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

Fingerprint

Cumulants
Homomorphisms
Frobenius
Homomorphism
Combinatorics
Ball
FKG Inequality
Random variable
Symmetric Product
Group Representation
Commutative Algebra
Billiards
Group Theory
Representation Theory
Generating Function
Counterexample
Character
Lowest
Table
Restriction

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

Johnson, K. W. (2019). K-Characters and n-Homomorphisms. In Lecture Notes in Mathematics (pp. 271-286). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_8
Johnson, Kenneth W. / K-Characters and n-Homomorphisms. Lecture Notes in Mathematics. Springer Verlag, 2019. pp. 271-286 (Lecture Notes in Mathematics).
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Johnson, KW 2019, K-Characters and n-Homomorphisms. in Lecture Notes in Mathematics. Lecture Notes in Mathematics, vol. 2233, Springer Verlag, pp. 271-286. https://doi.org/10.1007/978-3-030-28300-1_8

K-Characters and n-Homomorphisms. / Johnson, Kenneth W.

Lecture Notes in Mathematics. Springer Verlag, 2019. p. 271-286 (Lecture Notes in Mathematics; Vol. 2233).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Johnson KW. K-Characters and n-Homomorphisms. In Lecture Notes in Mathematics. Springer Verlag. 2019. p. 271-286. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-030-28300-1_8