How close can we come to a parity function when there isn't one?

Consider a group G such that there is no homomorphism f : G → {±1}. In that case, how close can we come to such a homomorphism? We show that if f has zero expectation, then the probability that f(xy) = f(x)f(y), where x, y are chosen uniformly and independently from G, is at most 1/2(1 + 1/ √ d), where d is the dimension of G’s smallest nontrivial irreducible representation. For the alternating group An, for instance, d = n − 1. On the other hand, An contains a subgroup isomorphic to Sn−2, whose parity function we can extend to obtain an f for which this probability is 1/2(1 + 1/ ( n 2 ) ). Thus the extent to which f can be “more homomorphic” than a random function from An to {±1} lies between O(n) and Ω(n). The symmetric group Sn has a parity function, i.e., a homomorphism f : Sn → {±1}, sending even and odd permutations to +1 and −1 respectively. The alternating group An, which consists of the even permutations, has no such homomorphism. How close can we come to one? What is the maximum, over all functions f : An → {±1} with zero expectation, of the probability Pr x,y [f(x)f(y) = f(xy)] , where x and y are chosen independently and uniformly from An? We give simple upper and lower bounds on this quantity, for groups in general and for An in particular. Our results are easily extended to functions f : G → C, but we do not do this here. Our main result is the following: Theorem 1. Let G be a group, and let f : G → {±1} such that Ef = 0. Then Pr x,y [f(x)f(y) = f(xy)] ≤ 1 2 ( 1 + 1 √ d ) , where d = minρ6=1 dρ is the dimension of the smallest nontrivial irreducible representation of G. Thus if G is quasirandom in Gowers’ sense [1]—that is, if minρ6=1 dρ is large—it is impossible for f to be much more homomorphic than a uniformly random function. For An in particular, the dimension of the smallest nontrivial representation is d = n−1, so Prx,y[f(x)f(y) = f(xy)]−1/2 = O(1/ √ n). If f is a class function, i.e., if f is invariant under conjugation so that f(xyx) = f(y) for all x, y ∈ G, then we can tighten this bound from 1/ √ d to 1/d: