On exponents and Auslander-Reiten components of irreducible lattices

Abstract

Let G be a finite group, and let R be a complete discrete rank one valuation ring of characteristic zero with maximal ideal max R pR, and residue class field RapR of characteristic p b 0. The notion of the exponent of an RG-lattice L is due to J. F. Carlson and the first author [1]. In this note we use it to show that any nonprojective absolutely irreducible RG-lattice L with indecomposable factor module " L LapL lies at the end of its connected component V of the stable Auslander-Reiten quiver G s RG of the group ring RG. Since such lattices L belong to p-blocks B with non-trivial defect groups dB we also study some relations between the order of dB and the exponent expL.Introduction. Let G be a finite group and let R be a complete discrete rank one valuation ring with radical JR pR, quotient field K QR of characteristic zero, and residue class field F RapR of characteristic p b 0. Let n be the p-adic valuation of R. The group ring of G over R is denoted by RG. A finitely generated RG-module M is a lattice, if it is also a free R-module. Its rank over R is denoted by rkM.In [1] J. F. Carlson and the first author defined the exponent expM of an RG-lattice M as the least power p a of p such that the multiplication of the elements of M by p a factors through a projective RG-lattice. 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 .In this note we give two applications of these elementary results about the exponents of certain virtually irreducible lattices. They show that the notion of the exponent is a useful tool for proving general results on lattices.In the first section we show that a non projective absolutely irreducible RG-lattice L such that L LapL is indecomposable lies at the end of its connected component V of the stable Auslander-Reiten quiver G s RG of the group ring RG, see Theorem 1.3.In the second section we consider p-blocks B of a finite group G with defect group dB G D. Theorem 2.1 asserts that B contains a virtually irreducible RG-lattice L with exponent expL p if and only if jDj p.

Almost split sequences for Knorr lattices

“…The case when a = 1 follows from Proposition 4.9, which shows how restrictive the existence of a Knörr lattice of exponent π in a given block is. This extends a result of Jones, Kawata and Michler [9,Theorem 2.1].…”

Auslander-Reiten components for some Knörr lattices

“…We mention two things. First, the main result of [9] also makes this assumption. Second, if χ is an absolutely irreducible character of G with values contained in R, then Thompson points out in [15] that there is always an RG-lattice affording χ that remains indecomposable (mod π).…”

“…Turning now to almost split sequences, it is shown in [9] that such a sequence terminating in a non-projective absolutely irreducible lattice that remains indecomposable (mod π) has a middle term whose projective-free part is indecomposable. The first main result here, proved in Section 2, generalizes this to a larger family of Knörr lattices.…”