Finitely presented algebras and groups defined by permutation relations

Abstract

The class of finitely presented algebras over a field K with a set of generators a 1 , … , a n and defined by homogeneous relations of the form a 1 a 2 ⋯ a n = a σ ( 1 ) a σ ( 2 ) ⋯ a σ ( n ) , where σ runs through a subset H of the symmetric group Sym n of degree n , is introduced. The emphasis is on the case of a cyclic subgroup H of Sym n of order n . A normal form of elements of the algebra is obtained. It is shown that the underlying monoid, defined by the same (monoid) presentation, has a group of fractions and this group is described. Properties of the algebra are derived. In particular, it follows that the algebra is a semiprimitive domain. Problems concerning the groups and algebras defined by arbitrary subgroups H of Sym n are proposed.