ϴ-Positivity: A Unifying Framework for Lie Groups

In a paper recently published in Inventiones mathematicae, our director Anna Wienhard and Olivier Guichard from the University of Strasbourg introduce a new mathematical framework that generalizes the influential theory of total positivity in split real semisimple Lie groups, expanding its applicability to a broader class of Lie groups. 

A matrix is total positive if all of its minors are positive. Introduced in the first half of the 20^th century, total positive matrices have played an important role in various fields of mathematics, ranging from graph theory, stochastic processes, game theory to electric networks and physics. 

The theory has been generalized widely by George Lusztig who introduced the total positive semigroup of a general split real simple Lie group. Lusztig’s theory of total positivity has become very influential in representation theory and combinatorics. Building on this foundational work, the new concept of ϴ-positivity provide a unified theory of positivity that includes Lusztig’s total positivity on split real Lie groups as well as the theory of Lie semigroups in Lie groups of Hermitian type. 

The study identifies four families of simple Lie groups that admit positive structures, two of which represent groundbreaking discoveries with no previously known positive structures.

Applications and Broader Impact
The motivation for this work originated in higher-rank Teichmüller theory, particularly in the study of Hitchin and maximal representations. Positivity now provides a unifying underlying structure for higher-rank Teichmüller spaces and opens the door to new conjectures, some of which have already been proven in subsequent work of the authors and others. In it’s simplest form, it replaces the semigroup of positive real numbers by the semigroup of positive definite symmetric matrices. 

The generalization of positivity has far-reaching implications. It provides a new perspective on Hermitian Lie groups of tube type, but also new deep connections to representation theory, (non-commutative) cluster algebras and to special classes of supersymmetric quantum field theories in physics. 

Original publication
Guichard, Olivier & Wienhard, Anna (2025): Generalizing Lusztig’s total positivity, Inventiones mathematicae. 2025, DOI: 10.1007/s00222-024-01303-y