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 ...

**6**

votes

**0**answers

121 views

### Macdonald's “Symmetric Functions and Hall Polynomials” Section 1.5 Example 9

I'm trying to follow Example 9 in Section 1.5 of the 2nd edition of Macdonald's book "Symmetric Functions and Hall Polynomials". I have trouble with understanding some points.
Before stating my ...

**3**

votes

**0**answers

86 views

### Shifted schur function and holonomic

Now let us denote by $\Lambda^{*}(n)$ the algebra of polynomials in $x_{1},\ldots,x_{n}$ that become symmetric in new variables
$$ x_{i}'=x_{i}-i+c, \ i \in 1,\ldots,n.$$
Here c is a arbitrary fixed ...

**6**

votes

**0**answers

158 views

### A binomial determinant formula: a new variant

In a previous MO question, the OP asks a proof for $\det_{1\leq i,j\leq n}\left(\binom{i}{2j}+\binom{-i}{2j}\right)=1$. Subsequently, Gjergji Zaimi generalized the problem to
$$\det_{1\le i,j\le n}\...

**13**

votes

**0**answers

131 views

### Generalization of Newton's identities to Schur functions

In some recent work, I've stumbled across the following identity for $\lambda \vdash n$:
$$
n s_\lambda = \sum_{k=1}^n p_k \sum_{\mu \nearrow_k \lambda} (-1)^{\mathrm{ht}(\lambda/\mu)} s_\mu.
$$
Here, ...

**1**

vote

**0**answers

48 views

### 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 $\...

**1**

vote

**0**answers

41 views

### 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
...

**7**

votes

**1**answer

128 views

### 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 ...

**1**

vote

**0**answers

46 views

### 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 $g(\cdot)$. ...

**1**

vote

**1**answer

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 ...

**3**

votes

**0**answers

44 views

### Bounds related to Monomial Symmetric Functions

Two of the well studied bases for symmetric functions over $\mathbb{Q}$ are the monomial symmetric functions $\{ m_{\lambda} \}_{\lambda \text{ a partition}}$ and the power sum symmetric functions $\{ ...

**0**

votes

**2**answers

186 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....

**4**

votes

**2**answers

243 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 $1 \le m \le n$...

**3**

votes

**1**answer

238 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 $...

**2**

votes

**0**answers

43 views

### 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 ...

**7**

votes

**1**answer

325 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 $(x_1,\cdots, x_n)$ ...

**2**

votes

**0**answers

111 views

### 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 =...

**6**

votes

**0**answers

80 views

### 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 ...

**13**

votes

**1**answer

412 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 ...

**6**

votes

**0**answers

64 views

### 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 ...

**4**

votes

**1**answer

81 views

### 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 ...

**5**

votes

**0**answers

118 views

### 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 ...

**0**

votes

**4**answers

375 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 ...

**1**

vote

**1**answer

70 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\...

**5**

votes

**0**answers

62 views

### 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 ...

**39**

votes

**3**answers

1k views

### 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 ...

**0**

votes

**0**answers

92 views

### 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 $b_n$, $...

**3**

votes

**1**answer

185 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 $M_{n\times n}$...

**4**

votes

**1**answer

200 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{...

**2**

votes

**1**answer

281 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 ...

**7**

votes

**0**answers

173 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 ...

**3**

votes

**2**answers

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 ...

**14**

votes

**1**answer

287 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 $...

**6**

votes

**0**answers

88 views

### 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]...

**1**

vote

**0**answers

106 views

### 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}...

**6**

votes

**1**answer

279 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!} \...

**5**

votes

**0**answers

284 views

### 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 ...

**12**

votes

**1**answer

233 views

### 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_{\...

**1**

vote

**0**answers

83 views

### 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-...

**3**

votes

**0**answers

165 views

### 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 $\...

**14**

votes

**3**answers

412 views

### 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,...

**2**

votes

**0**answers

79 views

### 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[...

**3**

votes

**1**answer

230 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 $s_{\lambda}(X) s_{\mu}(Y)$...

**24**

votes

**5**answers

3k views

### 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 ...

**1**

vote

**1**answer

122 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 ...

**10**

votes

**1**answer

329 views

### 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 ...

**7**

votes

**1**answer

779 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 ...

**2**

votes

**0**answers

94 views

### 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 ...

**7**

votes

**1**answer

251 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_\...

**2**

votes

**1**answer

148 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{...

**3**

votes

**1**answer

221 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 ...