# Questions tagged [symmetric-functions]

Symmetric functions are symmetric polynomials, in finitely many, or countably infinitely many variables. They arise in the representation theory of symmetric groups and in the polynomial representation theory of general linear groups. Bases of the ring of symmetric functions are indexed by integer partitions. Schur functions, elementary symmetric functions, complete symmetric functions, and power sum symmetric functions are the most commonly used bases.

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### A particular family of symmetric functions (sums of powers of sums of subsets)

For any $m,k$ define $$f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k.$$ Do these symmetric polynomials have a name and any theory?
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### Combinatorial reciprocity for symmetric functions

I am wondering whether a certain instance of combinatorial reciprocity (in the sense of Stanley's classic paper "Combinatorial Reciprocity Theorems"), concerning symmetric functions, is ...
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### Relationship between Jack polynomials and root systems

I don't know much about algebraic combinatorics; however, I came across the Wikipedia page for Jack polynomials and read that they form an orthogonal basis in a space of symmetric polynomials, with ...
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### Identities involving Littlewood–Richardson coefficients?

I am not aware of that many identities that involve several Littlewood–Richardson coefficients. One recent identity, is a generating function as sum of squares of LR-coefficients, due to Harris and ...
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### Tanglegrams and functional equations of M. Somos

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the ...
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### Proving an optimization problem from continuous input to binary is optimal

Suppose we have a function $f(x,y,z)$ where the inputs are uniform from 0 to 1. The output is either $+1$ or $-1$. And there is a partial symmetry $f(x,y,z) = f(z,y,x)$. Tell me what the minimum of ...
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### Polynomial invariant — from product formula to monomial expansion

Context This question deals with the polynomial invariant denoted by $H_{n}$ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....
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### Expanding the zonal polynomial $Z_\lambda(x/(1-x))$

Schur polynomials $s_\lambda(x)$ have a determinantal expression. Using that, I know how to write the polynomial $s_\lambda(\frac{x}{1-x})=s_\lambda(\frac{x_1}{1-x_1},\frac{x_2}{1-x_2},...)$ as an ...
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### Maximally independent polynomial families with row symmetry

Introduction: In the 1-dimensional case, given $m$-variables $$\mathbf{x} = (x_1,x_2,\dots,x_m)^T,$$ the elementary symmetric polynomials $(e_k(\mathbf{x}))_{k=1}^m$ give a "symmetric basis",...
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### Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
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### proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"

I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...
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### Typo in Stanley, Enumerative combinatorics II, Cor. 7.23.9?

In Stanley, EC2, we have the following statement: I think there is a typo in the first sum after "generating function", and that $[n]_q!$ should be replaced by $(1-q)(1-q^2)\dotsb (1-q^n)$, ...
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### Are top Brauer characters bounded?

Let $p_\lambda$ be power sum symmetric functions. Let $s_\lambda$ and $o_\lambda$ be irreducible characters of the unitary and orthogonal groups $U(N)$ and $O(N)$, respectively (the $s$ are the Schur ...
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### "Kronecker Product" for quasi-symmetric functions

Recall that the Kronecker product $s_\lambda * s_\mu$ of two Schur functions $s_\lambda$ and $s_\mu$ is the symmetric function whose expansion (in terms of Schur functions) is given by \begin{equation}...
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Let $T_k(x_1,\ldots,x_n)$ be the Todd polynomials, $e_k(x_1,\ldots,x_n)$ the elementary symmetric polynomials and $p_k(x_1,\ldots, x_n)$ the power sums of degree $k$. We have the following generating ...