Pointed groups and construction of modules
Let \({\mathcal O}\) be a complete discrete valuation ring with residue field k of characteristic p. Let A be an \({\mathcal O}\)-algebra, finitely generated and free as an \({\mathcal O}\)-module. Let \(A^*\) be the group of invertible elements of A. Let G be a finite group. A is an interior G-algebra if there is a group homomorphism \(G\to A^*\) and G acts by inner automorphisms, giving A an \({\mathcal O}G\)-module structure. For an \({\mathcal O}G\)-module M and a subgroup H (\(\leq G)\), \(M^ H\) is the \({\mathcal O}H\)- submodule of M of those elements fixed by H. \({\mathcal P}(A)\) is the set of \(A^*\)-conjugacy classes of primitive idempotents of A. A pointed group \(H_{\beta}\) on A consists of a subgroup H together with \(\beta\in {\mathcal P}(A^ H)\); \(\beta\) is a point of H on A. For instance, if \(A={\mathcal O}G\), then \(A^ G\) is the centre of \({\mathcal O}G\) and \(\beta\) is a set containing one primitive central idempotent of \({\mathcal O}G\) (i.e. a block). Again for an \({\mathcal O}G\)-module M, one has an interior G- algebra structure on \(A=End_{{\mathcal O}}(M)\); \(A^ G=End_{{\mathcal O}G}(M)\) and a primitive idempotent \(i\in \beta \in p(A^ G)\) is the projection endomorphism onto an \({\mathcal O}G\)-indecomposable direct summand of \(M.\) P\({}_{\gamma}\) is called local if \(i\in \gamma\) does not lie in the sum of images of the trace functions from various \(A^ K\) for proper subgroups K of P; then P is necessarily a p-subgroup of G. Pointed subgroups are partially ordered by inclusion of subgroups and ``inclusion of idempotents''. A maximal local pointed subgroup of \(H_{\beta}\) is called a defect pointed group of \(H_{\beta}\); then H acts transitively on the set of defect pointed groups of \(H_{\beta}\). When \(A={\mathcal O}G\), a defect pointed group \(P_{\gamma}\) of the block \(H_{\beta}\) corresponds to the defect group of the block. When \(A=End_{{\mathcal O}}(M)\), the defect pointed group \(P_{\gamma}\) of the indecomposable direct summand \(G_{\alpha}\) of M gives its vertex and source. Let \({\mathcal M}({\mathcal O}G)\) be the Green ring consisting of \(\pm\) combinations (virtual modules) of indecomposable (\({\mathcal O}\)-free) \({\mathcal O}G\)-module isomorphism classes. The residual Green ring \({\mathcal R}{\mathcal M}({\mathcal O}G)\) is the quotient of \({\mathcal M}({\mathcal O}G)\) by the ideal generated by those module classes induced up from proper subgroups of G. For each subgroup \(H\leq G\), there is the restriction map \({\mathcal M}({\mathcal O}G)\to {\mathcal M}({\mathcal O}H)\). Composition gives ring homomorphisms \(rd_ H: {\mathcal M}({\mathcal O}G)\to {\mathcal R}{\mathcal M}({\mathcal O}H)\). It is known that a virtual module is determined by the totality of its residuals (images) in \({\mathcal R}{\mathcal M}({\mathcal O}H)\) with \(H/O_ p(H)\) cyclic. However elements of \({\mathbb{Q}}\otimes_{{\mathbb{Z}}}{\mathcal M}({\mathcal O}G)\) are also separated. Even in the virtual module case it is difficult to determine whether a set of residuals has arisen from an element of \({\mathcal M}({\mathcal O}G)\) (as distinct from an element of \({\mathbb{Q}}\otimes {\mathcal M}({\mathcal O}G))\). One motivation of this paper is the determination of such conditions on a set of residuals to give a construction of a virtual module or element of \({\mathcal M}({\mathcal O}G)\). Such constructions for characters were carried out by the author [in Math. Z. 166, 117-129 (1979; Zbl 0387.16006)]. The required conditions arise from a suitable generalization of Brauer's Second Main Theorem. Let M be an \({\mathcal O}G\)-module, \(H_{\beta}^ a \)pointed group on \({\mathcal O}G\) and, if \(i\in \beta\), write \(\chi^{\beta}\) for the character or trace of the \({\mathcal O}H\)-module \(i\cdot M\). Let u be a p- element of G and s a \(p'\)-element in \(C_ G(u)\). Then \[ (1)\quad \chi^{\alpha}(us)=\sum_{\epsilon}\phi^{\alpha}_{\epsilon}(s)\chi^{\epsilon}(u), \] where the sum is over the set of local points (blocks) \(\epsilon\) of \(\) (i.e. the irreducible \({\mathcal O}\)-modules). The values \(\chi^{\epsilon}(u)\) correspond to the generalized decomposition numbers \((d^ P_{ij}\) \((P=))\) of Brauer and \(\phi^{\alpha}_{\epsilon}(s)\) correspond to values of the modular characters of \(C_ G(u)\). In particular, for \(s=1\), (1) gives (2) \(\chi^{\delta}(u)=\sum_{\epsilon}m^{\delta}_{\epsilon}\chi^{\epsilon}(u)\), for any pointed subgroup \(Q_{\delta}\subset G_{\alpha}\), (2) defining the numbers \(m^{\delta}_{\epsilon}\). - In the above cited paper, (1) and (2) were derived for the character of an A-module M, where A is an interior G-algebra. It was shown that for any local pointed subgroup \(Q_{\delta}\) of \(G_{\alpha}^ a \)virtual character \(\lambda^{\delta}\) of Q is chosen so that identities (2) hold, then (3) equalities (1) define a virtual character of G. - In the above considerations of characters, subgroups \(H=P\times K=\times \) of G arose. In the Green ring case, all subgroups H with \(H/O_ p(H)\) cyclic must be considered. (1), (2) and (3) are generalized with \({\mathbb{C}}\)-value central functions of G replaced by the Green ring over \({\mathbb{Q}}\), virtual characters by virtual modules and values of virtual characters by residuals of virtual modules in \({\mathcal R}{\mathcal M}({\mathcal O}H)\) for \(H/O_ p(H)\) cyclic. The fact that H is not the direct product of a p-group and a \(p'\)-cyclic group causes the difficulty. This necessitates a calculus of \(k^*\)- central extensions of subgroups and of decomposition modules whose residuals replace the generalized decomposition numbers. The second last section concerns questions of uniqueness of the ``generalized decomposition numbers''. The last section gives a local formula for computing the usual bilinear form on the Green ring.
scientific article; zbMATH DE number 4075370
scientific article