0
votes
0answers
17 views

Properties of the 'forgotten' symmetric polynomials

In I.G. Mcdonald "Symmetric Functions and Hall Polynomials" pg.22, the forgotten symmetric functions 'f' are introduced very briefly as the result of applying an involution w to the monomial symmetric ...
0
votes
0answers
25 views

Looking for general proof of a sum of an additive form of elementary symmetric polynomials

For sake of avoiding complicated general formulation I try to formulate in the special case of a set of 3 numbers $M=\{a_1,a_2,a_3\}$ with e.g. $a_i\in\mathbb R$. The sum I am looking for is in this ...
1
vote
1answer
46 views

How many terms are in $\sum \alpha_1^{a_1}\alpha_2^{a_2}\cdots \alpha_r^{a_r}\alpha_{r+1}\alpha_{r+2}\cdots \alpha_s$

Suppose that $\alpha_1, \cdots, \alpha_n$ be $n$ roots of the polynomial equation $p(x)=0$ of degree $n$. I was studying on symmetric polynomial and have come accross of several problems on like $\sum ...
1
vote
0answers
24 views

Necessary condition for symmetric sums

It is easy to see that if a function $f(x_1,x_2,x_3)$ can be written in the form: $$ f(x_1,x_2,x_3) = g(x_1,x_2) - g(x_1,x_3) + g(x_2,x_3) $$ for some function $g$, then we have: $$ f(x_1,x_2,x_3) - ...
4
votes
0answers
68 views

Combinatorics and symmetric functions

(The actual questions in this posting are at the bottom.) Occasionally someone asks here how to show that every nonempty finite set has just as many subsets of odd cardinality as of even cardinality ...
1
vote
1answer
78 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
6
votes
0answers
101 views

Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} ...
4
votes
0answers
125 views

The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
8
votes
0answers
200 views

Symmetric polynomials

I've got a seemingly simple question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots ...
1
vote
1answer
171 views

Decomposition of products of monomial symmetric polynomials into sums of them

I'm trying to make sense of the answer given in: this question I am stuck at the phrase 'where the partitions $\gamma$ result from adding, respectively, from $\alpha$ all distinct partitions obtained ...
5
votes
2answers
196 views

Derivative of Schur function

In his answer to http://mathoverflow.net/questions/129854, R. Stanley says that the partial derivative (over the relevant x[i]) of the Schur function of a partition lambda of n equals the sum the ...
3
votes
1answer
45 views

How to compute the weights of $\Gamma_{3,1}$ the irrep of $\mathfrak{sl}_3\Bbb C$

I am wondering about a combinatorial formula for computing the weights of the irreducible representations $\Gamma_{a,b}$ of $\mathfrak{sl}_3\Bbb C$. By $\Gamma_{a,b}$ I mean the irrep that has highest ...
2
votes
1answer
64 views

What is the degree of the fourier expansion

Let $ f:\{-1,1\}^3 \rightarrow \{-1,1\} $ , $f(x)= \operatorname{sgn}(x_1+x_2+x_3)$; (Majority function), then Fourier expansion of $f$ is $f(x)= \frac{1}{2} ...
7
votes
2answers
247 views

Can $e_n$ always be written as a linear combination of $n$-th powers of linear polynomials?

User Eric Gregor and I were talking in chat and he mentioned this question and postulated the possibility of an approach through symmetric polynomials. After some thinking, I came to this: ...
9
votes
1answer
215 views

What are the analogues of Littlewood-Richardson coefficients for monomial symmetric polynomials?

The product of monomial symmetric polynomials can be expressed as $m_{\lambda} m_{\mu} = \Sigma c_{\lambda\mu}^{\nu}m_{\nu}$ for some constants $c_{\lambda\mu}^{\nu}$. In the case of Schur ...
3
votes
3answers
212 views

Do these special power functions generate all homogeneous symmetric polynomials?

Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree. My question is as follows. To be succinct, let's say we have four ...
2
votes
1answer
164 views

An identity on symmetric polynomial

In the polynomial algebra $\mathbb{C}[X_1, X_2,\ldots, X_n]$, we define a set of symmetric polynomials as follows $h_i(X_k, X_{k+1}, \ldots, X_n)$ = sum of all monomials of total degree $i$ in the set ...
6
votes
2answers
286 views

Number of terms in a monomial symmetric polynomial

Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many ...