SOME HOMOLOGY REPRESENTATIONS FOR GRASSMANNIANS IN CROSS-CHARACTERISTICS

摘要

Let F be the finite field of q elements and let P(n, q) denote the projective space of dimension n - 1 over F. We construct a family H-(k,i)~n of combinatorial homology modules associated to P(n, q) for coefficient fields of positive characteristic co-prime to q. As FGL(n, q)-representations these modules are obtained from the permutation action of GL(n,q) on the Grassmannians of F~n. We prove a branching rule for H-(k,i)~n and use this to determine the homology representations completely. Our results include a duality theorem and the characterization of H-(k,i)~n through the standard irreducibles of GL(n,q) over F. Bibliography: 14 titles.