Non symmetric Cauchy kernels for the classical groups

Amy M. Fu (1) , Alain Lascoux (2)

1 CFC - Center for Combinatorics [Nankai]
2 LIGM - Laboratoire d'Informatique Gaspard-Monge

We give non-symmetric versions of the Cauchy kernel and Littlewood's kernels, corresponding to the types A, B, C and D, of the classical groups. Defining two families of key polynomials (one of them being the Demazure characters), we show that these new kernels are diagonal in the basis of key polynomials. We define scalar products such that the two families of key polynomials are adjoint to each other.

Mots clés: Cauchy kernel; Littlewood's kernels; Classical groups; Key polynomials; Isobaric divided differences

Domaines: Informatique et langage [cs.CL]

Amy M. Fu, Alain Lascoux. Non symmetric Cauchy kernels for the classical groups. Journal of Combinatorial Theory, Series A, 2009, 116 (4), pp.903-917. ⟨10.1016/j.jcta.2008.10.007⟩. ⟨hal-00622767⟩