关于有限群表示范畴上几乎可裂序列的注记

摘要

在有限群局部表示理论中,Green对应相当重要,由此可得到一些有趣的应用,本文给出了几乎可裂序列的Green对应.证明了如下结果:设X是不可分解非投射kG-模,Y是相应的不可分解非投射kL-模.那么(i)O→Ω2(X)→(X⊕U)0→X→O是可裂正合列当且仅当0→Ω2(Y)→(Y⊕U)0→Y→0是可裂正合列;(ii)0→Ω2(X)→(X⊕U)0→X→0是几乎可裂正合列当且仅当0→Ω2(Y)→(Y⊕U)0→Y→0是几乎可裂正合列.%An application of Green correspondence is given by using almost split sequences. In the local representation theory of groups, the Green correspondence is important. The main conclusion of this paper is that for an indecomposable non-projective kG-module X and its corresponding indecomposable non-projective kL-module Y, (i)0→ Ω2 (X)→(X(○)U)0→X→0 is split if and only if 0→Ω2 (Y)→(Y(○)U)0→Y→0 is split; (ii)0→Ω2 (X)→(X(○)U)0→X →0 is almost split if and only if 0→Ω2 (Y)→(Y(○)U)0→Y→0 is almost split.