The representations of wreath products via double centralizing theorems (Q1080941)

A famous discovery of I. Schur was that via a double centralizing property the ordinary irreducible polynomial representations \(\{\lambda\}\) of \(GL_ m({\mathbb{C}})\) can be obtained from the ordinary irreducible representations [\(\lambda\) ], \(\lambda\vdash n\), of the symmetric groups \(S_ n\). This close connection is very explicit, e.g. the polynomials which yield the entries of the representing matrices \(\{\lambda\}\) (M), \(M\in GL_ m({\mathbb{C}})\) can be expressed in terms of the elements of the matrices [\(\lambda\) ](\(\pi)\), \(\pi \in S_ n\). Hence it is important to consider other situations where double centralizing can be applied, too. The present author uses his own extensions of Schur's method inverting the roles of \(GL_ m\) and \(S_ n\) and the corresponding generalizations in order to get information on the representations of \(A\wr S_ n\) and \(A\wr A_ n\), A a semisimple algebra, \(A_ n\) the alternating group. The ground field \({\mathbb{F}}\) is assumed to be algebraically closed and of characteristic zero. Let A be an associative algebra, spanned over \({\mathbb{F}}\) by its units (e.g. a semisimple algebra). Denote by \(A\wr S_ n\) the group algebra of \(S_ n\) over the ring \(A\otimes...\otimes A\). A complete set of inequivalent irreducible representations of \(A\wr S_ n\) is obtained: \[ \{N_{<\lambda >}| \quad <\lambda >=(\lambda (1),...,\lambda (t)),\quad \lambda (i)\vdash m_ i,\quad \sum^{t}_{i=1}m_ i=n\}, \] the dimensions are given, and a complete list of irreducible representations of \(A\wr A_ n\) is derived.