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On representations of twisted group rings. (English) Zbl 1068.16035

Let \(G\) be a finite group acting on a field \(L\) with fixed field \(K\). The twisted group ring \(L\wr G\) carries the multiplication \((\sigma y)(\tau z)=\sigma\tau y^\tau z\) for all \(\sigma,\tau\in G\), \(y,z\in L\). This ring is also sometimes referred to as the skew group ring of \(G\) with coefficients in \(L\). Let the characteristic of \(L\) and the order of the kernel \(N\) of the operation of \(G\) on \(L\) be coprime. Let \(S\) be a discrete valuation ring with field of fractions \(K\), with maximal ideal generated by \(\pi\) and with integral closure \(T\) in \(L\). It is computed the colength of \(T\wr G\) in a maximal order in \(L\wr G\). In the case when \(S/\pi S\) is finite, the \(S/\pi S\)-dimension of the center of \(T\wr G/\text{Jac}(T\wr G)\) is computed. If this quotient is split semisimple, this yields a formula for the number of simple \(T\wr G\)-modules, generalizing Brauer’s formula.

MSC:

16S35 Twisted and skew group rings, crossed products
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
20C10 Integral representations of finite groups

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