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The Minor problem, namely the study of the spectrum of a principal submatrix of a Hermitian matrix taken at random on its orbit under conjugation, is revisited, with emphasis on the use of orbital integrals and on the connection with branching coefficients in the decomposition of an irreducible representation of U(n), resp. SU(n), into irreps of U(n-1), resp. SU(n-1).
We prove that a finite group $G$ has a normal Sylow $p$-subgroup $P$ if, and only if, every irreducible character of $G$ appearing in the permutation character $({bf 1}_P)^G$ with multiplicity coprime to $p$ has degree coprime to $p$. This confirms a
In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These coefficients are of i
We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area one. It then jumps once, a fixed distance $d$, i
This is an introduction to some aspects of Fomin-Zelevinskys cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in
We describe the algebraic ingredients of a proof of the conjecture of Frenkel and Ip that the category of positive representations $mathcal{P}_lambda$ of the quantum group $U_q(mathfrak{sl}_{n+1})$ is closed under tensor products. Our results general