K-Characters and n-Homomorphisms

Research output: Chapter in Book/Report/Conference proceedingChapter

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 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.

Original languageEnglish (US)
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Verlag
Pages271-286
Number of pages16
DOIs
StatePublished - Jan 1 2019

Publication series

NameLecture Notes in Mathematics
Volume2233
ISSN (Print)0075-8434
ISSN (Electronic)1617-9692

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

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