Young diagrammatic methods in the homology of groups
It was shown by H. Weyl that the irreducible representations of the classical groups GL(n,\({\mathbb{C}})\), Sp(n,\({\mathbb{C}})\), Sp(2n,\({\mathbb{C}})\) and \(SO(2n+1,{\mathbb{C}})\) are essentially in one to one correspondence with Young diagrams of depth at most n. Then K. Koike and I. Terada generalized the Littlewood-Richardson Rule for decomposing tensor products of representations for linear groups into sums of irreducible components to the symplectic and orthogonal groups. In the paper under review, the correspondence between representations and diagrams and the work of Koike and Terada are exploited to determine the homology of the groups \(SL(n,p^ 2)\), \(Sp(2n,p^ 2)\), \(SO(2n+1,p^ 2)\) with \({\mathbb{Z}}/p{\mathbb{Z}}\) coefficients in dimensions 1 through 5, with the exception of the group \(H_ 5(SL(n,p^ 2),{\mathbb{Z}}/p{\mathbb{Z}})\), for which only bounds are determined. In addition, the p primary component of the homology of the groups \(SL(n,p^ 2)\), \(Sp(2n,p^ 2)\), \(SO(2n+1,p^ 2)\) with \({\mathbb{Z}}\) coefficients is found in dimensions 1 through 5, with the exception of the group \(H_ 5(SL(n,p^ 2),{\mathbb{Z}})\), for which only bounds are determined. These groups are found by computing the relevant entries and differentials in certain spectral sequences involving the corresponding classical groups and the associated Lie algebras.
scientific article; zbMATH DE number 4142310
scientific article