K-Characters and n-Homomorphisms

This chapter discusses two situations where the combinatorics behind k-characters appears with no apparent connection to group representation theory. In geometry a Frobenius n-homomorphism is defined essentially in terms of the combinatorics of k-characters. Buchstaber and Rees generalized the result of Gelfand and Kolmogorov which reconstructs a geometric space from the algebra of functions on the space and used Frobenius n-homomorphisms which arise naturally from k-characters. Incidentally they show that given commutative algebras A and B, with certain obvious restrictions on B, a homomorphism from the symmetric product Sⁿ(A) to B arises from a Frobenius n-homomorphism. The cumulants for multiple random variables may be considered as an alternative method to understand higher connections between distributions. For example, if N billiard balls move randomly on a table, the nth cumulant determines the probability of n balls simultaneously colliding. The FKG inequality can be interpreted as an inequality for the lowest cumulant of the random variables f₁ and f₂ (the case n = 2). Richards examined how the inequality could be extended to the higher cumulants. Although there are counterexamples to a direct extension, he stated a result for modified cumulants to which he gave the name “conjugate” cumulants, in the cases n = 3, 4, 5. Sahi subsequently explained that Richards’ definitions could be incorporated in a more general setting by giving a generating function approach. Richards stated a theorem but Sahi later indicated a gap in the proof, so it remains a conjecture, although Sahi proved a special case.