By Anthony Bak (auth.), R. Keith Dennis (eds.)
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Extra resources for Algebraic K-Theory: Proceedings of a Conference Held at Oberwolfach, June 1980 Part II
Following Let G be finite and let r be prime to the order of G. diagram commutes for any subgroup K(G) H- = Then the H of G: x K i (~r)H ! I (Xi ~r)m K( )H = × K i where m permutes same conclusion x K, (x ~r)m j the ~ ( j ) t h Proof. factors as ~r the indexing representations factor via "(x ~i-r) . which sends the jth factor to J , ~r if ~r(vj) c S O and via ~-r if ~r(vj) c S O . 6 with ~ = U(n), we conclude R(U(n)), = x (~r ~ ~r )vP(in) " O If p ~ R+(H,U(n)) U(n) p ÷ U(ni). ~r :B~(G) The conclusion The by the map [~r~(in)]H = [e~r(in)]H = × (i ~ a)~P~r(in) for of H.
We need the following observations. 2. if b(v,v) = 0 for all v ~ V. and admits a symplectic forms are skew symmetric, Let (V,b) be an irreducible and conversely orthogonal basis [14, Thm 19]. if char F # 2. F[G]-module. (i) If (ii) If char F = 2 and V e S+, then hV is irreducible (iii) If char F = 2 and V s S@, then 2(V,b) ~ hV. (iv) If char F = 2 and V is non-trivial, char F ~ 2, then (V,b) ~) (V,-b) ~ hV. Identify V with V* via and 3(V,b) ~ (V,b) ~ hV. then b is alternate and V e S± if and only if V admits an invariant nonsingular Proof.
Then O(G) G is equivalent as a permutative category to the weak product of one copy of O for each Ui, one copy 46 of ~ for each rVj, and one copy of equivalent ~ for each rc'W k. to the weak product of correspondingly Proof. Regard Therefore KO(G) G is indexed copies of KO, K, and KSp. O(G) G as the category of orthogonal their isomorphisms. Via M + M ~F Z representations on objects and f + f ® Z is Ui, Vj, or W k and F is R, C, or H, we obtain functors O,~, and ~ to O(G) G. 7. 4. symplectlc K-theory, to the weak product where where of one copy of KSp for each qcUi, one copy of K for each qVj, and one copy of K0 for each Wk; compare Kawakubo §4.