This section is based on joint work with Sutherland. Recall that dn denotes the self-map of Pn,k defined by reflection in the last coordinate hyperplane. We say that Pn, k is neutral (elsewhere outsimple) if dn ≃ 1, and define S-neutral similarly. If n and k are both odd then Pn,k is neutral, as remarked in §7. If n is even then dn has degree −1 on the integral homology Hn−1,(Pn,k) = Z, and so Pn,k is not S-neutral. Thus the interest resides in the case when n is odd and k even. Notice that Pk+1,k = Pk is neutral for all even values of k. In the course of §6 we have already proved

Proposition (21.1). Suppose that Pn,k is S-neutral, where n is odd and k even. Then Pm+n,kand Pm+k−n,kare S-neutral, whenever m ≡ 0 mod âk.

Here âk is as in (1.10). Now consider Vn,k as a Z2 -space under the outer automorphism which changes the sign of the last row and column of each matrix. Since the inclusion Pn,k → Vn,k is a Z2 - map it follows at once from (3. 4) that Pn,k is neutral if Vn,k is neutral and n ≥ 2k.