559 citations · 1.2k across the 37 of their papers we have counts for
16 papers · 1 filter
Incidences and tilings
Sergey Fomin, Pavlo Pylyavskyy
We show that various classical theorems of real/complex linear incidence geometry, such as the theorems of Pappus, Desargues, Möbius, and so on, can be interpreted as special cases…
Long mutation cycles
Sergey Fomin, Scott Neville
A mutation cycle is a cycle in a graph whose vertices are labeled by the quivers in a given mutation class and whose edges correspond to single mutations. For any fixed , w…
Introduction to Cluster Algebras. Chapter 7
Sergey Fomin, Lauren Williams, Andrei Zelevinsky
This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been p…
Universal quivers
Sergey Fomin, Kiyoshi Igusa, Kyungyong Lee
We show that for any positive integer , there exists a quiver with vertices and edges such that any quiver on vertices is a full subquiver of a quiver…
Introduction to Cluster Algebras. Chapters 1-3
Sergey Fomin, Lauren Williams, Andrei Zelevinsky
This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutati…
Noncommutative Schur functions, switchboards, and Schur positivity
Jonah Blasiak, Sergey Fomin
The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a…
Subtraction-free complexity, cluster transformations, and spanning trees
Sergey Fomin, Dima Grigoriev, Gleb Koshevoy
Subtraction-free computational complexity is the version of arithmetic circuit complexity that allows only three operations: addition, multiplication, and division. We use cluster…
Tensor diagrams and cluster algebras
Sergey Fomin, Pavlo Pylyavskyy
The rings of SL(V) invariants of configurations of vectors and linear forms in a finite-dimensional complex vector space V were explicitly described by Hermann Weyl in the 1930s. W…
Root systems and generalized associahedra
Sergey Fomin, Nathan Reading
These lecture notes for the IAS/Park City Graduate Summer School in Geometric Combinatorics (July 2004) provide an overview of root systems, generalized associahedra, and the combi…
Generalized cluster complexes and Coxeter combinatorics
Sergey Fomin, Nathan Reading
We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes t…
Polytopal realizations of generalized associahedra
Frederic Chapoton, Sergey Fomin, Andrei Zelevinsky
In hep-th/0111053, a complete simplicial fan was associated to an arbitrary finite root system. It was conjectured that this fan is the normal fan of a simple convex polytope (a ge…
The Laurent phenomenon
Sergey Fomin, Andrei Zelevinsky
A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes---quite unexpectedly---it does. We suggest a unified trea…
Loop-erased walks and total positivity
Sergey Fomin
We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely,…
Finite posets and Ferrers shapes
Thomas Britz, Sergey Fomin
A survey on the Greene-Kleitman correspondence, with complete proofs, many of which are new.
Stratified spaces formed by totally positive varieties
Sergey Fomin, Michael Shapiro
By a theorem of A.Björner, for every interval in the Bruhat order of a Coxeter group , there exists a stratified space whose strata are labeled by the elements of $[u,v]…
Recognizing Schubert cells
Sergey Fomin, Andrei Zelevinsky
This paper focuses on the properties of Schubert cells as quasi-projective subvarieties of a generalized flag variety. More specifically, we investigate the problem of distinguishi…