nLab totally positive matrix

Totally positive matrices

A matrix $A$ over the field of real numbers is totally positive (resp. totally nonnegative) if every minor (= determinant of any submatrix) is a positive (resp. nonnegative) real number.

Total positivity implies a number of remarkable properties; for example all eigenvalues are distinct and positive.

Total positivity in Lie theory

George Lusztig discovered that this classical total positivity is a related to more general total positivity phenomena in the theory of Lie groups and quantum groups.

Literature and related entries

• EoM: total positivity
• George Lusztig, Total positivity in reductive groups, In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics 123; Birkhäuser 1994 doi
• cluster algebra
• many lecture notes listed at a minicourse page on total positivity
• M. Skandera, Introductory notes on total positivity, ps, June 2003.

Relation of positive Grassmannian and Kadomtsev-Petviashvili equation

• Yuji Kodama, Lauren Williams, KP solitons and total positivity for the Grassmannian, Invent. math. 198, 637–699 (2014) doi arXiv:1106.0023

• S. Launois, T. H. Lenagan, B. M. Nolan, Total positivity is a quantum phenomenon: the Grassmannian case, Memoirs of the Amer. Math. Soc. 1448 (2023) 123 p.; an earlier shorter version is arXiv:1906.06199

category: algebra