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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 …

Let $\mathcal{U}(N)$ be the unitary group. It is well known that $$ \int_{\mathcal{U}(N)} U_{ij} U^\dagger_{nm} \,dU=\delta_{im}\delta_{jn}\frac{1}{N},$$ where $dU$ is the Haar measure. More compl …

I am writing more than a year after the question was posted, only to spell out some more details regarding the calculation implicit in the solution presented by Suvrit, and to clarify the dependence o …

This is example 2 in page 116 of MacDonald's book, "Symmetric Functions and Hall Polynomials"

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 …

Schur functions are irreducible characters of the unitary group $\mathcal{U}(N)$. This implies the integration formulae $$ \int_{\mathcal{U}(N)}s_\lambda(AUA^\dagger U^\dagger)dU=\frac{|s_\lambda(A)|^ …

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

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 …

The Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ appear in the expansion of a product of Schur functions into Schur functions, $s_{\mu}(x)s_\nu(x)=\sum_\lambda c^\lambda_{\mu\nu}s_\lambda(x …

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 …

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 …

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, …

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.