##
**Pointed groups and construction of modules.**
*(English)*
Zbl 0658.20004

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 \(<u>\) (i.e. the irreducible \({\mathcal O}<u>\)-modules). The values \(\chi^{\epsilon}(u)\) correspond to the generalized decomposition numbers \((d^ P_{ij}\) \((P=<u>))\) 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 \(H=P\times K=<u>\times <s>\) 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.

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 \(<u>\) (i.e. the irreducible \({\mathcal O}<u>\)-modules). The values \(\chi^{\epsilon}(u)\) correspond to the generalized decomposition numbers \((d^ P_{ij}\) \((P=<u>))\) 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 \(H=P\times K=<u>\times <s>\) 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.

Reviewer: S.B.Conlon

### MSC:

20C20 | Modular representations and characters |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

### Keywords:

interior G-algebra; primitive idempotents; pointed group; trace functions; maximal local pointed subgroup; defect pointed groups; defect group; Green ring; residual Green ring; virtual module; Brauer’s Second Main Theorem; blocks; generalized decomposition numbers; modular characters; virtual character
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DOI

### References:

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