Alan Sokal | 2024-09-24 at 3:30 PM, at University of Pennsylvania, DRL room 4C6
This talk is organized jointly with the Penn/Temple probability seminar.
Abstract: A matrix \(M\) of real numbers is called totally positive if every minor of \(M\) is nonnegative. Gantmakher and Krein showed in 1937 that a Hankel matrix \(H = (a_{i+j})_{i,j \ge 0}\) of real numbers is totally positive if and only if the underlying sequence \((a_n)_{n \ge 0}\) is a Stieltjes moment sequence, i.e. the moments of a positive measure on \([0,\infty)\). Moreover, this holds if and only if the ordinary generating function \(\sum_{n=0}^\infty a_n t^n\) can be expanded as a Stieltjes-type continued fraction with nonnegative coefficients: \[\sum_{n=0}^{\infty} a_n t^n \;=\; \cfrac{\alpha_0}{1 - \cfrac{\alpha_1 t}{1 - \cfrac{\alpha_2 t}{1 - \cfrac{\alpha_3 t}{1- \cdots}}}}\] (in the sense of formal power series) with all \(\alpha_i \ge 0\). So totally positive Hankel matrices are closely connected with the Stieltjes moment problem and with continued fractions.
Here I will introduce a generalization: a matrix \(M\) of polynomials (in some set of indeterminates) will be called coefficientwise totally positive if every minor of \(M\) is a polynomial with nonnegative coefficients. And a sequence \((a_n)_{n \ge 0}\) of polynomials will be called coefficientwise Hankel-totally positive if the Hankel matrix \(H = (a_{i+j})_{i,j \ge 0}\) associated to \((a_n)\) is coefficientwise totally positive. It turns out that many sequences of polynomials arising naturally in enumerative combinatorics are (empirically) coefficientwise Hankel-totally positive. In some cases this can be proven using continued fractions, by either combinatorial or algebraic methods; I will sketch how this is done. There is also a more general algebraic method, called production matrices. In a vast number of cases, however, the conjectured coefficientwise Hankel-total positivity remains an open problem.
Alan Sokal is also giving a public lecture on Science and Ideology on Monday, September 23, 2024 at 5:15 PM in 402 Claudia Cohen Hall.