### Abstract

Let W be a finite reflection (or Coxeter) group and K: R^{2} → R. We define the concept of total positivity for the function K with respect to the group W. For the case in which W = G_{n}, 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)= e^{xy}, (x, y) ∈ ℝ^{2}, 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 language | English (US) |
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Pages (from-to) | 60-87 |

Number of pages | 28 |

Journal | Journal of Approximation Theory |

Volume | 82 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1995 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Numerical Analysis
- Mathematics(all)
- Applied Mathematics

### Cite this

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*Journal of Approximation Theory*, vol. 82, no. 1, pp. 60-87. https://doi.org/10.1006/jath.1995.1068

**Total positivity, finite reflection groups, and a formula of Harish-Chandra.** / Gross, Kenneth I.; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Total positivity, finite reflection groups, and a formula of Harish-Chandra

AU - Gross, Kenneth I.

AU - Richards, Donald

PY - 1995/1/1

Y1 - 1995/1/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=3042513106&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3042513106&partnerID=8YFLogxK

U2 - 10.1006/jath.1995.1068

DO - 10.1006/jath.1995.1068

M3 - Article

AN - SCOPUS:3042513106

VL - 82

SP - 60

EP - 87

JO - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

IS - 1

ER -