The class of finitely presented algebras over a field \(K\) with a set of generators \(a_1,\dots,a_n\) and defined by relations of the form \(a_1a_2\cdots a_n=a_{\sigma(1)}a_{\sigma(2)}\cdots a_{\sigma(n)}\), where \(\sigma\) runs through a subgroup \(H\) of the symmetric group \(S_n\), is considered. Such an algebra is denoted by \(K[S_n(H)]\), while \(S_n(H)\) stands for the monoid defined by the same monoid presentation.
First, using a result of Adyan, it is shown that \(S_n(H)\) is cancellative if and only if the stabilizers of 1 and of \(n\) in the group \(H\) are trivial, and in this case \(S_n(H)\) embeds in a group.
The main result of the paper asserts that the algebra \(K[S_n(H)]\) is semiprimitive provided that \(H\) is an Abelian group. Some further results on prime ideals of \(S_n(H)\) and consequences concerning certain primitive ideals of \(K[S_n(H)]\) are also proved.
This work is a continuation of earlier work of the first and the second author and of the reviewer on finitely presented algebras defined by permutation relations [J. Pure Appl. Algebra 214, No. 7, 1095-1102 (2010; Zbl 1196.16022); J. Algebra 324, No. 6, 1290-1313 (2010; Zbl 1232.16014); Contemp. Math. 499, 1-26 (2009; Zbl 1193.16019)].