On the semi-simplicity of the Hecke algebra of \((\mathbb{Z}/r\mathbb{Z})\wr{\mathfrak S}_ n\)
Let \(A\) be an integral domain and let \(q, u_1, u_2, \dots, u_r\) be elements of \(A\) with \(q\) invertible. The algebra over \(A\) defined by generators \(a_1, \dots, a_r\) satisfying the conditions: \((a_1- u_1) (a_1-u_2) \cdots (a_1-u_r)=0\), \((a_j-q) (a_j+1)=0\), \(2 \leq j \leq n\), \((a_1a_2)^2=(a_2 a_1)^2\), \(a_j a_{j+1} a_j=a_{j+1} a_j a_{j+1}\), \(2 \leq j \leq n- 1\), \(a_i a_j=a_j a_i\), \(|i-j|\geq 2\) is denoted by \({\mathfrak H}_{n,r} (A)\) and is called the Hecke algebra of \((\mathbb{Z}/ r\mathbb{Z}) \wr S_n\). In the case that \(A\) is a field \(F\), \({\mathfrak H}_{n,r} (F)\) is a finite-dimensional algebra with dimension \(r^n n!\) and in general it is not a semi-simple algebra. In the main theorem of this paper, the author proves that semi-simplicity of \({\mathfrak H}_{n,r} (F)\) is equivalent to the condition that \({\mathfrak H}_{n,r}(F)\) is a split semi-simple algebra which is equivalent to the condition that the parameters satisfy \(q^d u_i \neq u_j\) \((i \neq j\), \(d \in Z\), \(|d|< n)\) and \([n]! \neq 0\) where \([n]!=\prod^n_{j=1} [j]\), \([j]=1+q+\cdots+q^{j-1}\). Using the main theorem, in case that \({\mathfrak H}_{n,r} (F)\) is a semi- simple algebra all the irreducible \({\mathfrak H}_{n,r}(F)\)-modules are determined. In the final section of this paper, the author determines a set of irreducible representations for a certain non-semi-simple algebra \({\mathfrak H}_{n,r} (F)\).
scientific article; zbMATH DE number 665864