Young diagrammatic methods in the homology of groups

Abstract

In 1939, H. Weyl showed that the irreducible representations of the classical groups GL(n, C), Sp(2n, C) and SO(2n+1, C) are essentially in one to one correspondence with Young diagrams of depth at most n. In 1987, 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 this paper the correspondence between representations and diagrams and the work of Kioke and Terada are exploited to determine the homology of the groups SL( n, p 2), Sp(2 n, p 2), SO(2 n+1, p 2) with Z / p Z coefficients in dimensions 1 through 5, with the exception of the group H 5(SL(n,p 2), Z/p Z) , for which only bounds are determined. In addition, the p primary component of the homology of the groups SL( n, p 2), Sp(2 n, p 2), SO(2 n+1, p 2) with Z coefficients is found in dimensions 1 through 5, with the exception of the group H> 5(SL(n,p 2), Z) , for which only bounds are determined. These groups are found by computing the relevant entries and differentials in the spectral sequences H s(SL(n,p),H t(sl(n,p),R)) ⇒ H s+t(SL(n,p 2),R), H s(Sp(2n,p),H t(sp(2n,p),R)) ⇒ H s+t(Sp(2n,p 2),R), H s(SO(2n+1,p),H t(so(2n+1,p),R)) ⇒ H s+t(SO(2n+1,p 2),R) , where sl( n, p), sp(2 n, p), and so(2 n+1, p) are the additive modules of their respective Lie algebras and R= Z or Z/p Z . The methods used are the following. The groups H t ( g, k), for g= sl n , sp 2 n , or so 2 n+1 are of the form Λ i ( g)⊗ S j ( g), where Λ i () represents the ith exterior power and S j () the jth symmetric power of the relevant module. The modules Λ i ( g)⊗ S j( g) can be decomposed into their irreducible components via the Littlewood-Richardson Rule or the Kioke-Terada generalization. Once the irreducible components of the tensor products are known, the entries in the spectral sequences can be found by utilizing results if Friedlander and Parshall. The work of Friedlander and Parshall reduces the computations of the homology groups to calculations of generalized exponents. Once the relevant entries in the spectral sequences are known, the differentials in the spectral sequences must be determined. This is done by relating the relevant differentials to a known differential in a linear spectral sequence calculated by Evens and Friedlander via the hyperbolic map.