Finitely presented algebras defined by permutation relations of dihedral type. (Q2796984)

The class of finitely presented algebras over a field \(K\) with a set of generators \(a_1,\ldots,a_n\) 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.NEWLINENEWLINE The main result of the paper establishes a canonical form of elements of \(S_n(H)\) in case the cyclic group \(\langle(1,2,\ldots,n)\rangle\) is a subgroup of index \(2\) in \(H\). The proof is very long and technical. As an application, it is shown that the universal group of \(S_n(H)\) is a unique product group. Moreover, it is shown that the algebra \(K[S_n(H)]\) is semiprimitive and it is an automaton algebra in the sense of Ufnarovskii.NEWLINENEWLINE This paper 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)] and of the authors of the present paper [J. Pure Appl. Algebra 216, No. 5, 1033-1039 (2012; Zbl 1259.16024)], where the cases when \(H\) is the full symmetric group, the alternating group or an Abelian group, are considered.