Clarence, I think that this is the two-adic matrix representation that we want. The reflections are listed in the order in which they appear in your TURBO output. The element x satisfies x^2+x+2 and the matrices are mapped over the 2-adics by sending x to the 2-adic solution x' of this equation which is congruent to 0 mod 2. To check the integrality of the matrices, it's useful to know that x' = 0 mod 2 2 mod 4 2 mod 8 10 mod 16 26 mod 32 --- at least I hope so. My latest calculation says that the restriction of this representation to P(2,1) lies by some miracle in the proper subgroup. Writing this up may take a while, since we have to calculate explicitly the group of homotopy classes of self-equivalences of \zeta_2 = B(SU(2)xSU(2)xSU(2) /(Z/2)) and write out its action on the set [\zeta_3, \zeta_2] of homotopy classes of maps from \zeta_3 to \zeta_2. I'm pretty sure I know this, but writing out a formal proof might be painful. The matrices should be the hard part... maybe there's an easier way for the rest of it. Bill (c211) for i:1 thru 21 do display(conjugated_reflection[i]); [ - 1 - 1 0 ] [ ] conjugated_reflection = [ 0 1 0 ] 1 [ ] [ 0 0 - 1 ] [ 1 1 0 ] [ ] conjugated_reflection = [ 0 - 1 0 ] 2 [ ] [ - 2 - 1 - 1 ] [ - 1 0 0 ] [ ] conjugated_reflection = [ 0 - 1 0 ] 3 [ ] [ 2 1 1 ] [ x - 2 x x - 2 ] [ ----- - ----- ] [ 2 2 4 ] [ ] [ x + 2 ] conjugated_reflection = [ 1 0 ----- ] 4 [ 2 ] [ ] [ x ] [ - x - x - - ] [ 2 ] [ x 3 x + 2 ] [ - 0 ------- ] [ 2 4 ] [ ] [ x + 2 ] conjugated_reflection = [ - 1 - 1 - ----- ] 5 [ 2 ] [ ] [ x ] [ - x 0 - - ] [ 2 ] [ x + 2 x x + 2 ] [ - ----- - - - ----- ] [ 2 2 4 ] [ ] [ x ] conjugated_reflection = [ 1 0 - - ] 6 [ 2 ] [ ] [ x ] [ x x - ] [ 2 ] [ 1 1 1 ] [ ] conjugated_reflection = [ 0 - 1 0 ] 7 [ ] [ 0 0 - 1 ] [ - 1 0 - 1 ] [ ] conjugated_reflection = [ 0 - 1 0 ] 8 [ ] [ 0 0 1 ] [ - 1 0 0 ] [ ] conjugated_reflection = [ 0 0 - 1 ] 9 [ ] [ 0 - 1 0 ] [ - 1 0 0 ] [ ] conjugated_reflection = [ 2 1 1 ] 10 [ ] [ 0 0 - 1 ] [ - 1 - 1 - 1 ] [ ] conjugated_reflection = [ 0 0 1 ] 11 [ ] [ 0 1 0 ] [ 1 0 1 ] [ ] conjugated_reflection = [ - 2 - 1 - 1 ] 12 [ ] [ 0 0 - 1 ] [ x 3 x + 2 ] [ - - - ------- 0 ] [ 2 4 ] [ ] [ x ] conjugated_reflection = [ x - 0 ] 13 [ 2 ] [ ] [ x ] [ - 1 - - 1 ] [ 2 ] [ x + 2 x - 2 ] [ 0 ----- - ----- ] [ 4 4 ] [ ] [ x + 2 x + 2 ] conjugated_reflection = [ - x - 1 - ----- - ----- ] 14 [ 2 2 ] [ ] [ x x ] [ x + 1 - - ] [ 2 2 ] [ x - 2 x - 2 x ] [ ----- ----- - ] [ 2 4 2 ] [ ] [ x ] conjugated_reflection = [ - x - - - x ] 15 [ 2 ] [ ] [ x + 2 ] [ 1 ----- 0 ] [ 2 ] [ 3 x + 2 3 x + 2 ] [ x + 1 ------- ------- ] [ 4 4 ] [ ] [ x + 2 x ] conjugated_reflection = [ - x - 1 - ----- - - ] 16 [ 2 2 ] [ ] [ x x + 2 ] [ - x - 1 - - - ----- ] [ 2 2 ] [ x + 2 x + 2 x ] [ - ----- - ----- - - ] [ 2 4 2 ] [ ] [ x ] conjugated_reflection = [ x - x ] 17 [ 2 ] [ ] [ x ] [ 1 - - 0 ] [ 2 ] [ 3 x + 2 3 x + 2 ] [ - x - 1 - ------- - ------- ] [ 4 4 ] [ ] [ x x + 2 ] conjugated_reflection = [ x + 1 - ----- ] 18 [ 2 2 ] [ ] [ x + 2 x ] [ x + 1 ----- - ] [ 2 2 ] [ x 3 x + 2 ] [ - - 0 - ------- ] [ 2 4 ] [ ] [ x ] conjugated_reflection = [ - 1 - 1 - ] 19 [ 2 ] [ ] [ x ] [ x 0 - ] [ 2 ] [ x 3 x + 2 ] [ - ------- 0 ] [ 2 4 ] [ ] [ x ] conjugated_reflection = [ - x - - 0 ] 20 [ 2 ] [ ] [ x + 2 ] [ - 1 - ----- - 1 ] [ 2 ] [ x - 2 x + 2 ] [ 0 - ----- ----- ] [ 4 4 ] [ ] [ x x ] conjugated_reflection = [ x + 1 - - ] 21 [ 2 2 ] [ ] [ x + 2 x + 2 ] [ - x - 1 - ----- - ----- ] [ 2 2 ] (d211) done From mailrus!iuvax!ndmath!ndmath.UUCP!bill@wrath.cs.cornell.edu Fri Jul 21 14:57:23 1989 Received: from wrath.cs.cornell.edu (wrath.ARPA) by mssun7.MSI.CORNELL.EDU (4.0/1.1nn-Cornell_Mathematical_Sciences_Institute) id AA12719; Fri, 21 Jul 89 14:57:19 EDT Received: by wrath.cs.cornell.edu (5.61+2/1.91d) id AA27034; Fri, 21 Jul 89 14:56:25 -0400 Received: by mailrus.cc.umich.edu (5.59/1.0) id AA02029; Fri, 21 Jul 89 14:48:37 EDT Received: from ndmath by iuvax.cs.indiana.edu with UUCP (5.61+/1.3) id AA21541; Fri, 21 Jul 89 13:52:57 -0500 Received: from alibaba. (alibaba.math.nd.edu) by ndmath.math.nd.edu.UUCP (3.2/smail2.5) id AA10463; Fri, 21 Jul 89 13:43:22 EST Received: by alibaba. (3.2/smail2.5) id AA02952; Fri, 21 Jul 89 13:44:43 EST Date: Fri, 21 Jul 89 13:44:43 EST From: mailrus!iuvax!ndmath!bill@wrath.cs.cornell.edu (Bill Dwyer) Message-Id: <8907211844.AA02952@alibaba.> To: mailrus!mssun7.msi.cornell.edu!cww@wrath.cs.cornell.edu Subject: Reversal of fortune! Status: R Incredibly enough, the stuff seems to work out now. Let x be a solution of x^^2+x+2 and let p be a prime of Z[x] which contains x ( so that x is congruent to 0 mod p). My earlier problems were two fold: (1) I didn't correctly normalize the representation of GL(3,F_1) over Z[x] to agree with the standard representation of P(1,2) over Z[x]... I normalized it so that the things agreed on the unipotent radical of P(1,2), but but it was necessary to adjust the change of basis by a further diagonal matrix to get the representations to agree on P(1,2) itself. (2) You CAN'T actually do (1) in any case. In fact, when you normalized this representation of GL(3,F_2) you actually get matrices with entries in Q[x]... these matrices are p-adically integral, but NOT globally integral. A typical entry, for instance, might be x/2. This confused macsyma at least as much as it confused me; I'll send you another letter with the actual matrices. Bill