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.

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4 votes
0 answers
94 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
223 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 ...
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9 votes
0 answers
128 views

Relationship between Jack polynomials and root systems

I don't know much about algebraic combinatorics; however, I came across the Wikipedia page for Jack polynomials and read that they form an orthogonal basis in a space of symmetric polynomials, with ...
2 votes
1 answer
152 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 ...
0 votes
0 answers
34 views

Expansion with respect to spherical harmonics - symmetry

Assume a square-integrable function $f$ has the following expansion with respect to Spherical Harmonics: $$ f(\xi)=\sum_{\ell=3}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\xi)\tag 2 $$ where $Y_\...
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5 votes
0 answers
104 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
31 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 + ...
7 votes
1 answer
266 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(\...
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8 votes
0 answers
190 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
219 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 ...
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3 votes
0 answers
124 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 (*) ...
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6 votes
1 answer
319 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 $$...
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1 vote
0 answers
73 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=\...
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5 votes
1 answer
194 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
343 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 & ...
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3 votes
0 answers
64 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
81 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 $c_{\nu\mu}^{\lambda}$ ...
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0 votes
0 answers
58 views

Function with $\text{Sym}_k$-orbit as fiber / forgetting ordering of vectors

Fix $n,k >0$. Is there a continuous function $$ f: \oplus^k \mathbb{R}^n \longrightarrow \mathbb{R}^m$$ for some $m$ so that each fiber is essentially given by an orbit of the permutation group $\...
6 votes
1 answer
245 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
284 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)...
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5 votes
0 answers
716 views

A generalization of the difference of two squares identity

This question deals with finding explicit integer functions for the coefficients of the monomial expansion of the polynomial $$ Q \left( x_1, \ldots , x_n \right) = \prod_{\kappa_1, \ldots , \kappa_{n-...
3 votes
0 answers
218 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
456 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)....
6 votes
0 answers
169 views

Hall-Littlewood polynomials of non-dominant weights

$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let $$ R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...
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6 votes
0 answers
209 views

What do Macdonald polynomials hint about $\operatorname{Rep}(S_\infty)$?

$\DeclareMathOperator\Rep{Rep}$It is well-known that the Macdonald "$P$" polynomials deform the Jack "$J$" polynomials [1]. The latter have profound relations with representation ...
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3 votes
1 answer
278 views

Polynomial function defined recursively by a resultant - is it well defined?

Preliminaries Let $ n $ be an integer such that $ n \geq3 $. Denote $ \left[ n \right] \equiv \{1,2, \ldots ,n \} $. Let $ P $ be a non-empty subset of $ \left[ n \right] $ such that $ \left|P \right| ...
2 votes
0 answers
48 views

Chapter 2 Section 2 in Macdonald's Symmetric Functions and Hall Polynomials

Throughout this post $R$ denotes a discrete valuation ring with residue field $R/\frak{m}$ being finite. I'm reading Macdonald's exposition on the Hall Algebra on page 183 and trying to make sense of ...
3 votes
0 answers
51 views

LLT polynomials and graded Specht modules

In the paper ``Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras And Unipotent Varieties'' Lascoux, Leclerc and Thibon define polynomials which quantise the Schur functions. These $...
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1 vote
0 answers
86 views

Continuous decomposition of permutation-invariant set functions

The seminal machine learning paper Deep Sets (Zaheer et al., 2017) discusses representations of permutation-invariant functions on real tuples, or (multi)set functions. Given a countable set $X$ and a ...
10 votes
1 answer
204 views

$2$-adic valuation of Schur $P$-functions in the power-sum basis

For a partition $\lambda$, let $P_\lambda$ be the Schur $P$-functions (case $t=-1$ of Hall-Littlewood symmetric functions) and let $p_\lambda=p_{\lambda_1}p_{\lambda_1}\cdots p_{\lambda_k}$ be the ...
7 votes
2 answers
223 views

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 ...
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3 votes
0 answers
103 views

Multiplicities of irreducible $U(n)$-modules in the tensor product $V_{\lambda}\otimes V_{\mu}$

It is known that every irreducible representation of the unitary group $U(n)$ can be uniquely described by the non-increasing sequence $\lambda=(\lambda_1,\ldots,\lambda_n)$ of integers (denote the ...
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3 votes
0 answers
160 views

Polynomial invariant relating the circumradius and sides of a cyclic polygon

This question deals with the polynomial invariant which relates the circumradius and the squares of the sides of a cyclic polygon. This invariant is discussed briefly in the seminal paper On the Areas ...
1 vote
0 answers
112 views

A holonomic function and its singularity

The following series where $q_i , h$ are constant parameters. $G(z)$ is a rational function. $$F(x):=\sum_{d= 1}^\infty \sum_{k=1}^d (-1)^{d-k} \, s_{(k, 1^{d-k})}(\tfrac{q_1}{h}, \tfrac{q_2}{h}, \...
  • 643
6 votes
0 answers
146 views

Expanding the zonal polynomial $Z_\lambda(x/(1-x))$

Schur polynomials $s_\lambda(x)$ have a determinantal expression. Using that, I know how to write the polynomial $s_\lambda(\frac{x}{1-x})=s_\lambda(\frac{x_1}{1-x_1},\frac{x_2}{1-x_2},...)$ as an ...
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2 votes
1 answer
117 views

Maximally independent polynomial families with row symmetry

Introduction: In the 1-dimensional case, given $m$-variables $$\mathbf{x} = (x_1,x_2,\dots,x_m)^T,$$ the elementary symmetric polynomials $(e_k(\mathbf{x}))_{k=1}^m$ give a "symmetric basis",...
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3 votes
1 answer
147 views

Is there a Jacobi–Trudi formula for skew zonal polynomials?

Skew Schur polynomials are defined as $s_{\lambda/\mu}=\sum_\nu c^\lambda_{\mu\nu}s_\nu$, where the Littlewood–Richardson coefficients $c^\lambda_{\mu\nu}$ satisfy $s_\mu(x)s_\nu(x)=\sum_\lambda c^\...
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3 votes
1 answer
152 views

Different ways of defining the Chern character of a complex

Consider a finite complex $E$ of (holomorphic) vector bundles on a (complex) manifold $X$, i.e, the complex is of the form $$ 0 \to E_N \to E_{N-1} \to \dots \to {E_0} \to 0, $$ where the bundles are ...
2 votes
1 answer
183 views

Changing $S_2 \wr S_n$ for $S_n \wr S_2$ in the theory of zonal polynomials

The permutation group $S_{2n}$ has $H_{2,n}=S_2\wr S_n$ as a subgroup. The plethysm $h_n(h_2)=\sum_{\lambda\vdash n}s_{2\lambda}$ is well known. The zonal spherical functions $\omega_\lambda(g)=\frac{...
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6 votes
0 answers
383 views

This sum over partitions has unexpectedly nice denominators

Fix an integer $n\geq0$, a power series $\gamma \in \mathbb Q[[X]]$ with valuation 1, and a symmetric function $f$ (with coefficients in $\mathbb Q$). Now, consider the series $$ S_n = \sum_{\Lambda\...
  • 1,471
15 votes
1 answer
631 views

a Vandermonde-type of determinants summed over permutations

Let $S_n$ be the symmetric group. Consider $$D:=\sum_{\sigma\in S_n} \text{sgn}(\sigma)\cdot \det\begin{pmatrix}1 & a_{\sigma(1)}-0 & (a_{\sigma(1)}-0)^2 & \cdots & (a_{\sigma(1)}-0)^{...
  • 151
9 votes
0 answers
449 views

Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
  • 19.6k
4 votes
1 answer
217 views

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 ...
6 votes
1 answer
381 views

Typo in Stanley, Enumerative combinatorics II, Cor. 7.23.9?

In Stanley, EC2, we have the following statement: I think there is a typo in the first sum after "generating function", and that $[n]_q!$ should be replaced by $(1-q)(1-q^2)\dotsb (1-q^n)$, ...
1 vote
1 answer
74 views

Are top Brauer characters bounded?

Let $p_\lambda$ be power sum symmetric functions. Let $s_\lambda$ and $o_\lambda$ be irreducible characters of the unitary and orthogonal groups $U(N)$ and $O(N)$, respectively (the $s$ are the Schur ...
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5 votes
0 answers
74 views

Form on symmetric functions and their q,t- analogues

[Notations are as in Macdonald's Symmetric Functions and Hall Polynomials] The space of symmetric functions $\Lambda_{\mathbb{Q}}$ has a bilinear form defined by $ (p_\lambda, p_\mu)= z_\lambda \...
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2 votes
0 answers
67 views

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:...
  • 643
13 votes
1 answer
209 views

Recognizing algebraic independence among Schur polynomials

Given a set of integer partitions $\{\lambda_1, \lambda_2,\dots \lambda_n\}$. Are there combinatorial criteria for deciding whether the associated Schur polynomials $s_{\lambda_1}, s_{\lambda_2},\dots ...
5 votes
1 answer
277 views

"Kronecker Product" for quasi-symmetric functions

Recall that the Kronecker product $s_\lambda * s_\mu$ of two Schur functions $s_\lambda$ and $s_\mu$ is the symmetric function whose expansion (in terms of Schur functions) is given by \begin{equation}...
  • 1,625
10 votes
1 answer
801 views

Todd polynomials

Let $T_k(x_1,\ldots,x_n)$ be the Todd polynomials, $e_k(x_1,\ldots,x_n)$ the elementary symmetric polynomials and $p_k(x_1,\ldots, x_n)$ the power sums of degree $k$. We have the following generating ...
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