# Questions tagged [schur-functions]

The schur-functions tag has no usage guidance.

76
questions

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### Identities involving Littlewood–Richardson coefficients?

I am not aware of that many identities that involve several Littlewood–Richardson coefficients.
One recent identity, is a generating function as sum of squares of LR-coefficients,
due to Harris and ...

7
votes

1
answer

266
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### Harmonic flow on the Young lattice

Let me begin with some preliminary concepts: A positive real-valued function $\varphi: P \rightarrow \Bbb{R}_{>0}$ on a locally finite, ranked poset $(P, \trianglelefteq)$ is harmonic if
$\varphi(\...

4
votes

1
answer

219
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### Nonnegativity locus of Schur polynomials

Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...

16
votes

0
answers

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### Identity involving Schur polynomials, binomial coefficients and contents of partition

Let $C_{\lambda,\mu}$ be the coefficients defined as
$$ s_\lambda\left(\frac{x_1}{1-x_1},...,\frac{x_N}{1-x_N}\right)=\sum_{\mu\supset \lambda}C_{\lambda\mu}s_\mu(x_1,...,x_N),$$
where $s$ are the ...

9
votes

1
answer

284
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### Determinant connection between Schur polynomials and power sum polynomials

Let $f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots $ be a family of symmetric polynomials. For the partition $\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$ consider the determinant
$$
D_\lambda(f)...

11
votes

1
answer

493
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### Generating function for Schur polynomials

Consider the generating function
$$
G_n(x_1,x_2,\ldots,x_n, t_1,t_2,\ldots,t_n) =\sum_{\lambda}s_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1}t_2^{\lambda_2} \cdots t_n^{\lambda_n},
$$
where the sum ...

2
votes

1
answer

118
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### How to re-expand the sum of Schur function?

Consider next sum
\begin{eqnarray}
\label{PF_spindef}
Z = \sum_{r=0}^{N N_f} h^{2r} \ Q(r) .
\end{eqnarray}
and
\begin{equation}
Q(r) \ = \ \sum_{\sigma \vdash r} s_{\sigma}(1^{N_f}) \
s_{\sigma}...

11
votes

2
answers

301
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### Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...

7
votes

2
answers

223
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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 ...

1
vote

0
answers

66
views

### LGV scheme: Any situations where the weights shift differently for each path?

In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder
In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...

4
votes

0
answers

108
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### Tensors of minimal rank in Schur modules $S_{\lambda}V \subset V^{\otimes |\lambda|}$

It is well known that for a vector space $V$ with $\dim(V)=n+1$ the $GL(V)-$module $V^{\otimes d}$ splits as a sum of irreducible representations (with suitable multiplicities) $S_{\lambda}V$, where $\...

4
votes

1
answer

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### proof of result from Ian Macdonald's paper "A New Class of Symmetric Functions"

I'm currently working my way through Ian MacDonald's somewhat seminal 1988 paper entitled "A New Class of Symmetric Functions" in Seminaire Lotharingien B20a, pp. 131–171 (EuDML). I'm fine ...

2
votes

0
answers

67
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### Double Schur function expansion

In literature, I have seen the weighted Hurwitz number $N_{g,n}(d_1 , d_2 \ldots , d_n)$ which are symmetric number and they can be written as double Schur function expansion.
\begin{align} \label{eq:...

4
votes

2
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275
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### LGV scheme for lattice paths that move in non-unit spatial positive steps

In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here).
What do we mean by "time"? In the language of ...

14
votes

2
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797
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### Do you know an elegant proof for this expression for a Schur function?

I know that the identity
$$
s_\mu = \sum_{\mu-\lambda \text{ is a horizontal strip}} \;\sum_{\alpha\vdash|\lambda|} \frac{\chi^\lambda_\alpha}{z_\alpha} \prod_i(p_i-1)^{a_i}
$$
holds.
Here $\alpha=1^{...

6
votes

0
answers

185
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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 ...

0
votes

0
answers

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### Theorem 5.3 ([Okounkov-01]) in Borodin and Gorin's lecture note

In this lecture note: https://arxiv.org/pdf/1212.3351.pdf, Theorem 5.3(P28):
Suppose that the $\lambda \in \mathbb{Y}$ is distributed according to the Schur measure $\mathbb{S}_{\rho_1; \rho_2}$. ...

1
vote

0
answers

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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 ...

1
vote

1
answer

239
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### 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 ...

2
votes

1
answer

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### 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|...

1
vote

0
answers

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### 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
$$...

2
votes

0
answers

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### 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 ...

2
votes

0
answers

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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 ...

4
votes

1
answer

181
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### 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, ...

2
votes

0
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### 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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vote

0
answers

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### 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 $\lambda =[d]$ ...

2
votes

0
answers

77
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
$$
\...

13
votes

1
answer

354
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### 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.
...

10
votes

1
answer

284
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### 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)|^...

6
votes

0
answers

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### 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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### 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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votes

0
answers

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### 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, ...

5
votes

2
answers

342
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 $\...

3
votes

0
answers

190
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 &= \...

2
votes

1
answer

879
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 ...

0
votes

0
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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 $b_n$, $...

4
votes

1
answer

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### 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{...

11
votes

1
answer

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### 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 ...

13
votes

1
answer

680
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### 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?

3
votes

0
answers

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### 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,\...

3
votes

1
answer

463
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 $s_{\lambda}(X) s_{\mu}(Y)$...

2
votes

1
answer

219
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### 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 ...

3
votes

0
answers

186
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(\...

7
votes

1
answer

199
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}\...

3
votes

0
answers

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### 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 ...

8
votes

2
answers

475
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 $\lambda =(\lambda_1,\dots,\lambda_n)$ ...

2
votes

0
answers

100
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### 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 ...

11
votes

1
answer

759
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### 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 ...

6
votes

0
answers

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### 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 ...

2
votes

0
answers

276
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### 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 ...