arXiv subjects
Publications and source records attributed to Francois Descouens.
We present an overview of the implementation of symmetric functions in MuPAD-Combinat. We also explain how to interface C++ programs with MuPAD in order to make efficient research during concrete work sessions.
We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We mainly use the combinatorial interpretation of Haglund, Haiman and Loehr giving the expansion of the modified Macdonald polynomials on the monomial basis.
The aim of this note is to give some factorization formulas for different versions of the Macdonald polynomials when the parameter t is specialized at roots of unity, generalizing those existing for Hall-Littlewood functions.
Hall-Littlewood functions indexed by rectangular partitions, specialized at primitive roots of unity, can be expressed as plethysms. We propose a combinatorial proof of this formula using A. Schilling's bijection between ribbon tableaux and ribbon rigged configurations.
We describe a general algorithm for generating various families of ribbon tableaux and computing their spin polynomials. This algorithm is derived from a new matricial coding. An advantage of this new notation lies in the fact that it permits one to generate ribbon tableaux with skew shapes. This algorithm permits us to compute quickly big LLT polynomials in MuPAD-Combinat.
Using the action of the Yang-Baxter elements of the Hecke algebra on polynomials, we define two bases of polynomials in n variables. The Hall-Littlewood polynomials are a subfamily of one of them. For q=0, these bases specialize into the two families of classical Key polynomials (i.e. Demazure characters for type A). We give a scalar product for which the two bases are adjoint of each other.