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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 $\alpha,\beta\vdash n$ 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 …

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 $d_\lambda$ with $\lambda\vdash n$ be the dimensions of irreducible representions of the permutation group $S_n$. Let $C^{\nu}_{\lambda,\mu}$ be the standard Littlewood-Richardson coefficients. I …

I realize now that, in the original quantity $$ F_n(\mu)=\frac{1}{n!(n+m)!}\sum_{\lambda\vdash n}\sum_{\nu\vdash n+m}C^{\nu}_{\lambda,\mu} d_\lambda d_\nu M_\nu^2 \delta_{D(\nu),D(\mu)},$$ I should ha …

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 …

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 $\chi_\lambda(\mu)$ 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 …

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 …

For $M=2$ we get a "solution" as follows. For given $\pi_1$ and $\pi_2$ we want to know how many permutations $\pi$ are there such that $\pi\pi_1$ has $c_1$ cycles and $\pi\pi_2$ has $c_2$ cycles. Let …

It is widely known that $$ \frac{1}{n!}\sum_{\pi\in S_n}\chi_\lambda(\pi)\chi_\mu(\pi)=\delta_{\lambda,\mu},$$ where $S_n$ is the permutation group and $\chi$ are its irreducible characters. In exer …

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

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 …

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 …

It is well known that irreducible characters of the symmetric group satisfy orthogonality relations, $$ \sum_{\mu \in P(n)} \chi_\mu^\lambda\chi_\mu^\omega=z_\lambda\delta_{\lambda,\omega},\quad \su …