Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals

Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of \gl(n) as the dimension of a weight space of semi-invariants. Using this, we prove their saturation and investigate when they are nonzero. We also show that for certain partitions the associated stretched polynomials satisfy the same conjectures as single Littlewood-Richardson coefficients. We then provide a polytopal description of this multiplicity and show that its positivity may be computed in strongly polynomial time. Finally, we remark that similar results hold for certain other generalized Littlewood-Richardson coefficients.Comment: 28 pages, comments welcom

Why Does a Kronecker Model Result in Misleading Capacity Estimates?

Many recent works that study the performance of multi-input multi-output (MIMO) systems in practice assume a Kronecker model where the variances of the channel entries, upon decomposition on to the transmit and the receive eigen-bases, admit a separable form. Measurement campaigns, however, show that the Kronecker model results in poor estimates for capacity. Motivated by these observations, a channel model that does not impose a separable structure has been recently proposed and shown to fit the capacity of measured channels better. In this work, we show that this recently proposed modeling framework can be viewed as a natural consequence of channel decomposition on to its canonical coordinates, the transmit and/or the receive eigen-bases. Using tools from random matrix theory, we then establish the theoretical basis behind the Kronecker mismatch at the low- and the high-SNR extremes: 1) Sparsity of the dominant statistical degrees of freedom (DoF) in the true channel at the low-SNR extreme, and 2) Non-regularity of the sparsity structure (disparities in the distribution of the DoF across the rows and the columns) at the high-SNR extreme.Comment: 39 pages, 5 figures, under review with IEEE Trans. Inform. Theor

Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kaehler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N x N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard "supersymmetry" approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard-Stratonovich transformation in the "bosonic" sector. The method, suggested recently by one of us, is shown to be capable of calculation when reinforced with a generalization of the Itzykson-Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat-Heckman localization principle for integrals over non-compact homogeneous Kaehler manifolds. In the limit of large $N$ the asymptotic expression for the correlation function reproduces the result outlined earlier by Andreev and Simons.Comment: 34 page, no figures. In this version we added a few references and modified the introduction accordingly. We also included a new Appendix on deriving our Itzykson-Zuber type integral following the diffusion equation metho