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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 $ …

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 Sch …

During a calculation, I have met a function on partitions, $F(\lambda)$, which seems to evaluate to positive integers. I have a procedure for computing it, but I was hoping for a formula, so I thought …

Let $\alpha,\beta\vdash n$ and define the polynomial $$f_{\alpha,\beta}(x)=\sum_{\lambda \vdash n}\chi_\lambda(n)\chi_\lambda(\alpha)\chi_\lambda(\beta)x^{\ell(\lambda)-1},$$ where $\chi_\lambda$ are …

This conjecture has now been proved by Valentin Bonzom, Guillaume Chapuy and Maciej Dołęga in their paper $b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and O(N)-BGW integral.

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 inf …

Lassalle and Schlosser have obtained in Inversion of the Pieri formula for Macdonald polynomials some recurrence relations for MacDonald polynomials $P_\lambda(x;q,t)$ which can be restricted to Jack …

I haven't found anything about Jacobi–Trudi for skew zonal polynomials. For usual zonal polynomials, Kerov has shown (Generalized Hall–Littlewood symmetric functions and orthogonal polynomials) that, …

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^\la …

A magic square of size $n$ and sum $k$ is a $n\times n$ matrix with non-negative integer elements, whose rows and columns all sum to $k$. A permutation matrix is a magic square of sum $1$. Every magic …

(I have looked more carefully at this theory of "universal characters" mentioned by Stanley and am updating this answer according to what I learned. All of this was contained in the answer by Stanley, …

Let $\lambda$ be a partition, represented by a usual Young diagram in which $1\le i\le \ell(\lambda)$ labels the rows and, for each $i$, $1\le j\le \lambda_i$ labels the columns. For each box $\square …

Let $\chi_\lambda(\mu)$ be the usual characters of the irreducible representations of the permutation group $S_n$. The normalized character is the quotient $\chi_\lambda(\mu)/f^\lambda$, where $f^\lam …

I would appreciate some help proving a conjecture related to combinatorics and representation theory. Given an integer partition $\lambda\vdash n$, define a polynomial in $N$ whose roots are the negat …

$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Schur functions $s_\lambda(x)$ with $\lambda\vdash n$ are simultaneously the irreducible characters of the unitary group $\U(N)$ and proportional to …