Total positivity, finite reflection groups, and a formula of Harish-Chandra

Kenneth I. Gross, Donald Richards

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

Let W be a finite reflection (or Coxeter) group and K: R2 → R. We define the concept of total positivity for the function K with respect to the group W. For the case in which W = Gn, the group of permutations on n symbols, this notion reduces to the classical formulation of total positivity. We prove a basic composition formula for this generalization of total positivity, and in the case in which W is the Weyl group for a compact connected Lie group we apply an integral formula of Harish-Chandra (Amer. J. Math.79 (1957), 87-120) to construct examples of totally positive functions. In particular, the function K(x, y)= exy, (x, y) ∈ ℝ2, is totally positive with respect to any Weyl group W. As an application of these results, we derive an FKG-type correlation inequality in the case in which W is the Weyl group of SO(5).

Original languageEnglish (US)
Pages (from-to)60-87
Number of pages28
JournalJournal of Approximation Theory
Volume82
Issue number1
DOIs
StatePublished - Jan 1 1995

Fingerprint

Total Positivity
Reflection Group
Weyl Group
Finite Group
Correlation Inequalities
Lie groups
Analytic group
Coxeter Group
Integral Formula
Permutation
Formulation
Chemical analysis

All Science Journal Classification (ASJC) codes

  • Analysis
  • Numerical Analysis
  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "Let W be a finite reflection (or Coxeter) group and K: R2 → R. We define the concept of total positivity for the function K with respect to the group W. For the case in which W = Gn, the group of permutations on n symbols, this notion reduces to the classical formulation of total positivity. We prove a basic composition formula for this generalization of total positivity, and in the case in which W is the Weyl group for a compact connected Lie group we apply an integral formula of Harish-Chandra (Amer. J. Math.79 (1957), 87-120) to construct examples of totally positive functions. In particular, the function K(x, y)= exy, (x, y) ∈ ℝ2, is totally positive with respect to any Weyl group W. As an application of these results, we derive an FKG-type correlation inequality in the case in which W is the Weyl group of SO(5).",
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Total positivity, finite reflection groups, and a formula of Harish-Chandra. / Gross, Kenneth I.; Richards, Donald.

In: Journal of Approximation Theory, Vol. 82, No. 1, 01.01.1995, p. 60-87.

Research output: Contribution to journalArticle

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