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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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 ...
pallab1234's user avatar
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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 ...
Tom Copeland's user avatar
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3 votes
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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 ...
Shaul Zemel's user avatar
3 votes
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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 ...
eti902's user avatar
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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 ...
Tom Copeland's user avatar
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14 votes
1 answer
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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(...
Gjergji Zaimi's user avatar
1 vote
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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 ...
Paul Broussous's user avatar
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 ...
Richard Stanley's user avatar
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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). ...
matha's user avatar
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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 ...
Shaul Zemel's user avatar
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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 ...
Antoine Labelle's user avatar
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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?
Mihawk's user avatar
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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 ...
Matt Samuel's user avatar
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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. ...
Tom Copeland's user avatar
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7 votes
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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:\...
Q. Zhang's user avatar
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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 ...
Max Alekseyev's user avatar
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 $k$, ...
nichehole's user avatar
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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 $$ \...
Giulio R's user avatar
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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 ...
Giulio R's user avatar
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12 votes
3 answers
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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. ...
Timothy Chow's user avatar
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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. $x_i^2 \le x_{i-1}x_{i+1}$). ...
René Gy's user avatar
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1 answer
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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 ...
Albert's user avatar
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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 ...
ArB's user avatar
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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 ...
Nick Belane's user avatar
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{...
PalmTopTigerMO's user avatar
8 votes
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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 ...
babu_babu's user avatar
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0 answers
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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 ...
richrow's user avatar
  • 357
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 ...
Trevor K's user avatar
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) \, = \, \...
Jeanne Scott's user avatar
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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{...
Dan Petersen's user avatar
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?
Brendan McKay's user avatar
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 ...
Sam Hopkins's user avatar
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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 ...
Per Alexandersson's user avatar
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 ...
Antoine Labelle's user avatar
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 + ...
Per Alexandersson's user avatar
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(\...
Jeanne Scott's user avatar
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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_\...
Richard Stanley's user avatar
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 ...
Tobias Fritz's user avatar
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3 votes
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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 (*) ...
GGT's user avatar
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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 $$...
Marcel's user avatar
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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=\...
GiS's user avatar
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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 ...
Nathaniel Johnston's user avatar
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 & ...
Jeanne Scott's user avatar
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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 \ |\ ...
Antoine Labelle's user avatar
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 $c_{\nu\mu}^{\lambda}$ ...
eti902's user avatar
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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 ...
T. Amdeberhan's user avatar
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)...
Leox's user avatar
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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}...
PalmTopTigerMO's user avatar
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 ...
Kunal Marwaha's user avatar
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)....
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