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 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 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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References
V.M. Buchstaber, E.G. Rees, A constructive proof of the generalized Gelfand isomorphism. Funct. Anal. Appl. 35(4), 257–260 (2001)
V.M. Buchstaber, E.G. Rees, The Gel’fand map and symmetric products. Sel. Math. (N.S.) 8, 523–535 (2002)
V.M. Buchstaber, E.G. Rees, Frobenius n-homomorphisms, transfers and branched coverings. Math. Proc. Camb. Philos. Soc. 144, 1–12 (2008)
C.M. Fortuin, P.W. Kasteleyn, J. Ginibre, Correlation inequalities on some partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971)
I.M. Gelfand, A.N. Kolmogorov, On rings of continuous functions on topological spaces. Dokl. Akad. Nauk SSSR 22, 11–15 (1939); English transl., Selected works of A.N. Kolmogorov, Mathematics and Mechanics, vol. I (Kluwer, Dordrecht 1991), pp. 291–297
D.V. Gugnin, Polynomially dependent homomorphisms and Frobenius n-homomorphisms. Proc. Steklov Inst. Math. 266, 59–90 (2009)
D.V. Gugnin, Topological applications of graded Frobenius n-homomorphisms. Tr. Mosk. Mat. Obs. 72(1), 127–188 (2011); English transl., Trans. Moscow Math. Soc. 72(1), 97–142 (2011)
D.V. Gugnin, Topological applications of graded Frobenius n-homomorphisms II. Trans. Moscow Math. Soc. 73(1), 167–172 (2012)
D.St.P. Richards, Algebraic methods toward higher order probability inequalities II. Ann. Probab. 32, 1509–1544 (2004)
G.-C. Rota, J. Shen, On the combinatorics of cumulants. J. Comb. Theory A 91, 283–304 (2000)
S. Sahi, Higher correlation inequalities. Combinatorica 28, 209–227 (2008)
T.P. Speed, Cumulants and partition lattices. Aust. J. Stat. 25, 378–388 (1983)
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Johnson, K.W. (2019). K-Characters and n-Homomorphisms. In: Group Matrices, Group Determinants and Representation Theory. Lecture Notes in Mathematics, vol 2233. Springer, Cham. https://doi.org/10.1007/978-3-030-28300-1_8
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