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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 …

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

(EDIT: I have removed the denominators I had in a previous version as they were superfluous) The $N\times N$ determinant $$D(a,\vec{b})=\det\left((2N+a+b_j-i-j)!\right)$$ has the nice form $$D(a,\v …

The functions $p^*_k(x)=\sum_{i=1}^N ((x_i-i)^k-(-i)^k)$ are analogues of power sum symmetric functions, called shifted symmetric by Okounkov and Olshanski. Define $p^*_{(k_1,k_2,...)}=p^*_{k_1}p^*_{k …

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 …

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 …

Zonal polynomials may be expressed in terms of power sums as $$Z_\lambda=n!\sum_\nu \frac{1}{z_\nu }2^{n-\ell(\nu)}\omega_\lambda(\nu)p_\nu,$$ with usual notation in which $\omega_\lambda(\nu)$ are th …

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 $J_\lambda^{(\alpha)}(x)$ 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 $ …

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 …

In my recent work I have been led to consider the following type of permutation factorizations. Let $\pi_1$ and $\pi_2$ be two fixed permutations, with disjoint support, i.e. $\pi_1$ acts on $\{1,.. …

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 $\chi_\lambda(\mu)$ 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 have found a solution myself, at least in the case when $a$ and $b$ are partitions. The determinant can be written as $$ D=\det((x_i+y_j)!)=\det\left( \int z^{x_i+y_j}e^{-z}dz\right)$$ We resort t …

Let us call a partition odd if all its parts are odd, and let $Odd(n)$ be the set of all odd partitions of $n$, e.g. $Odd(6)=\{(5\,1),(3\, 3),(3\,1^3),(1^6)\}$. Let $H(n)$ denote the set of all hook …