Total Positivity
Abstract.
A totally positive matrix is a real matrix all of whose
minors are positive. My lectures will give a broad survey of the
relation between total positivity and combinatorics:
- Total positivity and networks: I will discuss the parametrization
of totally positive matrices dating back to Loewner and Whitney, and
the work of Brenti and Lindström-Gessel-Viennot relating total
positivity to non-intersecting paths in networks. I will then discuss
the generalization of total positivity to Grassmannians in the works
of Lusztig and Postnikov.
- Total positivity and statistical mechanics: there is a close
relation between total positivity and certain statistical mechanical
models (dimer model, electrical networks, Ising model) on planar
graphs. I will survey some of these connections, following works of
Postnikov, Curtis-Ingerman-Morrow, de Verdier-Gitler-Vertigan,
Kenyon-Wilson, Lam, Lis, Galashin-Pylyavskyy.
- Total positivity and combinatorial topology: in recent years an
analogy between totally positive spaces and the theory of convex
polytopes has been appearing. I will talk about the motivations from
poset topology (e.g. work of Björner, Fomin-Shapiro, Hersh) and
scattering amplitudes (Arkani-Hamed and Trnka), and some new results
of Galashin, Karp, and myself in this direction.