Algebraic Topology Waterloo 1978: Proceedings of a by G. Carlsson, R. James Milgram (auth.), Peter Hoffman, Victor

By G. Carlsson, R. James Milgram (auth.), Peter Hoffman, Victor Snaith (eds.)

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Extra resources for Algebraic Topology Waterloo 1978: Proceedings of a Conference Sponsored by the Canadian Mathematical Society, NSERC (Canada), and the University of Waterloo, June 1978

Example text

Then (a) follows from the Hasse-Schilling norm theorem [S-E]. To show (b), note that the fixed set of Hence. since Di = M2(~) ~K9" T is one dimensional. by assumption, we must have that T "" the "symplectic" involution: T (a b) (d -b) = c But L+(M (F) 2 T sp d ) = L-(M (F) 2 -c , a T ord ) o where T ord is the usual transpose and (b) follows. 17 (b). with the same relation as occurs in 25 §4. - q An application: Ll' (Z(n» The representation ring for n for a finite a finite 2-group. 2-~. Recall that any irreducible representation of a finite p-group is monomial.

4: fixing Let fix set " x E k(a,b) Let w E k(x/ so 4 non-trivially on Let space over fA (y) treA) , NeA) under 1 < as > then there is a type I(B) involution ,). be - treA)y + NeA) both belong to k U E K(A)l Now, take ,(U)A k ,(A U) C , A E F, A if. K then Let 2 we have x E k(x) C K < a ,b > , so a type II involution on be the fixed set of k w-l,(y)w ,'(y) K, with fixed field F Y then i f follows. Proposition 6. 5: Let, or x. f : Proof: > 1 K Then F K < a,b > U A = -A U , u -h(U) so -A(U + LeU»~.

Recall that any irreducible representation of a finite p-group is monomial. 2 -1 -1 - P i ) , q;,(p i-I) q)(p i + P i ) , q;(p 2i 2 2 2 2 26 If the center of (f) is of the last type then is of the second type then If the center 21. l(iii). Z/ . 1 (i) where -1 9:(p i+ P i ) 2 2 (a) 0 (b) '2 I is is the Quaternion algebra with center D 1 , invariants '2 at all infinite primes and at all finite primes i > 2 at i 2 , 0 otherwise 2 (it) -1 M2 (Q:(P i+ P . » 21. - Pi» 21. g. [Se] • at CD primes and case (b) it is M2 (9: p) What (i) M2 (F p ) iI:z (i,j) otherwise.

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