# Questions tagged [symmetric-functions]

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 partitions. Schur functions, elementary symmetric functions, complete symmetric functions, and power sum symmetric functions are the most commonly used bases.

324
questions

0
votes

0
answers

64
views

### Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...

0
votes

1
answer

193
views

### Unpacking the plethystic substitution $h_n[n\mathbf{z}]$ in a paper by Aval, Bergeron and Garsia

I'm not familiar with the formalism of plethysm, so I need some help in unpacking the plethystic substitution $h_n[n\mathbf{z}]$ found in eqns. 5.6 and 5.9 of "Combinatorics of labelled ...

3
votes

1
answer

186
views

### Tangent space of a GIT quotient of $GL_{N}$

Let $G:=\operatorname{GL}_{N}$ act on its Lie algebra $\mathfrak{g}:=\mathfrak{gl}_{N}$ by conjugation. Then it acts naturally on the associated ring $\mathcal{O}(\mathfrak{g})$ of (algebraic or ...

3
votes

0
answers

91
views

### Plethysm and wreath product

I am looking for a proof about the link between plethysm and wreath product. It is a well-known fact, being use extensively in many papers, but I can't find a good reference. Everything that follows ...

1
vote

0
answers

53
views

### Factorization of the symmetric function identity $E(t)=1/H(t)$ with the refined Euler characteristic polynomials of associahedra / Lagrange inversion

I've come across two matrix identities, flagged with daggers below, relating the two sets of elementary and complete homogeneous symmetric polynomials/functions via the two sets of refined Lah and ...

14
votes

1
answer

609
views

### Is this generalized version of plethysm Schur positive?

Question: Suppose that $f(x_1, x_2, \dots x_n)$ is a polynomial with nonnegative integer coefficients. For each permutation $\sigma\in S_n$, let $f_{\sigma}$ denote $f(x_{\sigma(1)}, \dots, x_{\sigma(...

1
vote

0
answers

64
views

### Hall-Littlewood polynomials with sage

I need to make some computations with Sage involving Hall-Littlewood polynomials. I couldn't find any satisfying information in the Sage manuals/tutorials that I found on the internet. What I found is ...

13
votes

1
answer

553
views

### A symmetric function related to sums of square roots

Let $x_1,x_2,\dots,x_n$ be indeterminates (say over $\mathbb{Q}$). For
every sequence $\epsilon=(\epsilon_1, \dots,\epsilon_n)\in\{-1,1\}^n$
define $$ y_\epsilon = \sum_i \epsilon_i \sqrt{x_i}. $$ Let ...

11
votes

1
answer

457
views

### Is the appearance of Schur functions a coincidence?

The Schur functions are symmetric functions which appear in several different contexts:
The characters of the irreducible representations for the symmetric group (under the characteristic isometry).
...

7
votes

0
answers

116
views

### The quotient of a higher Specht polynomial over the corresponding regular Specht polynomial

I'll need some notation before I can phrase my question, so please bare with me for a little. I'll try to get there as fast as possible (it's also my first MO question...).
Let $\lambda$ be a ...

4
votes

0
answers

96
views

### Representation-theoretic interpretation of double Schur polynomials

The Schur polynomials
$$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$
naturally appear as polynomial representatives for Schubert classes in ...

0
votes

0
answers

70
views

### Gessel-Viennot theorem

In the paper, page 76, why we need the condition that the subpath lying between lines y=-x and y=k+1 consists entirely of vertical steps?

4
votes

1
answer

443
views

### Is this simple symmetry of Littlewood-Richardson coefficients known?

Let $\lambda$ be a partition with at most $p$ parts, let $\mu$ be a partition with at most $q$ parts, and let $\nu$ be a partition with at most $p+q$ parts. Let $m\geq \nu_1$ be an integer. We denote ...

1
vote

0
answers

192
views

### Classical and free cumulants, symmetric functions, and inverses (references), related to associahedra, parking functions, noncrossing partitions

Looking for references for one or more of the following four sets of partition polynomials 1a) through 4a), particularly those which present geometric / topological combinatorial interpretations.
...

7
votes

1
answer

317
views

### Decomposition of a tensor product of representations of $\mathrm{GL}_l(\mathbb{C})$ and decomposition of Littlewood-Richardson numbers?

For a positive integer $m$, denote $T(m)=\{(\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\lambda_1\ge \lambda_2\ge\dots \ge\lambda_m\}$ and $T^+(m)=\{ (\lambda_1,\dots,\lambda_m)\in \mathbb{Z}^m:\...

7
votes

0
answers

78
views

### Generalization of Lucas sequences to order 3 (and above)

For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The ...

17
votes

2
answers

2k
views

### Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers

Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$.
Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...

3
votes

1
answer

115
views

### Cauchy identity for Jack functions

There are two versions of Cauchy identity for Schur functions, namely
$$
\sum_{\lambda}s_\lambda(\underline x)s_\lambda(\underline y) = \prod_{i,j=1}^n\frac 1{1-x_iy_j}\ ,\qquad {\rm (1)}
$$
and
$$
\...

4
votes

2
answers

432
views

### About Cauchy identity for Schur polynomials

(This was originally posted here, https://math.stackexchange.com/questions/4687466/cauchy-identity-for-schur-functions, and I am reposting it here as it seems to be more appropriate.)
PRELIMINARY.
The ...

12
votes

3
answers

831
views

### Symmetric version of Hilbert's seventeenth problem?

Artin's solution to Hilbert's seventeenth problem tells us that a multivariate polynomial $f$ takes only non-negative values over the reals if and only if it is a sum of squares of rational functions.
...

9
votes

0
answers

269
views

### Inequality for symmetric polynomial functions of log concave variables

Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$).
...

4
votes

1
answer

165
views

### Frobenius series for the $S_n$-module $\mathbb{Q}[X]$

I'm reposting this question, by recommendation of a moderator.
I'm reading Haiman's article titled Conjectures on the quotient ring by diagonal invariants. In what follows, all vector spaces and ...

5
votes

0
answers

136
views

### How did Macdonald come up with $q,t$-Kostka polynomials?

The $q,t$ Kostka polynomials are defined to be the coefficients of the big Schur $s_\lambda[X(1-t)]$ in the expansion of the integral form Macdonald polynomials $J_\mu[X;q,t]$. The integral form ...

4
votes

2
answers

244
views

### Can every symmetric function be factorized through symmetric polynomials?

A symmetric function is a function $f:\mathbb R^n\to \mathbb R$ such that $f(x_1,\ldots,x_n)=f(\sigma(x_1,\ldots,x_n))$ for every permutation $\sigma\in S_n.$
The most commonly encountered symmetric ...

2
votes

0
answers

122
views

### "Symmetrize" a (balanced) hypergeometric 4F3

Let $a,b,c,d$ be positive integers such that $a+b+c+d=2^{n}$ with $n \ge 2$.
Denote
$$
N \equiv a+b+c+d=2^{n}
$$
Consider the balanced hypergeometric series
$$
\frac{\operatorname{\Gamma}\left( \frac{...

8
votes

0
answers

210
views

### Alternate proof in Fulton–Harris of representation theoretic version of Littlewood–Richardson rule

$\DeclareMathOperator\Ind{Ind}$Let $d = d_1 + d_2$ with $d_1$, $d_2$ positive integers. Let $\lambda$ be a partition of $d_1$ and $\mu$ a partition of $d_2$, so that the Young symmetrizer construction ...

7
votes

0
answers

128
views

### Asymptotic character theory of unitary groups via shifted Schur functions

In the paper "Shifted Schur Functions" http://arxiv.org/abs/q-alg/9605042 by Andrei Okounkov and Grigori Olshanski it is said that one of the motivations for that paper was the asymptotic ...

9
votes

2
answers

752
views

### Using Schur-Weyl duality

I am trying to gain a better understanding of Schur-Weyl duality specifically applied to symmetric functions. My motivating example is trying to understand the Frobenius character of the multilinear ...

5
votes

0
answers

112
views

### Sum of Schur functions associated to self-conjugate partitions

The $\tau$-function $H^\circ \big(t ;\vec{x} \big)$ associated with counting simple Hurwitz numbers is the formal power series
\begin{equation}
(\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, =
\,
\...

6
votes

0
answers

141
views

### Inequality for support of plethysm: "slope" of partitions

Let $\lambda$ be an integer partition. We define $$\newcommand{\slope}{\mathrm{slope}}\slope(\lambda) = \begin{cases}\ell(\lambda)/\vert\lambda\vert & \lambda \neq 0 \\ 0 & \lambda=0.\end{...

4
votes

0
answers

120
views

### A particular family of symmetric functions (sums of powers of sums of subsets)

For any $m,k$ define
$$ f_{m,k}(x_1,\ldots,x_n) = \sum_{1\le i_1<i_2<\cdots<i_m\le n} (x_{i_1}+\cdots+x_{i_m})^k. $$
Do these symmetric polynomials have a name and any theory?

7
votes

1
answer

384
views

### Combinatorial reciprocity for symmetric functions

I am wondering whether a certain instance of combinatorial reciprocity (in the sense of Stanley's classic paper "Combinatorial Reciprocity Theorems"), concerning symmetric functions, is ...

2
votes

1
answer

249
views

### 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 ...

6
votes

1
answer

176
views

### Action of $\widehat{\mathfrak{sl}_2}$ on symmetric functions with $\mathbb{Z}_{(2)}$ coefficients

It is known that there is a representation of the affine Lie algebra $\widehat{\mathfrak{sl}_q}$ (over $\mathbb{Z}$) on the algebra of symmetric functions, where the action of the Chevalley generators ...

1
vote

0
answers

43
views

### Kostka coefficients for cylindric skew Schur functions using Kostant's partition function?

There is a classical formula for computing Kostka coefficients,
using a signed sum over permutations:
$$
K_{\lambda,\mu} = \sum_{\sigma \in S_n}
(-1)^{inv(\sigma)} \mathfrak{P}\left( \sigma(\lambda + ...

8
votes

1
answer

347
views

### 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(\...

8
votes

0
answers

223
views

### Scalar products on symmetric functions behaving like the Macdonald scalar product

The Macdonald symmetric functions (or Macdonald polynomials)
$P_\lambda(x)$ are orthogonal with respect to the Macdonald scalar
product
$$ \langle p_\lambda,p_\mu\rangle =
\delta_{\lambda\mu}z_\...

4
votes

1
answer

302
views

### 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 ...

3
votes

0
answers

132
views

### Expansion in Schur function of negative binomial exponent

I want to know if there exist a known expansion or can be derived of the polynomial
$$ \prod_{i=1}^{m}\prod_{j= 1}^{n}(1-z(x_i + y_i))^{-w} \tag{*}$$
in terms of Schur function. That is asking for (*) ...

6
votes

1
answer

334
views

### Identity involving Jack polynomials at $x^{-1}$

Let $J_\lambda^{(\alpha)}(x)$ be the Jack polynomials in $N$ variables, with a normalization such that the coefficient of the monomial polynomial $m_\lambda$ is equal to 1.
They satisfy the identity
$$...

1
vote

0
answers

75
views

### Determine whether a set generates a residue field of an invariant ring

Fix two positive integers $m>n$.
Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$.
Denote $R=\...

5
votes

1
answer

240
views

### Dimension reduction for non-negativity of elementary symmetric polynomials

Fix integers $1 \leq k \leq n$ and suppose $\mathbf{x} \in \mathbb{R}^n$ is such that $e_j(x_1,x_2,\ldots,x_n) \geq 0$ for all $1 \leq j \leq k$, where $e_j$ is the $j$-th elementary symmetric ...

10
votes

0
answers

378
views

### Has anyone met this "$q$-character" table for $S_4$?

Is anyone aware of the following $q$-character table for the
symmetric group $S_4$?
\begin{array}{|c|c|c|c|c|c|}
\hline
\mathrm{conj}\backslash\mathrm{rep}
& 2+1+1 & 3+1 & ...

3
votes

0
answers

67
views

### Subrings of the ring of symmetric functions

While experimenting with symmetric functions, I noticed the following equality of subrings of the ring of symmetric functions:
$$\mathbb{Z}[(n-1)!\cdot p_n \ |\ n \ge 1] = \mathbb{Z}[n!\cdot h_n \ |\ ...

2
votes

0
answers

113
views

### Yamanouchi ribbon tableaux?

Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions.
The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ ...

6
votes

1
answer

256
views

### Tanglegrams and functional equations of M. Somos

Recent references on the matter at hand include, a lecture slide The Konvalinka-Amdeberhan conjecture
and plethystic inverses and a preprint on Counting tanglegrams with species by I. Gessel; the ...

9
votes

1
answer

512
views

### 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)...

5
votes

0
answers

967
views

### A generalization of the difference of squares identity

Let us find explicit integer functions for the coefficients of the monomial expansion of
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...

3
votes

0
answers

487
views

### Proving an optimization problem from continuous input to binary is optimal

Suppose we have a function $f(x,y,z)$ where the inputs are uniform from 0 to 1. The output is either $+1$ or $-1$. And there is a partial symmetry $f(x,y,z) = f(z,y,x)$.
Tell me what the minimum of ...

1
vote

1
answer

607
views

### Polynomial invariant — from product formula to monomial expansion

Context
This question deals with the polynomial invariant denoted by $ H_{n} $ in Maksym Fedorchuk and Igor Pak's 2004 paper Rigidity and polynomial invariants of convex polytopes (sections 7.6 and 9)....