On the expression of a symmetric function in terms of the elementary symmetric functions

Definition Of

The theorem that any rational symmetric function of n variables x1, x2, … xn is expressible as a rational function of the n elementary symmetric functions, Σx1, Σx1x2, Σx1x2x3, etc., is usually proved by means of the properties of the roots of an equation. It is obvious, however, that the theorem has no necessary connection with the properties of equations; and the object of this paper is to give an elementary proof of the theorem, based solely on the definition of a symmetric function.

Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets

10.37236/1264 ◽

1995 ◽

Vol 3 (2) ◽

Cited By ~ 3

Author(s):

Richard P. Stanley

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Partially Ordered Sets ◽

Schur Function ◽

Ordered Sets ◽

Graded Poset ◽

Kostka Polynomials ◽

Relative Version ◽

Definition Of ◽

Formal Power

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power $q$, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions

Canadian Journal of Mathematics ◽

10.4153/cjm-1959-039-4 ◽

1959 ◽

Vol 11 ◽

pp. 383-396 ◽

Cited By ~ 60

Author(s):

Marvin Marcus ◽

Roger Purves

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Similarity Transformation ◽

Linear Transformations ◽

Image Position

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn let Er(A) be the rth elementary symmetric function of the eigenvalues of A. Our main result states that if 4 ≤ r ≤ n — 1 and Er(T(A)) = Er(A) for A ∈ Mn then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation:No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.

An Inequality for Elementary Symmetric Functions

Canadian Mathematical Bulletin ◽

10.4153/cmb-1972-024-4 ◽

1972 ◽

Vol 15 (1) ◽

pp. 133-135 ◽

Author(s):

K. V. Menon

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Geometric Mean ◽

Binomial Coefficient ◽

Image Position

Let Er denote the rth elementary symmetric function on α1 α2,…,αm which is defined by1E0 = 1 and Er=0(r>m).We define the rth symmetric mean by2where denote the binomial coefficient. If α1 α2,…,αm are positive reals thenwe have two well-known inequalities3and4In this paper we consider a generalization of these inequalities. The inequality (4) is known as Newton's inequality which contains the arithmetic and geometric mean inequality.

The asymptotic behavior of elementary symmetric functions on a probability distribution

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953301000193 ◽

2001 ◽

Vol 14 (3) ◽

pp. 237-248 ◽

Author(s):

V. S. Kozyakin ◽

A. V. Pokrovskii

Keyword(s):

Asymptotic Behavior ◽

Probability Distribution ◽

Symmetric Function ◽

Symmetric Functions ◽

Random Mappings

The problem on asymptotic of the value π(m,n)=m!σm(p(1,n),p(2,n),…,p(n,n)) is considered, where σm(x1,x2,…,xn) is the mth elementary symmetric function of n variables. The result is interpreted in the context of nonequiprobable random mappings theory.

On the Schur Expansion of Hall-Littlewood and Related Polynomials via Yamanouchi Words

10.37236/6732 ◽

2017 ◽

Vol 24 (1) ◽

Author(s):

Austin Roberts

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Macdonald Polynomials ◽

Macdonald Polynomial ◽

Gamma Delta ◽

Colored Graph ◽

Littlewood Polynomials ◽

Dual Equivalence ◽

Leading Term ◽

Definition Of

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram $\delta \subset {\mathbb Z} \times {\mathbb Z}$, written as $\widetilde H_{\delta}(X;q,t)$ and $\widetilde H_{\delta}(X;0,t)$, respectively. We then give an explicit Schur expansion of $\widetilde H_{\delta}(X;0,t)$ as a sum over a subset of the Yamanouchi words, as opposed to the expansion using the charge statistic given in 1978 by Lascoux and Schüztenberger. We further define the symmetric function $R_{\gamma,\delta}(X)$ as a refinement of $\widetilde H_{\delta}(X;0,t)$ and similarly describe its Schur expansion. We then analyze $R_{\gamma,\delta}(X)$ to determine the leading term of its Schur expansion. We also provide a conjecture towards the Schur expansion of $\widetilde H_{\delta}(X;q,t)$. To gain these results, we use a construction from the 2007 work of Sami Assaf to associate each Macdonald polynomial with a signed colored graph $\mathcal{H}_\delta$. In the case where a subgraph of $\mathcal{H}_\delta$ is a dual equivalence graph, we provide the Schur expansion of its associated symmetric function, yielding several corollaries.

On Determinants of Symmetric Functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500007343 ◽

1927 ◽

Vol 1 (1) ◽

pp. 55-61 ◽

Cited By ~ 4

Author(s):

A. C. Aitken

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

The Other ◽

The Difference ◽

Image Position

The result of dividing the alternant |aαbβcγ…| by the simplest alternant |a0b1c2…| (the difference-product of a, b, c, …) is known to be a symmetric function expressible in two distinct ways, (1) as a determinant having for elements the elementary symmetric functions C, of a, b, c, …, (2) as a determinant having for elements the complete homogeneous symmetric functions Hr. For exampleThe formation of the (historically earlier) H-determinant is evident. The suffixes in the first row are the indices of the alternant; those of the other rows decrease by unit steps. This result is due to Jacobi.

Algebraic Numbers as Product of Powers of Transcendental Numbers

Symmetry ◽

10.3390/sym11070887 ◽

2019 ◽

Vol 11 (7) ◽

pp. 887 ◽

Author(s):

Pavel Trojovský

Keyword(s):

Crucial Role ◽

Symmetric Functions ◽

Transcendental Number ◽

Algebraic Numbers ◽

Transcendental Numbers ◽

Definition Of

The elementary symmetric functions play a crucial role in the study of zeros of non-zero polynomials in C [ x ] , and the problem of finding zeros in Q [ x ] leads to the definition of algebraic and transcendental numbers. Recently, Marques studied the set of algebraic numbers in the form P ( T ) Q ( T ) . In this paper, we generalize this result by showing the existence of algebraic numbers which can be written in the form P 1 ( T ) Q 1 ( T ) ⋯ P n ( T ) Q n ( T ) for some transcendental number T, where P 1 , … , P n , Q 1 , … , Q n are prescribed, non-constant polynomials in Q [ x ] (under weak conditions). More generally, our result generalizes results on the arithmetic nature of z w when z and w are transcendental.

Schur and $e$-Positivity of Trees and Cut Vertices

10.37236/8930 ◽

2020 ◽

Vol 27 (1) ◽

Author(s):

Samantha Dahlberg ◽

Adrian She ◽

Stephanie Van Willigenburg

Keyword(s):

Linear Combination ◽

Connected Graph ◽

Symmetric Function ◽

Symmetric Functions ◽

Perfect Matching ◽

Schur Functions ◽

Connected Components ◽

Chromatic Symmetric Function ◽

Positive Linear Combination

We prove that the chromatic symmetric function of any $n$-vertex tree containing a vertex of degree $d\geqslant \log _2n +1$ is not $e$-positive, that is, not a positive linear combination of elementary symmetric functions. Generalizing this, we also prove that the chromatic symmetric function of any $n$-vertex connected graph containing a cut vertex whose deletion disconnects the graph into $d\geqslant\log _2n +1$ connected components is not $e$-positive. Furthermore we prove that any $n$-vertex bipartite graph, including all trees, containing a vertex of degree greater than $\lceil \frac{n}{2}\rceil$ is not Schur-positive, namely not a positive linear combination of Schur functions. In complete generality, we prove that if an $n$-vertex connected graph has no perfect matching (if $n$ is even) or no almost perfect matching (if $n$ is odd), then it is not $e$-positive. We hence deduce that many graphs containing the claw are not $e$-positive.

Lyndon Words and Transition Matrices between Elementary, Homogeneous and Monomial Symmetric Functions

10.37236/1044 ◽

2006 ◽

Vol 13 (1) ◽

Author(s):

Andrius Kulikauskas ◽

Jeffrey Remmel

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Transition Matrices ◽

Combinatorial Interpretations ◽

Lyndon Words

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

Probability and the elementary symmetric functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100047885 ◽

1973 ◽

Vol 74 (1) ◽

pp. 133-139 ◽

Author(s):

J. Denmead Smith

Keyword(s):

Symmetric Function ◽

Symmetric Functions ◽

Random Variables ◽

Independent Random Variables ◽

Congruent Modulo

Let p be a prime, and suppose that x1,…,xN are independent random variables which take the values 0, 1,…,p − 1 with probabilities s0, sl…,sp−1 where s0+…+sp−1 = 1 and 0 < sk < 1 for each k. PN(n) denotes the probability that the elementary symmetric function σr(x1,…,xN) = ∑x1…,xr of the rth degree in the variables x1,…,xN is congruent, modulo p, to a prescribed integer n.