# 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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### Continuous decomposition of permutation-invariant set functions

The seminal machine learning paper Deep Sets (Zaheer et al., 2017) discusses representations of permutation-invariant functions on real tuples, or (multi)set functions. Given a countable set $X$ and a ...
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### $2$-adic valuation of Schur $P$-functions in the power-sum basis

For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...
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### About the sum of rectangular power sums

Let $n \geq 1$ be an integer and consider the symmetric function $$D_n = \sum_{d|n} p_d^{n/d},$$ where $p_{d}$ are the power-sum symmetric functions. It can be checked up to $n=35$ that the symmetric ...
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### Multiplicities of irreducible $U(n)$-modules in the tensor product $V_{\lambda}\otimes V_{\mu}$

It is known that every irreducible representation of the unitary group $U(n)$ can be uniquely described by the non-increasing sequence $\lambda=(\lambda_1,\ldots,\lambda_n)$ of integers (denote the ...
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### Polynomial invariant relating the circumradius and sides of a cyclic polygon

This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon. This invariant is discussed briefly in the seminal paper On the Areas ...
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### A doubt in the proof of theorem regarding chromatic symmetric functions

The following theorem is due to Stanley. I can't understand, in the proof of the theorem, how he got the bottom line equation from the equation above it. Kindly explain it to me. Thank you.
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### Question regarding elementary symmetric functions

How to prove the following equalities? $$\sum_{k\ge1}\sum_{\substack{J \subseteq \mathbb N \\ J = \{i_1,\dots,i_k\}}}\big((-1)^{k-1}\sum_{j=1}^{k}x_{i_j}\big) = x_1 + x_2 + \cdots = e_1.$$ More ...
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### A conjecture about sums over partitions arising from Hilbert scheme of points

$\DeclareMathOperator{\leg}{\operatorname{leg}}\DeclareMathOperator{\arm}{\operatorname{arm}}$The following situation arose from the study of some localization computations on Hilbert schemes of ...
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### Generalization of symmetric functions

A $n$-variable function $f$ is a symmetric function if $$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$ for every permutation $\sigma \in S_n$. In particular, if $f$...
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### Derivations for symmetric functions

A symmetric function is a formal power series in infinitely many variables $x_1,x_2,\dots$ invariant under the permutation of variables (as opposed to a polynomial). Let $\Lambda$ denote the algebra ...
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### Combinatorics and geometry underlying a refined Pascal matrix/Newton identities

The partition polynomials of OEIS A263633 give the coefficients of the power series/o.g.f of the multiplicative inverse (reciprocal) of a power series/o.g.f. and so give the Newton identities for ...
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### Interaction of plethysm with other operations

The plethysm $s_{\nu}[s_{\mu}]$ of two symmetric functions is the character of the composition of Schur functors $S^{\nu}(S^{\mu}(V))$. We know that this operation is linear and multiplicative in its ...
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### Quick ways to compute transition matrices for classical symmetric function bases

I am trying to implement quick algorithms for computing the transition matrices involving the monomial, power-sum, elementary, complete homogeneous and Schur polynomials. There are several relations ...
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### On a certain expansion in term of Schur functions

This question is related to this other one A Schur positivity conjecture related to row and column permutations by Richard Stanley (thanks to Sam Hopkins for letting me know about it). Consider a ...
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### Conjugation of bosonic and fermionic

We use the notation from semi-infinite wedge formalism $\bigwedge^{\infty/2}V$ with vector space $V$ generated by $$\left\{\underline{s}\mid s \in \mathbb{Z}+\frac12\right\}$$, we consider the charge ...
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### Cut and Join for Hurwitz number with multiple spin

Let me introduce some background of cut and join equation for spin Hurwitz number with the completed cycle as mentioned in https://arxiv.org/pdf/1103.3120.pdf We fix two partition $\mu$ and $\nu$ of ...
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### Generalization of elementary symmetric polynomials

The elementary symmetric polynomials (ESPs) are defined as - \begin{align*} E_{1}^{1} &= X_1, \\ E_{1}^{2} &= X_1 + X_2, \\ E_{2}^{2} &= X_1 X_2, \\ E_{2}^{3} &= X_1 X_2 + X_1 X_3 + ...
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### On the value of a skew Schur function at the identity

(Asked in MSE but got no response) The generating function $\frac{1}{(1-t)^N}=\sum_k {N+k-1\choose k}t^k=\sum_k h_k(1)t^k$ and the Jacobi-Trudi formula $s_{\lambda/\mu}=\det(h_{\lambda_i-i-\mu_j+j})$ ...
This question might be too lightweight here but on math.SE it did not receive any feedback since May 2, so... Polarization works both ways. Not only can you represent any homogeneous polynomial $f$ of ...