559 citations · 1.2k across the 37 of their papers we have counts for
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Coordinate rings and birational charts
Sergey Fomin, George Lusztig
Let be a semisimple simply connected complex algebraic group. Let be the unipotent radical of a Borel subgroup in . We describe the coordinate rings of (resp., …
Cluster algebras: Notes for the CDM-03 conference
Sergey Fomin, Andrei Zelevinsky
This is an expanded version of the notes of our lectures given at the conference "Current Developments in Mathematics 2003" held at Harvard University on November 21--22, 2003. We…
Cluster algebras III: Upper bounds and double Bruhat cells
Arkady Berenstein, Sergey Fomin, Andrei Zelevinsky
We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an…
Cluster algebras I: Foundations
Sergey Fomin, Andrei Zelevinsky
In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
Totally nonnegative and oscillatory elements in semisimple groups
Sergey Fomin, Andrei Zelevinsky
We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F.R.Gantmacher, M.G.Krein, A.Whitney, C.Loewner, M.Gasca, and J.M.Pena to the…
Double Bruhat cells and total positivity
Sergey Fomin, Andrei Zelevinsky
We study intersections of opposite Bruhat cells in a semisimple complex Lie group, and associated totally nonnegative varieties.