Virtually Irreducible Lattices

Abstract

To Bertram Huppert on his 60th birthdayRepresentation theory of finite groups has to a large extent been character theory, i.e. the study of traces. The present paper stands in that tradition but instead of looking at traces of group elements in a given representation, it focuses on traces of endomorphisms of that representation. The group G comes in by the backdoor: first, multiplication with a conjugacy class sum is an endomorphism and second, much more important, for every subgroup H of G one has a map T% from the //-endomorphisms into the G-endomorphisms; the trivial observation that tr Tu(a) = \G : H\ tr a is exploited extensively.The starting point of the investigation was the following problem. Let M be an /?G-lattice affording an irreducible character %; let D be a vertex and 5 a £>-source of M, so 5 is an indecomposable /?D-lattice with vertex D. Which (if any) additional properties of S can be deduced from the fact that % is irreducible? Such additional properties are highly desirable, since the class of all finitely generated indecomposable RD-lattices is far too large to be useful.A partial answer to this question is given in this paper: S will be 'virtually irreducible' (defined in 1.2 below). A systematic study of virtually irreducible lattices shows that this class of lattices is closed under standard constructions like the Heller operator, taking duals, the Green correspondence, and taking sources. Under additional assumptions, it is also reasonably well-behaved under induction and restriction from and to subgroups. These results are developed in the first three paragraphs. There were some tempting sidetracks which I could not resist. In 1.12, a short proof is given for Ito's theorem that the degree of an irreducible character divides the index of a normal abelian subgroup. In 2.11, it is shown that an irreducible character which vanishes on all elements of order p will vanish on all elements with an order divisible by p.In § 4, Clifford Theory is used to study the vertices of virtually irreducible lattices. The main result is 4.15 which states that the defect group D of a block is abelian if and only if all virtually irreducible lattices in that block have D as a vertex. This characterization of blocks with abelian defect group is loosely related to Brauer's height 0 conjecture. However, it is not true in general that all virtually irreducible lattices in a block with abelian defect group are of height 0; a counter-example is given in 4.20.Finally, in § 5, a version of Frobenius reciprocity is given for virtually irreducible lattices (5.3), and in 5.8, it is shown that (rk U) p ^(rkM) p , if U | M ® X for some X and M is virtually irreducible.

On exponents and Auslander-Reiten components of irreducible lattices

“…Therefore (*) implies that trt k Á jW X Tj Á p kp mÀ1 X Hence ntrt nrkS, because pBk. As 1 À x p 0 mod pR, t is not invertible in End RV S. Hence S is not a virtually irreducible RV-lattice by Definition 1.2 of [3], a contradiction.…”

“…Let S be a source of L. Then S is a virtually irreducible RV-lattice by Corollary 2.5 of [3]. Hence expS expL p by Lemma 3.1 of [1].…”

“…P r o o f. Let L be a virtually irreducible RG-lattice of the p-block B with expL p and vertex vxL G V. Then Corollary 4.11 of [3] asserts that there is a defect group D of B with center Z j 1 such that Z % C D V % V % D.…”

“…If M is a virtually irreducible RG-lattice in the sense of R. Knörr [3], then expM p m for some non-negative integer m, because nexpM nrk R M njGj by Proposition 4.3 of [1]. Important examples of virtually irreducible RG-lattices are the R-forms of absolutely irreducible characters c of G. If such an R-form M belongs to a p-block B of G with defect d, and c has height htc h, then expM p dÀh .…”

Contributions to Modular Representation Theory of Finite Groups

“…The following beautiful result of Knorr [57] is perhaps the most general contribution to Brauer's Height zero conjecture. Theorem 4.…”