Group algebras and semigroup algebras defined by permutation relations of fixed length

Abstract

Let H be a subgroup of Sym n, the symmetric group of degree n. For a fixed integer l ≥ 2, the group G presented with generators x1, x2,…,xn and with relations xi1xi2⋯xil = xσ(i1) xσ(i2)⋯xσ(il), where σ runs through H, is considered. It is shown that G has a free subgroup of finite index. For a field K, properties of the algebra K[G] are derived. In particular, the Jacobson radical 𝒥(K[G]) is always nilpotent, and in many cases the algebra K[G] is semiprimitive. Results on the growth and the Gelfand–Kirillov dimension of K[G] are given. Further properties of the semigroup S and the semigroup algebra K[S] with the same presentation are obtained, in case S is cancellative. The Jacobson radical is nilpotent in this case as well, and sufficient conditions for the algebra to be semiprimitive are given.