Abstract
Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.
Highlights
We consider simple Lie algebras, i.e., the infinite families An, Bn, Cn and Dn and exceptional algebras G2, F4, E6, E7 and E8
In [7] we studied a generalization of C– and S– functions of Weyl groups of An using immanants of a certain matrix A with exponential entries
Immanants are functions defined on the set of squared matrix of order n. They are related to irreducible characters of the symmetry group Sn, among them the standard determinant and permanent
Summary
We consider simple Lie algebras, i.e., the infinite families An, Bn, Cn and Dn and exceptional algebras G2, F4, E6, E7 and E8. Immanants are functions defined on the set of squared matrix of order n They are related to irreducible characters of the symmetry group Sn, among them the standard determinant and permanent. In the paper we reviewed the fact that the C– (S–)functions correspond to the permanent (determinant) of the matrix A and studied non-trivial immanants These new special functions possess some of the properties of orbit functions, they are not invariant with respect to the Weyl group An. These new special functions possess some of the properties of orbit functions, they are not invariant with respect to the Weyl group An Another way to define these functions is to use irreducible characters of the Weyl group of An directly.
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