Total positivity, finite reflection groups, and a formula of Harish- Chandra
Let \(W\subset O(n)\) be a finite reflection group acting on \(\mathbb{R}^n\). For any region \(D\subset \mathbb{R}^2\) and each function \(K: D\to \mathbb{R}\) put \[ D_W K(s, t):= \sum_{w\in W} \text{det } w \prod_{j=1}^n K \bigl(s_j, w(t)_j \bigr) \] for \(s,t\in \mathbb{R}^n\) with \((s_j, w(t)_j)\in D\) for all \(w\in W\) and \(j=1, \dots,n\). Generalizing the classical concept of total positivity due to S. Karlin, the authors say that \(K\) is totally positive definite with respect to \(W\) if \(D_W K(S, t)\geq 0\) for all admissible \(s\), \(t\) contained in the fundamental Weyl chamber in \(\mathbb{R}^n\). In case of \(W= S_n\), the symmetric group of order \(n\), this notion reduces to the classical one. For this generalized concept of total positivity a basic composition theorem is given. Moreover, if \(W\) is the Weyl group of a compact connected Lie group, then an integral formula of Harish-Chandra is used to construct examples of totally positive definite functions. In particular it turns out that \(K(x,y):= e^{xy}\) \((x,y\in \mathbb{R})\) is totally positive definite with respect to any Weyl group \(W\). Finally, as an application, an FKG-type correlation inequality is derived for the Weyl group of \(SO(5)\).
scientific article; zbMATH DE number 782144