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Register a new userOn the Minor Problem and Branching Coefficients
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Authors
Jean-Bernard Zuber
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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).
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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 prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.
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 importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies $a_{ij}a_{ji}ge 4$ for all $i,j$, then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the T-equivariant cohomology of flag varieties G/P and Bott-Samelson varieties Gamma_ii(G).
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 $d$, in a random direction. What shape should the lawn be to maximise the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc shaped lawn is not optimal for any $d>0$. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that for $ d < pi^{-1/2}$ the optimal lawn resembles a cogwheel with $n$ cogs, where the integer $n$ is close to $ pi ( arcsin ( sqrt{pi} d /2 ) )^{-1}$. We find transitions to other shapes for $d gtrsim pi^{-1/2}$.
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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 generalize those of Ponsot and Teschner in the rank 1 case of $U_q(mathfrak{sl}_2)$. In higher rank, many nontrivial features appear, the most important of these being a surprising connection to the quantum integrability of the open Coxeter-Toda lattice. We show that the closure under tensor products follows from the orthogonality and completeness of the Toda eigenfunctions (i.e. the q-Whittaker functions), and obtain an explicit construction of the Clebsch-Gordan intertwiner giving the decomposition of $mathcal{P}_lambda otimes mathcal{P}_mu$ into irreducibles.
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