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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 ...

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Frobenius coordinate expansion of character

Let $\lambda$ be the partition of integer $d$. The Frobenius coordinate of $\lambda$ is given $$ (a_1 ,\ldots,a_{d(\lambda)}|b_1,\ldots,b_{d(\lambda)}),$$ where $d(\lambda)$ denote the diagonal of $\...
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Diagonal operator and infinite wedge space formalism

Let $\bigwedge^{\infty /2}V$ denote semiinfinte wedge space. The followin article section 2 gives a good description about the space and the operator on it. https://arxiv.org/pdf/math/0207233.pdf ...
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Reference for Kakutani result on power sum bases of symmetric functions

Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
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Necessary conditions for a function to be represented as symmetric tensor product

Let $f(\cdot,\cdot)$ be any function of 2 arguments. Suppose also that the following equation holds $$f(y,x)+f(x,y)=g(x) h(y) +g(y)h(x) \quad \forall x, y$$ for arbitrary $h(\cdot)$ and $g(\cdot)$. ...
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1answer
28 views

Does stability of equilibrium point preserved by permutation matrix (symmetry)?

Given the following differential equations: \begin{equation} \begin{aligned} \dot{x}_1 &= f_1(x_1,\ldots,x_n) \\ \vdots \\ \dot{x}_n &= f_n(x_1,\ldots,x_n) \end{aligned} \end{equation} In ...
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171 views

A symmetric polynomial inequality

I improve my previous question. Because this conjecture is exactly natural development of A Muirhead Like Inequality and Muirhead's Inequality so I think the conjecture is true. But I can not prove it....
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235 views

Combination power elementary symmetric polynomial inequality

Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows: Inequality 1: Let $n>2$ and $1 \le m \le n$...
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234 views

A generalization of Newton-Girard Identities

Let $x_1, ..., x_n$ be formal variables. One variant of the Newton-Girard identities expresses $$\sum_{\pi \in S_n} x_{\pi(1)} x_{\pi(2)} \cdots x_{\pi(k)}$$ as a polynomial in the power sums of the $...
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Quasi-symmetric functions and determinants

In the field of symmetric functions, determinants show up all the time. For example, Jacobi-Trudi, Giambelli identity, definition of Schur polynomials as a quotient of determinants, and so on. This ...
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315 views

A Muirhead Like Inequality

I am looking for a proof of the inequality as follow: Let $n$ be an integer number $n \ge 2$ and $x_1, \cdots, x_n$ and $y_1,\cdots, y_n$ are nonegative real numbers such that $(x_1,\cdots, x_n)$ ...
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An inequality related to Power sum and elementary symmetric polynomial and majorizes

Power sum and elementary symmetric polynomial Let $x_1,. . . , x_n$ be variables, denote for $k \ge 1$ by $p_k(x_1,\dots,x_n)$ the $k-th$ power sum: $$ p_k(x_1,\dots,x_n)=\sum\nolimits_{i=1}^nx_i^k =...
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Decomposition of even symmetric polynomials and Euler numbers

Let's denote the even part of a polynomial $p$ by $E[p]$, which means only taking into account the monomials in $p$ which are even in all the arguments. Now let's consider the even part of the ...
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366 views

Character theoretic proof of the Littlewood–Richardson rule?

The Littlewood–Richardson coefficients are the multiplicities $$ c(\lambda,\mu,\nu)= \dim_{\mathbb{C}}\operatorname{Hom}_{S_n}(S(\nu),S(\lambda/\mu)) $$ and the Littlewood–Richardson rule says that ...
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Expansion of elementary symmetric function in Jack's?

Consider the expansion $$ e_\mu(x) = \sum_\lambda c_{\mu\lambda}(\alpha) J_\lambda^{(\alpha)}(x) $$ where $J_\lambda^{(\alpha)}(x)$ are the integral-form Jack polynomials (the ones with $n!$ as ...
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Quasi-symmetric generalizations of classical symmetric functions

I am looking for quasi-symmetric versions of the classical $e_\lambda$ and $h_\lambda$ (the elementary and complete homogeneous symmetric functions). Is there some reference for this? I am aware of ...
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Explicit form of raising and lowering operators in spherical gl(n) DAHA

I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$. Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...
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356 views

Show that sets are equal

Let $X=\{x_1,x_2,...,x_n\}$ and $Y=\{y_1,y_2,...,y_n\}$ be sets over a finite field $F$ with $p=char(F)>2$. Assume $$x_1^k+x_2^k+...+x_n^k=y_1^k+y_2^k+...+y_n^k,\ 1\leq k\leq n$$ I wanna ...
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67 views

On two types of shifted symmetric power sums

In the ring of shifted symmetric functions $\Lambda^*$ there are many ways to generalize the symmetric power sums. First of all, we have the functions $$p^*_k=\sum_{i=1} \left((x_i-i+1/2)^k-(-i+1/2)^k\...
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Interpretation of Hilbert/Frobenius series shift

Let $V = \oplus_{i\geq 0} V^i$ be a graded vector space. Recall that the Hilber series is defined as $$F(q) = \sum_{i\geq 0} q^i dim(V^i),$$ or if we have a graded $S_n$-module, $M$, we can ...
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Characterizing positivity of formal group laws

The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
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Methods to get Holonomic functions

Let $a_n$ be a holonomic sequence. By definition, that means there exists a linear differential equation of finite order which annihilates $F(x)$, where $F(x):=\sum a_n x^n$. Similarly let $b_n$, $...
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170 views

Symmetric functions of eigenvalues

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}$ be a smooth function, which is symmetric under the action of the symmetric group (acting on $\mathbb{R}^n$ by permuting the variables). Let $M_{n\times n}$...
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197 views

Decompostion of hook schur function in terms of cauchy product of holonomic functions

Let $s_{\lambda}$ denote the schur function and $\lambda$ is the partion of an integer. The schur function written in power sum symmetric basis apper as following. $\chi$ denote the character. \begin{...
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1answer
278 views

Generalized Newton Identities

I learnt a lot of new words (Hall-Littlewood, Jack and Macdonald polynomials) but unfortunately everything I dug up is written without a single example and I still don't know the answer to a very ...
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169 views

Characterizing $n$-exceptions of the ring of symmetric polynomials

(Also in Mathematics Stack Exchange: https://math.stackexchange.com/questions/2528000/characterizing-n-exceptions-on-the-ring-of-symmetric-polynomials) We say that an homogeneous symmetric polynomial ...
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281 views

Bounded Degree in Ring of Symmetric Functions

The Ring of Symmetric Functions over a commutative ring $R$, $\Lambda$, is the subring of the ring of formal power series $R[[x_1, x_2, \dots]]$ such that $f \in \Lambda$ if $f$ is invariant under ...
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1answer
278 views

Inequalities on elementary symmetric polynomials

I have recently come across the following result. Let $0 < d \leq n$. Given any vector $x \in \mathbb{R}^n$ that satisfies $e_{d-1}(x) = 0$, show that $$|x_1 \cdots x_d| \leq |e_d(x)|$$ where $...
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Natural maps between Schur functors: understanding the image

Let $V$ be a finite dimensional representation of symmetric group $\mathbb{S}_n.$ Consider a natural map $$\pi \colon \Lambda^2 V \otimes \Lambda^2 V \longrightarrow \Lambda^4 V.$$ Let $[\Lambda^2 V]...
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Holonomic generating function

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define \begin{align} B(d)&= \frac{1}{d!h^{d-1}...
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277 views

Rational generating function and recursion

Let $\lambda$ denote a hook of size $d$ and $c(\Box)$ the content of $ \Box \in \lambda $. Let $ \text{Hooks}(d) $ be the set of hooks with $d$ boxes. Define \begin{align} B(d)&= \frac{1}{d!} \...
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Internal tensor product of strict polynomial functors: is there a more explicit definition?

In the paper Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors, arXiv:1203.0311v4, Krause defines something that he calls an "internal tensor product" on the category of ...
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What is currently known or conjectured about q,t-Kostka polynomials?

The $q,t$-Kostka polynomials $K_{\lambda,\mu}(q,t)$ appear as the change of basis coefficients between Macdonald polynomials $H_\mu(x;q,t)$ and Schur functions $s_\lambda(x)$: $$H_\mu(x;q,t)=\sum_{\...
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Hook-content polynomial 2

Recently I have proven the following identity \begin{align} \sum_{\lambda\in \text{different hook of size d}} \frac{1}{d!} (-1)^{ht(\lambda)-1} \, \dim \lambda \, \prod_{\Box \in \lambda} \frac{1}{1-...
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Hook-content polynomial

Let $\lambda$ denote the hook of size $d$ for a fixed integer $d>0$. For $d=2$ there are two kinds of hooks for $d=3$ there are 3 different kind and so on. $c(\Box)$ denote the content of the $\...
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Schubert calculus expressed in terms of the cotangent space of the Grassmannians

Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,...
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Partial Flag Varieties and Quotients of Symmetric Polynomials

$\def\Q{\mathbf Q}\DeclareMathOperator{\Gr}{Gr}$First, consider a Grassmannian $\Gr(k, N)$ of $k$-dimensional subspaces in an $N$-dimensional space. It is known that its cohomology ring is $$H_k=\Q[...
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206 views

Product of Schur functions

Given two sets of variables $X=\{x_1,\cdots,x_n\}$, $Y=\{y_1,\cdots,y_m\}$, and two partitions $\lambda$ and $\mu$. Is there a formula for the product of the Schur functions $s_{\lambda}(X) s_{\mu}(Y)$...
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Understanding a quip from Gian-Carlo Rota

In the chapter "A Mathematician's Gossip" of his renowned Indiscrete Thoughts, Rota launches into a diatribe concerning the "replete injustice" of misplaced credit and "forgetful hero-worshiping" of ...
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1answer
120 views

Cauchy identity, with sum restricted over partitions with first part $\leq n$

The Cauchy Identity $$ \sum_{\nu}s_{\nu}(x)s_{\nu}(y) = \prod_{j,k=1}^{\infty}\frac{1}{1-x_{j}y_{k}} $$ expresses the sum over all integer partitions of the product of pairs of Schur polynomials as ...
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Cauchy identity in three sets of variables?

The Cauchy identity states that $$ \prod_{i,j} \frac{1}{1-x_i y_j} = \sum_\lambda s_\lambda(x) s_\lambda(y), $$ where $s_\lambda(x)$ is the Schur function. Is there a known decomposition of the ...
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759 views

A combinatorial expression of Hall-Littlewood polynomials

This is related to the question Hall-Littlewood functions and functions on the nilpotent cone, and arises in the construction of Coulomb branches of gauge theories. The motivation is explained at the ...
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Macdonald polynomials: existence and specializations

I am reading Macdonald's Symmetric Functions and Orthogonal Polynomials lecture notes, and have several related questions: In both the type A case (chapter 1) and the general irreducible root system ...
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1answer
240 views

Details about plethysm

I'm currently working on plethysm, i.e. the character of the composition $S^\lambda(S^\mu(V))$ of the Schur functors $S^\lambda$ and $S^\mu$ on a complex vector space $V$. We note this character $s_\...
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1answer
147 views

special values of symmetric functions at powers of $\frac1j$

Let $e_n(x_1,x_2,x_3,\dots)$ denote the $n$-th elementary symmetric function in the infinite variables $x_1,x_2,x_3,\dots$. Let $u$ and $v$ be the roots of $z^2-6z+1=0$. Question. Let $x_j=\frac1{...
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1answer
212 views

Singular locus of zero set of elementary symmetric polynomial

Let $\sigma_{m, r}$ be the degree-$r$ elementary symmetric polynomial in $m$ variables. Let $X_{m, r}$ be the zero set of $\sigma_{m, r}$ and $S_{m, r}$ its singular locus. I.e., $S_{m,r}$ is the set ...
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Lagrange interpolation vs homogeneous symmetric polynomials?

This question is a follow-up on another MO query here. Question. For $r\geq$ an integer, is it true that there exists homogeneous symmetric polynomial $P_r(x_1,\dots,x_n)$ with positive ...
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Only finitely many values of the symmetric functions of $1/1,1/2,\ldots,1/n$ are $2$-adic integers (?)

For integers $n \geq k \geq 1$ let $$H(n,k) := \sum_{1 \leq i_1 < \cdots < i_k \leq n} \frac1{i_1 \cdots i_k}$$ be the $k$-th elementary symmetric function of $\tfrac1{1},\tfrac1{2}, \ldots, \...
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Inequality with symmetric polynomials [closed]

How is proving this inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$? This is intuitively right, but I could not to solve it.
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On finding simpler symmetries to differential equations

I have developed a differential equation for the variation of a star's semi-major axis with respect to its eccentricity. It is as follows: $$\frac{dy}{dx}=\frac{12}{19}\frac{y\left(1+\left(\frac{73}{...
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Clustering Properties of Jack Polynomials at negative rationals

I'm trying to confirm or deny the truthfulness of a statement. Also I'm asking about a reference for the proof of the statement if it is true. Following Feigin et al, we say a partition $\lambda$ is $...