Permutation Representations
Studies in advanced mathematics · 2006
In Memory of Irving Reiner By a permutation representation (G, X) we mean a group G together with a left G-set X.The orbit set G \ X has as its elements the orbits [x] Gx for x X.The question which we shall address is to what extent the permutation representation can be recovered from information about orbit sets alone.Its motivation comes from homotopy theory and we shall indicate below how our remarks apply there.Evidently from G \ X alone we cannot reconstruct G and X.But for any set U the U-th power of X, i.e., the set X v of functions x: U X, is again a G-set, with (gx)i g(xi) for g G, U. Furthermore if F: U V then xf: X V Xv, the composition with f, is a G-equivariant map.Thus U G\X u, f G\X / defines a functor Orb(G, X)" Sets p Sets, the orbit-functor of (G, X).We shall see that from this functor we can indeed reconstruct, up to a suitable equivalence, the permutation representation. I. Orbital functorsWe shall adopt the following conventions.A natural number n is the set (0,1,...,n-I} of its predecessors, so that 0=.If f: mm' and g" n n' are maps of natural numbers then f+g: m+nm'+n' is the ordinal sum in the obvious sense.For any set W, _W: 0 W and W: W 1 are the unique maps; we shall also on occasion write W for the identity map.For sets U, V, A: U U V denotes the generalized diagonal map.If F: Sets p Sets then the natural transformation dg: F(U V) (FU) V is defined by prjdg F(U j) where j" 1 V is an element of V.