The symmetric-functions tag has no wiki summary.

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### Uniform generation of Symmetric Plane Partitons

In the conclusion of An Involution Principle-Free Proof of Stanley's Hook-Content Formula Krattenthaler notes that the techniques of the paper might be useful for finding bijective proofs of the ...

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### Character sums over a fixed subset of skew tableaux

Let $f(\lambda)$ count the number standard young tableaux of shape $\lambda\vdash n$ and $\lambda=(\lambda_1,\cdots,\lambda_r)$. Let $\mu \vdash k$ be a partition for $k<n$. It is a consequence of ...

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### $S_k(x+y)-S_k(x)-S_k(y)$ where $S_k$ is symmetric polynomial

Let $S_k$ be the $k$-th elementary symmetric polynomial of $n$ variables. How can I rewrite $$S_k(x+y)-S_k(x)-S_k(y)$$ by just using $x,y,S_1,S_2,\cdots S_{k-1}$ where $x=(x_1,x_2,\cdots,x_n)$ and ...

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### Dimension of the span of all partial derivatives of a given symmetric polynomial $f$ and the polynomial $E(f)$

I need some help on the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ ...

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### how should this multinomial identity be written?

question if it is correct, how is the identity tagged (4) below usually written, and can the use of conjugate partitions be avoided?
motivation i apologize for the length of this question - it's as ...

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### Other Variant of Schur Polynomials/Functions

We know that an important variant of the Schur Polynomial is the shifted Schur Polynomials that was developed by Okounkov & Olshanski.
The question here is: is there any other variant of Schur ...

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### Permutation-invariant matrix representation

The question guide says that Mathoverflow is for research level mathematics. While I do not perform research in mathematics (I study quantum chemistry), I believe this question is research-level ...

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### Generating function for $t$-residues of partitions using Heisenberg + $\hat{sl_t}$ representation theory

Recall that for $t\geq2$, a partition is a $t$-core if none of its hooklengths is divisible by $t$. It is known that the $t$-cores are parametrized by ${\mathbb Z}^{t-1}$. More precisely, let ...

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### Dynamics of RSK

There is a way of viewing the RSK correspondence as a map (in fact, bijection) $A \overset{RSK}\longrightarrow \widehat{A}$ from $n\times n$ matrices with entries $\mathbb{N}$ to (weak) reverse plane ...

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### Deforming a basis of a polynomial ring

The ring $Symm$ of symmetric functions in infinitely many variables is well-known to be a polynomial ring in the elementary symmetric functions, and has a $\mathbb Z$-basis of Schur functions ...

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### Efficiently computing (plethysm-like?)substitutions of symmetric functions

This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...

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### counting how many boxes from a given Young tableau contribute to hook length made out of two YTs

Think of a Young diagram as a collection of rows with numbers of elements
$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_d \geq \mu_{d+1}=0$ (and $\mu_k=0$ for $k>d$) and define for $s=(i,j)$ (where $i$ ...

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### Littlewood-Richardson coefficients for Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions over $\mathbb{Q}(\alpha)$.
We define a scalar product $\langle \cdot,\cdot\rangle_\alpha$ on $\Lambda$ by setting $\langle ...

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### On a positivity property of Hall-Littlewood polynomials

Here's the new, more thought through version.
Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...

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### Counting a Modified Class of Standard Young Tableau

Let $\lambda=(\lambda_1,\cdots,\lambda_n)$ be a partition, with $|\lambda|:=N$. Attach an extra box to $\lambda$ to the right end of the $r$'th row. In coordinate form, the last box on row $r$ has ...

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### computation with Hilbert scheme of $n$ points on $\mathbb C^2$ [closed]

How can we show that
$$\sum_{n = 0}^\infty q^n \operatorname{char}_T S^n(\mathbb C[x,y])=
\prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$
where $\operatorname{char}_T V$ denotes the character ...

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### A class of matrix determinants between Wronskians and Vandermondes

Update: see below
Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions ...

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### Shift-invariant symmetric functions in representation theory?

The connection between symmetric functions and representation theory is well-known.
Now consider the subspace of symmetric functions that are shift-invariant,
that is, functions satisfying ...

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### Comultiplication on Schur functors (& functions). Can it be seen from categorical perspective ?

Consider category of vector spaces. Consider functors from it to itself.
They actually form an algebra - since vector spaces can be added and tensor multiplied.
Question Is there co-product on this ...

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### Isotypic components of the action of the symmetric group on polynomials

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ decomposes as a direct sum of isotypic components for the action of the symmetric group $S_n$. The isotypic component of the trivial representation is ...

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### A particular specialization of symmetric polynomials: is it bijective?

Let $\Lambda^d_n$ the space of symmetric polynomials
in $n$ variables, with maximum 'partial degree' of each variable $d$.
A basis for this space is the set of symmetrized monomials $m_\lambda$,
where ...

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### Reference or a short argument that a certain subset generates the ring of p-typical symmetric functions under plethysm

Let p be a prime and$\Lambda_p$ be the subring of the ring of symmetric functions $\Lambda$ (over $\mathbb{Z}$) such that $$x \in \Lambda_p$$ iff there is an $i \in \mathbb{N}$ such that $p^ix \in ...

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### Anti-arithmetic product of symmetric functions: (why) is it integral?

This is an analogue of MathOverflow question #138148. Indeed it is so analogous that I wrote the following by copypasting said question and making the necessary changes.
For every commutative ring ...

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### AM-GM interpolation in the limit

Given a sequence of positive numbers $X = (x_1, x_2, \dots)$, define the $k$th elementary symmetric mean of the first $n$ entries to be
$$ S(X, n, k) := \frac{\displaystyle\sum_{1 \leq i_1 < ...

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### Is the “renormalized third comultiplication” on $\mathbf{Symm}$ integral?

Background:
For any commutative ring $R$, let $\mathbf{Symm}_R$ be the ring of symmetric functions in countably many variables $x_1$, $x_2$, $x_3$, ... over $R$. ("Symmetric functions" really means ...

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### The relation on the set of functions

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$),
let $\mathcal{F}$ be the set of ...

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### How do I find coefficients of a product expansion

Any power series $f(t) = 1 + t \mathbb{Z}[[t]]$ can be uniquely expanded in the following two ways:
$$1 + \sum_{i=1}^\infty f_i t^i =
\prod_{i=1}^\infty (1-t^i)^{-n_i}$$
Here, the $f_i$ and $n_i$ ...

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### An identity for elementary symmetric functions

Trying to understand a result in a representation theoretical paper, I realized that it implies the following elementary identity for symmetric functions. My question is whether this identity is true, ...

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### Closed-for expression for Newton-Girard symmetric polynomials with 0/1 variables

There are $n$ Bernoulli $s_i\in\left\{0,1\right\}$, $i=1,...,n$ with equal marginals $\Pr(s_i=1)=\theta$ $\forall i$ so that E$(s_i)=\theta$. Their standardized mean deviations are
\begin{equation*}
...

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### A Simple Bijective Proof Of Stanley's Hook-Content Formula for Hook Shapes

this is my first post on math overflow so I hope it goes well. I believe I have a fairly simple bijective proof for Stanley's Hook-Content Formula in the case of hook shapes. I wanted to see if ...

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### A formula on Kronecker coefficients

Accidentally, I proved the following formula for the Kronecker coefficients using some obscure method.
$$g_{lm^n,mn^l}^{m^{nl}}=1,\ \forall l,m,n\in\mathbb{N},$$
where $n^m$ is the rectangle ...

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### Littlewood-Richardson rule for Schubert polynomials

What is the current state of the problem of finding a combinatorial rule for multiplying two Schubert polynomials? Is the problem still open?

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### Why are the power symmetric functions sums of hook Schur functions only?

One interesting fact in symmetric function theory is that the power symmetric function $p_n$ can be written as an alternating sum of hook Schur functions $s_{\lambda}$:
$$
p_n = \sum_{k+\ell = n} ...

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### Power sums and Jack symmetric functions

Let $\Lambda$ be the algebra of symmetric functions in infinitely many variables over $\mathbb{C}$.
The $n$-th power sum symmetric function $p_n$ is defined (formally) as
\begin{equation}
p_n=\sum_i ...

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### Reference request: Grothendieck groups of Hecke algebras at root of unity and symmetric functions

Let $\zeta$ be an $\ell^{\text{th}}$ root of unity, and consider $H_n(\zeta)$, the (finite) Hecke algebra of type A. One can consider a dual pair of Hopf algebras arising from this data, denoted ...

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### Prove or disprove $ \int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > \int_{0}^{\infty} \int_{-\infty}^{-x} f(x)f(y)dydx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_{0}^{\infty} f(x) > 1/2$. I want to prove or disprove the following:
$$
\int_{0}^{\infty} \int_{-x}^{0} f(x)f(y)dydx > ...

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### Can you find linear recurrence relation for dimensions of invariant tensors?

Let $V$ be a finite dimensional highest weight representation of a (semi)-simple Lie algebra. For each $n\ge 0$ take $a_n$ to be the dimension of the space of invariant tensors in $\otimes^n V$.
In ...

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### A generalization of Macdonald functions?

I am interested in finding a set of functions $f(z_1,\cdots ,z_k;q,\,t)$, conjecturally polynomials, which depend on two parameters $(q,t)$ and an integer $k$, and are orthogonal under the following ...

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### Algorithm to find symmetric function given specialization

I have a symmetric function f(c1,c2,c3,c4,c5) which, when c1 < c2 < c3 < c4 < c5, has the form p1(c1)+p2(c2)+p3(c3)+p4(c4)+p5(c5), where the p_i's happen to be polynomials of degree ...

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### Roots of the derivative as symmetric functions of the roots of the polynomial

Let $p(t)=(t^2-a_1^2)\ldots(t^2-a_n^2)$ be an even polynomial with distinct real non-zero roots. Can the roots of its derivative $p'(t)$ be expressed nicely (e.g. as rational symmetric functions) in ...

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### Expressing power sum symmetric polynomials in terms of lower degree power sums

Is there an explicit formula expressing the power sum symmetric polynomials
$$p_k(x_1,\ldots,x_N)=\sum\nolimits_{i=1}^N x_i^k = x_1^k+\cdots+x_N^k$$
of degree $k$ in $N < k$ variables entirely ...

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### Constructing the sum of two pairs of symmetric polynomials

I have a question about generating a certain set of symmetric polynomials. I believe that what I'm looking for is known result, but I'm not 100% sure.
Suppose that I have two sets of indeterminates ...

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### Hopf algebra structure on the ring of quasisymmetric functions

I'm looking for a particular description of the Hopf algebra structure on the ring of quasisymmetric functions.
Let me illustrate by giving this kind of description for the Hopf algebra of symmetric ...

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### Arithmetic product of symmetric functions: why is it integral?

For every commutative ring $A$, let $\mathbf{Symm}_A$ be the ring of symmetric functions over $A$. Let $\mathbf{Symm}$ without a subscript denote $\mathbf{Symm}_{\mathbb{Z}}$.
We can define a ...

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### universality of Macdonald polynomials

I have been recently learning a lot about Macdonald polynomials, which have been shown to have probabilistic interpretations, more precisely the eigenfunctions of certain Markov chains on the ...

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### Degree principles for non-symmetric polynomials

A theorem of Timofte says that a symmetric polynomial inequality of degree $d$ holds on $\mathbb{R}^{n}_{+}$ if and only if it holds for all vectors in $\mathbb{R}^{n}_{+}$ with at most $\max\{\lfloor ...

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### Jack symmetric functions and their inner products

I have some questions regarding Jack polynomials. I use the notation of of I.G. Macdonald's book "Symmetric Functions and Hall polynomials".
Let $\Lambda$ be the ring of symmetric functions over ...

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### References request: representations of Heisenberg algebra.

Let $p_1, p_2, \ldots$, be the power sum symmetric functions. Let $p_n^* = n \frac{\partial}{\partial p_n}$. Then $$ p_n^* p_m - p_m p_n^* = \delta_{m, n} 1. $$
Where could I find this result in some ...

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### Plethysm of $\mathrm{QSym}$ into $\mathrm{QSym}$: can it be defined?

I will denote by $\Lambda$ the ring of symmetric functions, and by $\mathrm{QSym}$ the ring of quasisymmetric functions (both in infinitely many variables $x_1$, $x_2$, $x_3$, ..., both over $\mathbb ...

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### Cyclically symmetric functions

Where can I learn about the invariant theory associated with actions of cyclic groups (as opposed to symmetric groups)?
E.g., do the functions $x+y+z$, $xy+yz+zx$, and $x^2y+y^2z+z^2x$ generate the ...