Nilpotent elements in the Green ring
Let k be a field of characteristic p and G a finite group. Theorem 2.1 is a generalization of a theorem of Landrock for the absolutely irreducible case: Suppose that M and N are absolutely indecomposable kG-modules. Then \(M\otimes N\) has the trivial module k as a direct summand if and only if the following two conditions are satisfied: (i) \(M\cong N^*\), (ii) \(p\nmid \dim N\). Moreover if k is a component of \(N^*\times N\) then it has multiplicity one. Section 2 also gives a generalization of 2.1 to Scott modules. M is absolutely p-divisible if, for every extension field \(k_ 1\) of k, every direct summand of \(k_ 1\otimes_ kM\) has k-dimension divisible by p. Let a(G;p) be the linear span in the Green ring a(G) \((A(G)={\mathbb{C}}\otimes_{{\mathbb{Z}}}a(G))\) of such module classes M. Then 2.1 readily implies that a(G;p) is an ideal. Furthermore A(G)/A(G;p) has no nilpotent elements. A new proof that A(G,Cyc) has no nilpotent elements is given. Finally for an indecomposable M a short exact sequence \(0\to \Omega (M)\to Y\to \Omega^{-1}(M)\to 0\) is obtained, which is split if and only if \(p| \dim M.\) In Section 3 a general technique is given for producing an infinity of nilpotent and idempotent elements in the Green ring. Previous methods of Zemanek are replaced by calculations in the cohomology ring of certain modules. In Section 4 the conditions required for the techniques of Section 3 are verified for \(G\cong {\mathbb{Z}}/(p^ r)\times {\mathbb{Z}}/(p^ s)\) by detailed calculations. These are given more scope by realizing G as a quotient of a p-group, etc.
scientific article; zbMATH DE number 3989621
scientific article