Künzer, Matthias On representations of twisted group rings. (English) Zbl 1068.16035 J. Group Theory 7, No. 2, 197-229 (2004). 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. Reviewer: Kanat Abdukhalikov (Fukuoka) Cited in 1 Document 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 Keywords:twisted group rings; representations; discrete valuation rings; colengths; Brauer formula; maximal orders; numbers of simple modules Software:MeatAxe PDFBibTeX XMLCite \textit{M. Künzer}, J. Group Theory 7, No. 2, 197--229 (2004; Zbl 1068.16035) Full Text: DOI arXiv References: [1] Benz H., J. Number Theory 20 pp 282– (1985) [2] L. Bonami.On the structure of skew group rings. Algebra-Berichte 48 (Verlag Fischer, 1984). · Zbl 0537.16005 [3] Brauer R., Z. 63 pp 406– (1956) [4] R. Brauer and C. Nesbitt. On the modular characters of groups. Ann. of Math. (2) 42 (1941), 556-590. · Zbl 0027.15202 [5] L. le Bruyn, M. Van den Bergh and F. Van Oystaeyen. Graded orders (Birkh user, 1988). [6] Chalatsis A., J. Algebra 124 pp 1– (1986) [7] G. H. Cli , W. Plesken and A. Weiss. Order-theoretic properties of the center of a block. In The Arcata conference on representations of finite groups, Proc. Symp. Pure Math. 47 (American Mathematical Society, 1987), pp. 413-420. [8] Cli G. H., Lecture Notes in Math. 1142 pp 96– (1984) [9] C. W. Curtis and I. Reiner. Methods of representation theory, vol. 1 (Wiley, 1981). [10] C. W. Curtis and I. Reiner. Methods of representation theory, vol. 2 (Wiley, 1987). [11] A. Fr hlich. Galois module structure of algebraic integers. Ergebnisse der Math. (3) 1, (Springer-Verlag, 1983). [12] G. Karpilovsky. Symmetric and G-algebras (Kluwer, 1990). · Zbl 0705.16001 [13] Klshammer B., J. Algebra 72 pp 1– (1981) [14] M. K nzer. Ties for the integral group ring of the symmetric group. Ph.D. thesis, University of Bielefeld (1999). http://www.mathematik.uni-bielefeld.de/@kuenzer. [15] Nakayama T., Japan. J. Math. pp 12– (1935) [16] J. Neukirch. Algebraische Zahlentheorie (Springer-Verlag, 1992). · Zbl 0747.11001 [17] Noether E., J. Reine Angew. Math. 167 pp 147– (1932) [18] Noether E., Indust. 148 pp 5– (1934) [19] Plesken W., Lecture Notes in Math. pp 1026– (1983) [20] I. Reiner. Maximal orders (Academic Press, 1975). · Zbl 0305.16001 [21] M. Ringe. The C Meat-Axe, GAP package (1993). [22] J.P. Serre.Linear representations of finite groups (Springer-Verlag, 1977). · Zbl 0355.20006 [23] Speiser A., . Math. Ann. 77 pp 546– (1916) [24] J. Thevenaz.G-algebras and modular representation theory (Clarendon Press, 1995). [25] Williamson S., Nagoya Math. J. 23 pp 103– (1963) [26] Williamson S., Nagoya Math. J. 25 pp 165– (1965) [27] Williamson S., Nagoya Math. J. 28 pp 85– (1966) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.