Enumeration in the Chevalley algebras

Abstract

We consider the maximal nilpotent subalgebra NΦ(K) of the Chevalley algebra of the type Φ over an arbitrary field K. Our purpose is enumeration of ideals of NΦ(K) that are invariant under the subgroup D of all diagonal automorphisms. Connections between ideals of NΦ(K) and normal subgroups of the unipotent subgroup of the Chevalley group of the type Φ over K are used. Finally, we use methods of integral representation of sums and of listing lattice paths. We also discuss some other problems of the enumeration of ideals.