*K*-Characters and *n*-Homomorphisms

## Abstract

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 *S*^{n}(*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 *n*th 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 *f*_{1} and *f*_{2} (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.

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