On dimensions of block algebras.
Let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(P\) and source algebra \(A\). The author shows that \(\dim A\geq |P|\cdot l(B)\cdot u^2_A\) where \(l(B)\) is the number of simple \(B\)-modules and \(u_A\) is the greatest common divisor of the dimensions of the simple \(A\)-modules. Moreover, \(\dim A=|P|\cdot l(B)\) if and only if \(A\) is isomorphic to the group algebra of \(P\rtimes E\) for some Abelian \(p'\)-subgroup \(E\) of \(\Aut(P)\). These results -- suggested by a theorem of Brauer and discussions with this reviewer -- improves on an earlier theorem by the author [Arch. Math. 89, No. 4, 311-314 (2007; Zbl 1140.20010)].
scientific article; zbMATH DE number 5707700