On two notions of total positivity for generalized partial flag varieties of classical Lie types

TL;DR

This paper explores when Lusztig's total positivity aligns with Plucker positivity in classical Lie types, extending known results from type A to types B and C, and clarifying their geometric and combinatorial structures.

Contribution

It characterizes symplectic and odd-orthogonal partial flag varieties where Lusztig's total positivity matches Plucker positivity, broadening understanding beyond type A.

Findings

Identifies conditions for total positivity equivalence in types B and C

Extends previous type A results to additional Lie types

Provides geometric insights into partial flag varieties

Abstract

For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Plucker coordinates. This coincidence underpins the rich interaction between matroid theory, tropical geometry, and the theory of total positivity. Bloch and Karp furthermore characterized the (type A) partial flag varieties for which the two notions of positivity similarly coincide. We characterize the symplectic (type C) and odd-orthogonal (type B) partial flag varieties for which Lusztig's total positivity coincides with Plucker positivity.