# Questions tagged [schur-functions]

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59
questions

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### What is an orthogonal form?

I am reading the article: Algebro - Geometric applications of Schur S- and Q-polynomials
On page 179, they said about an orthogonal form $\psi$ on $W$. There is no definition of an orthogonal form on ...

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209 views

### Schur polynomials with zeros in an infinite geometric progression

Let $a_1,...,a_n\in \overline{\mathbb{Q}}^\times$ be algebraic numbers, such that $\frac{a_i}{a_j}$ is not a root of unity for $i\neq j$. Furthermore, let $m\in\mathbb{N}$, $m>1$.
Question: Is ...

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138 views

### Finding Littlewood-Richardson coefficients without using identities

The Littlewood-Richardson coefficients $C^{R}_{QP}$ for some partitions $R, Q, P$ can usually be dealt with using identities like for example $$C^{R}_{QP} = 0 \quad \text{ if } \quad|R| \neq |Q| + |P|...

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85 views

### Calculation of complete homogeneous symmetric functions [closed]

Say you have a complete homogeneous symmetric function $$h_4 = \sum_{1\leq i \leq j \leq k \leq l}q^{-i}q^{-j}q^{-k}q^{-l},$$ where $i = 1, 2, 3, \ldots$. There are 7 cases to consider, given by
$$...

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106 views

### Have these polynomials been studied? (Perhaps as generalizations of Schur polynomials in vector variables?)

For $n\geq 3$, let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb{Z}^2$ be a collection of points in the plane with integer coordinates $\mathbf{a}_i = (a_{1i},a_{2i})$ where each $a_{1i} > 0$. For ...

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59 views

### Restricted Cauchy identity

Is there some reference for sums like:
$$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)t^{|\nu|}$$
$$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)\cdot|\nu|$$
(summation ...

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133 views

### Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...

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56 views

### Annihilator of the of the generating function not holonomic

The following is a generating function in $x,h$ with infinite parameters
$q_1,q_2\ldots,$ and $w_1, w_2,\ldots$.
$$\Psi(x, h)= \sum_{d=0}^{\infty} s_{(d)} (q_1, q_2, \ldots) \exp \bigg( \sum_{r=1}^{\...

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23 views

### Extension of definition of Holonomic closure

My question is about finding the annihilator of a series. Let me begin with what is known and then ask my question. Let $s_d(\frac{q_1}{h},\ldots )$ denote schur function for partition $\lambda =[d]$ ...

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71 views

### Schur function on unit circles

Define $T^d$ as following
$$ T^d = \left\{(t_1,\cdots,t_d)\in\mathbb{C}^{d}\mid |t_i|= 1 \mbox{ for all } i\right\}
$$
For any partition $\lambda\vdash n$,The Schur function is defined
$$
\...

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300 views

### Is there a Giambelli identity with dual representations?

For natural numbers $a,b$ with $b\leq n-1$, let $V_{ (a|b)}$ be the irreducible representation of $GL_n$ with highest weight vector $(a+1, 1^b, 0^{n-b-1})$ where the exponentiation denotes repetition. ...

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239 views

### Integral of product of Schur functions

Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae
$$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^...

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143 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 ...

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99 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 ...

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172 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, ...

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144 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 $\...

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115 views

### Evaluating derivatives of Schur polynomials

Given an arbitrary partition $\lambda$ and an integer $N$ (the number of variables), is there any further way to evaluate the following derivative of the Schur polynomial?
\begin{align}
A &= \...

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375 views

### An upper bound for a vector with given norm 1 and norm 2

Suppose $X = (x_1, \ldots , x_n)$ is given and we know that $x_i$'s are nonnegative, $\sum_{i=1}^n x_i = n$ and $\sum_{i=1}^n x_i^2 = m $. Just by this information, is it possible to find a vector ...

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107 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 $b_n$, $...

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238 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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215 views

### Is there a geometric interpretation of skew Schur functions?

Consider the cohomology ring of the Grassmannian of k-planes in complex n-space. It has a standard presentation as a quotient of the ring of symmetric functions. In this presentation, the Schur ...

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475 views

### Why do we care about Schur Positivity

Some of the most important open problems in Algebraic Combinatorics concern the Schur positivity of classes of symmetric functions. Why is this an important property to have?

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208 views

### Dimension of roots of irreducible Schur polynomial on unit circle

Let $s_\lambda(x_1,\ldots,x_n)$ be a Schur polynomial in $\mathbb{C}[x_1,\ldots,x_n]$ with $\lambda=(\lambda_1\geq\lambda_2\geq\ldots\geq\lambda_n=0)$ and $\gcd(\lambda_1+n-1,\lambda_2+n-2,\dots,\...

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320 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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157 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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147 views

### Shifted Schur functions

Let's fix the ground field $\mathbb{C}$.
In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 Andrei Okounkov and Grigori Olshanski introduce a special basis for the center $Z(\...

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163 views

### Sum of the ratios of Schur functions

There are simple expressions for the sums of linear and quadratic combinations of Schur functions over all partitions (including the empty one)
$$
\sum_\lambda s_\lambda(x)=\prod_{i}\frac{1}{1-x_i}\...

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353 views

### Proofs that the Plücker relations generate the ideal of the Grassmannian

Some context: The $(k,n)$-Grassmannian is the set of $k$-dimensional subspaces of an $n$-dimensional vector space $V$. It can be realized as a projective variety via the Plücker embedding, and the set ...

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324 views

### An identity related to partitions into $n$ parts and Schur polynomials

While working with Schur polynomials I found what seems like a nice identity, and I wonder if it has a simple proof.
Notation: Suppose $d,n\in\mathbb{N}$, and $\lambda =(\lambda_1,\dots,\lambda_n)$ ...

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100 views

### Bounding Schur polynomials of a particular shape

Consider Schur polynomials $s_\lambda$ with $\lambda = (2m, m, m, \ldots, m, 0)$ and $\ell(\lambda) = n$ (that is, $\lambda$ has $n$ rows). Here $m \gg n$, which, for the sake of concreteness, let's ...

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678 views

### Plugging $1-x$ into Schur polynomials

I have a symmetric Laurent polynomial $f$ in $k$ variables expressed as a linear combination of Schur polynomials. I'd like to know what happens when I make the substitution $p(x_1,\ldots,x_k)\mapsto ...

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208 views

### a variational problem related to weighted logarithmic capacity

Consider the following multiple contour integral:
$$ \Phi_\lambda := \oint \ldots \oint \prod_{1 \le j < k \le n} (z_j^{-1} - z_k^{-1}) \prod_{j=1}^n \prod_{k=1}^n (1 - z_j x_k)^{-1} \prod_{j=1}^n ...

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230 views

### what is the link between plethysm in regular representation of the symmetric group and plethysm in Schur functions.

I am trying to understand first how one can define the plethysm say $s_\lambda \circ s_\mu$ as a module in the regular representation of the symmetric group.
1)How is it connected to the plethysms ...

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199 views

### Reference Request: Irreducibles of the regular representation of the permutation group is absolutely irreducible

I am writing a paper(physics) where I am using the fact that the irreducible's of the regular representations of the permutation group are absolutely irreducible in the following sense.
If $V$ is an ...

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352 views

### Schur polynomial, change of variable

Let $k=(k_1,k_2,k_3,k_4)\in \mathbb{N}^4$ and let $s_k(x_1,x_2,x_3,x_4)$ be the Schur polynomial on $GL_4$.
Question 1: If I replace $x_3$ with $x_1$ and $x_4$ with $x_2$, can $s_k(x_1,x_2,x_1,x_2)$ ...

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842 views

### Bounding Schur symmetric polynomials on the unit circle

Recall the Schur polynomial in $n$ variables, indexed by the partition $\lambda$, with $\ell(\lambda) \leq n$, is given by
\begin{equation}
s_\lambda(x_1,\ldots, x_n) = a_{\lambda + \delta}(x_1, \...

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268 views

### Dickson/determinant type polynomial (updated)

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation}
...

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316 views

### Most computationally efficient Littlewood-Richardson rule

There are many, many different versions of the Littlewood-Richardson rule: the original characterization via Yamanouchi words, Remmel's version, a description via the Poirier-Reutenauer bialgebra, the ...

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321 views

### 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 $\{S_\...

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338 views

### Staircase Schur functions squared

Let $\Delta_n$ be the staircase-shaped partition $(n-1,n-2,\dots,1)$. Are there any non-obvious combinatorial objects that index $s_{\Delta_n}^2$? Here, $s_\lambda$ is the Schur function indexed by ...

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203 views

### Explicit basis/weight vectors for irreducibles inside the plethysm $Sym^m(\bigwedge^p \mathbf(V))$

This is a follow up to this question about finding the multiplicities of irreducible representations restricted to Young diagrams of 2-columns or less, inside the plethysm $Sym^m(\bigwedge^p \mathbf(V)...

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944 views

### 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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369 views

### 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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186 views

### Optimization problem involving Multivariate Normal

I use $\phi(t)$ to describe the standard normal distribution density and $\Phi(t)$ as the normal distribution CDF and would like to prove that for all
$n\geq3$, the function:
$$h(\mu_{1},\ldots,\...

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229 views

### Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as
\begin{equation}
p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~,
\end{...

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1k views

### Every antisymmetric (alternating) polynomial is divisible by Vandermonde product

I am looking for a proof of the statement:
Every antisymmetric (alternating) polynomial is divisible by Vandermonde product, yielding a symmetric polynomial in the result.
The statement is really ...

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**1**answer

581 views

### principal specialization of projective Schur functions

Is there a nice/known formula for the (some) principal specialization of a projective (P or Q) Schur polynomial? That is, I am talking about a projective analogue of Stanley's hook content formula $$...

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1k views

### Generalization of Cauchy's identity

Let $ s_{\lambda} $ be the Schur function associated to the partition $ \lambda $.
Cauchy's identity (as in Macdonald) states that
$$
\sum_{\lambda} s_{\lambda}(X)s_{\lambda}(Y) = \prod_{i,j}(1-...

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351 views

### Schur polynomials in the Chern classes as direct images

Let $E\to X$ be a rank $r$ holomorphic vector bundle on a $n$-dimensional compact complex manifold. Then, it is well known that one can recover the Segre classes of $E$ as follows.
Let $\pi\colon P(E)\...

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578 views

### Schur functors generalization to “Jack”, “Hall-Littlewood”, “Macdonald” functors ?

Schur functors are functors from the category of vector spaces to itself.
If we take an operator $M: V->V$ and apply a Schur functor to it and then calculate trace $Tr(M^{\Lambda})$ we will get ...