Finitely presented algebras and groups defined by permutation relations

The class of finitely presented algebras over a field KK with a set of generators a1,…,ana1,…,an and defined by homogeneous relations of the form a1a2⋯an=aσ(1)aσ(2)⋯aσ(n)a1a2⋯an=aσ(1)aσ(2)⋯aσ(n), where σσ runs through a subset HH of the symmetric group SymnSymn of degree nn, is introduced. The emphasis is on the case of a cyclic subgroup HH of SymnSymn of order nn. 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 HH of SymnSymn are proposed.