Virtually irreducible lattices and Brauer correspondence of blocks
Let \(G\) be a finite group, \(R\) a complete discrete valuation ring of characteristic zero with valuation \(\nu\), field of fractions \(K\) and residue field \(\overline {R}\) of characteristic \(p>0\), which is assumed to be algebraically closed, and \(K\) is a splitting field for each subgroup of \(G\). An \(RG\)-lattice \(M\) is called virtually irreducible if for all \(\alpha\in \text{End}_{RG} (M)\), we have that \(\nu(\text{tr } \alpha)\geq \nu(\text{rk } M)\), and equality holds if and only if \(\alpha\) is invertible. This concept was introduced by \textit{R. Knörr} [Proc. Lond. Math. Soc., III. Ser. 59, 99-132 (1989; Zbl 0676.20006)], and has some nice properties, since a source or a Green correspondent of a virtually irreducible lattice is virtually irreducible, and a lattice affording an absolutely irreducible character is virtually irreducible. The principal aim of this paper is to prove analogues for virtually irreducible lattices of some classical theorems which relate the Brauer correspondence of blocks to irreducible characters. One of these classical results is the following: if the block \(b\) of a subgroup of \(G\) contains an irreducible character \(\zeta\) such that \(\zeta^ G\) is defined, then \(b^ G\) is defined and it contains \(\zeta^ G\). Its analogue for virtually irreducible lattices says that: if \(H\) is a subgroup of \(G\), \(b\) is a block of \(RH\), \(M\) is a virtually irreducible lattice in \(b\) such that \(M^ G= V\oplus W\) with \(V\) virtually irreducible and \(M\nmid W_ H\), then \(B^ G\) is defined and it contains \(V\), and moreover \(\nu(\text{rk } V)= \nu(\text{rk } M)+\nu | G: M|\). This theorem is applied to obtain a new proof of Brauer's first main theorem, and also to extend many of Knörr's results. Another main result is the following generalization of Brauer's third main theorem. Let \(B\) be a block of \(RG\), \(M\) an indecomposable \(RG\)-lattice in \(B\) of height zero, \(H\leq G\) and \(b\) a block of \(RH\) with defect group \(D\) such that \(C_ G (D)\leq H\). Then \(b^ G= B\) if and only if \(b\) contains an indecomposable summand of \(M_ H\) of height zero. A remarkable fact which is pointed out by the author is that the hypotheses of the new theorems are fulfilled in many more cases than the old ones.
generalization of Brauer's third main theorem
scientific article; zbMATH DE number 554266