Geodesic
Brauer Characters and Grothendieck Rings
Canadian journal of mathematics, Tome 27 (1975) no. 5, pp. 1025-1028

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Let G be a group of finite order g, A a splitting field of G of characteristic p (which may be 0) and R = AG the group algebra of G over A. In [2], the author studied some of the properties of the Grothendieck ring K(R) of the category of all finitely generated R-modules, and derived a number of consequences. This paper continues the study carried out in [2]. The study is concerned with the structure and irreducible representations of K(R). The ring K(R) is proved to be semisimple and the primitive idempotents of K(R) are explicitly constructed. If the ring K(R) is identified with the ‘algebra of representations', then Robinson's idempotent [3; 4; 5] follow from our description as a special case. 
Puttaswamaiah, B. M. Brauer Characters and Grothendieck Rings. Canadian journal of mathematics, Tome 27 (1975) no. 5, pp. 1025-1028. doi: 10.4153/CJM-1975-107-0
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[1] 1. Curtis, C. W. and Reiner, I., Representation theory of finite groups and associative algebras (Interscience, New York, 1962), pp. 144-145 and pp. 598–600. Google Scholar

[2] 2. Puttaswamaiah, B. M., Determination of Brauer characters, Can. J. Math. 26 (1974) 746–752. Google Scholar

[3] 3. De, G. Robinson, B., The algebras of representations and classes of finite groups, J. Mathematical Phys. 12 (1971), 2212–2215. Google Scholar

[4] 4. De, G. Robinson, B., Tensor product representations, J. of Algebra 20 (1972), 118–123. Google Scholar

[5] 5. De, G. Robinson, B., The dual of Frobenius’ reciprocity theorem, Can. J. Math. 25 (1973), 1051–1059. Google Scholar

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