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I think the essence is the central limit theorem. If you compute the traces of powers of your random matrix, they will be the sum of many independent random variables and will be Gaussian distributed …

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"

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 $P(n,\alpha)$ be the set of partitions of $n$ with signature $\alpha$. Such a set is invariant under your group, and any permutation of its elements is an automorphism. Your group is just $$G=\Pi_ …

This question was asked on MSE some time ago, here, but got no attention. The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)=Tr(R_\lambda(U))$. They are symmetric functio …

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 …

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgo …

The Oxford handbook of random matrix theory (Oxford University Press, 2011), edited by G. Akemann, J. Baik, P. Di Francesco, is an excellent reference, which covers a wide variety of properties and ap …