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
Filter by
Sorted by
Tagged with
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 ...
• 397
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 ...
• 9,817
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 ...
• 131
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 ...
• 835
1 vote
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 ...
• 9,817
609 views

• 960
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 ...
• 30.1k
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 $k$, ...
• 373
115 views

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 ...
• 229
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 ...
• 357
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 ...
• 93
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 (\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, = \, \...
• 1,837
141 views

• 15.5k
347 views

• 321
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 ...
• 5,630
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 & ...
• 1,837
67 views

• 546
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}...
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
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)....