an integer n ≥ 2, can be written as a polynomial in the prime factors of n. We then show how cyclotomic and complete homogeneous symmetric polynomials appear as factors of these polynomials when n has at most two distinct prime divisors.