Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials of Types A and C. Extended Abstract

A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type C, which are specializations of the corresponding Macdonald polynomials at q=0. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type A, so our work is a first step towards finding such a formula

Tema Con Variazioni: Quantum Channel Capacity

Channel capacity describes the size of the nearly ideal channels, which can be obtained from many uses of a given channel, using an optimal error correcting code. In this paper we collect and compare minor and major variations in the mathematically precise statements of this idea which have been put forward in the literature. We show that all the variations considered lead to equivalent capacity definitions. In particular, it makes no difference whether one requires mean or maximal errors to go to zero, and it makes no difference whether errors are required to vanish for any sequence of block sizes compatible with the rate, or only for one infinite sequence.Comment: 32 pages, uses iopart.cl

Hopf monoids from class functions on unitriangular matrices

We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyal's category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.Comment: Final Version, 32 pages, accepted in "Algebra and Number Theory