The papers listed below appear in (roughly) reverse chronological
order. All of the files are PostScript.
For most of the papers,
I would be glad to email the pdf file of the
final published version.
This is a preliminary draft of Chapters 1-3 of our forthcoming
We show that when the number of variables is fixed, the semiring complexity of
a Schur polynomial is logarithmic in the size of the indexing partition.
We study the phenomenon in which commutation relations for sequences
of elements in a ring are implied by similar relations for subsequences involving at most
three indices at a time.
We review and further develop a general approach to Schur positivity of
symmetric functions based on the machinery of noncommutative Schur functions.
We construct and study cluster structures in rings of
SL3-invariants of collections of
vectors, covectors, and matrices,
using combinatorics of webs on marked surfaces with boundary.
We explore the notion of subtraction-free arithmetic circuit
complexity, and compare it to other complexity measures.
We describe and explore cluster structures in classical rings of
SL3 invariants, and relate them to the web bases introduced
A brief and informal introduction to cluster algebras.
Total positivity serves as the main motivation.
We develop the combinatorics of labeled floor diagrams,
and apply it to enumeration of plane algebraic curves.
We construct geometric models for cluster algebras associated with
marked bordered surfaces, for any choice of coefficients of geometric
using generalized decorated Teichmüller spaces.
The cluster variables are interpreted as suitably renormalized lambda
lengths of tagged arcs.
Also available (but not recommended---read at your own risk):
preliminary version (May 2008, 64 pages).
We study cluster algebras associated with oriented bordered surfaces
with marked points.
We study the dependence of a cluster algebra on the choice of
We study a family of simplicial complexes
which generalize cluster complexes of finite type.
This leads to combinatorial algorithms for determining
Lecture notes for CDM-2003.
An introduction to cluster algebras in the historical order
We develop a new approach to cluster algebras based on the notion of
an upper cluster algebra, and apply it to the study of double Bruhat cells.
We characterize the singular values of an
Hermitian (or complex symmetric) matrix in terms of the singular
values of its off-diagonal block. We also characterize the eigenvalues of an
Hermitian matrix C=A+B in terms of the combined list of
eigenvalues of A and B.
We classify the cluster algebras of finite type.
This turns out to be another instance of
the Cartan-Killing classification.
We identify the cluster complex of such an algebra as the normal fan
of a generalized associahedron of the corresponding type.
We explicitly realize generalized associahedra as convex polytopes.
We prove the periodicity conjecture of
Al. B. Zamolodchikov in the theory of thermodynamic Bethe ansatz.
We also introduce and study a family of
simplicial complexes (generalized associahedra) associated
to arbitrary root systems.
We establish Laurentness of a large class of recursively defined
birational maps, proving conjectures by D. Gale, R. Robinson,
J. Propp, N. Elkies, and M. Kleber.
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
Combinatorial treatment of total positivity phenomena
associated with walks in planar directed weighted graphs (not necessarily acyclic),
or with Markov processes in planar domains.
We address the problem of distinguishing between different Schubert
cells using vanishing patterns of generalized Plücker coordinates.
quadratic Hopf algebras related to Schubert calculus (with C. Procesi,
230 (2000), 174-183.
We introduce and study certain quadratic Hopf algebras related to Schubert
calculus of the flag manifold.
operators and Yang-Baxter equations for Weyl groups (with
F. Brenti and A. Postnikov, 19 pages)
Mathematics Research Notices 1999:8, 419-441.
We study a family of operators acting in the span of a Weyl group which
provide a solution of the Yang-Baxter equations of corresponding type.
Bruhat cells and total positivity (with A. Zelevinsky,
45 pages, with color figures); the grayscale
version also available
Journal of the AMS 12
We study intersections of opposite Bruhat cells in a semisimple complex
Lie group, and associated totally nonnegative varieties.
Cf. P. Littelmann's
equivalence, jeu de taquin, and the Littlewood-Richardson rule
Appendix 1 to Chapter 7 in: R. P. Stanley,
Combinatorics, vol. 2, Cambridge University Press, 1999.
algebras, Dunkl elements, and Schubert calculus
(with A. N. Kirillov,
Advances in Geometry, Progress in Mathematics 172 (1999),
We suggest a new combinatorial construction for the cohomology ring
(ordinary or quantum) of the flag manifold.
the Frobenius map (with J.-Y. Thibon,
Petersburg Mathematical Journal 10 (1999), 545-552.
We express the generating function for the values of an arbitrary virtual
character of the symmetric group (or the Hecke algebra) in terms of its
Schubert Polynomials (with S. Gelfand and A. Postnikov)
Journal of the AMS 10
We compute the Gromov-Witten invariants of the flag manifold and derive
the quantum Monk's formula.
Interpolation of Symmetric Polynomials (with A. Barvinok)
Advances in Applied
Mathematics 18 (1997), 271-285.
We develop efficient algorithms for computing expansions of symmetric
polynomials into Schur functions.
words and plane partitions (with A. N. Kirillov,
of Algebraic Combinatorics 6 (1997), 311-319.
We prove an identity generalizing enumerative formulas of Stanley and
Macdonald related to reduced words for the element of maximal length in
the symmetric group.
of canonical bases and totally positive matrices
(with A. Berenstein
and A. Zelevinsky)
We provide: (i) explicit formulas for Lusztig's transition maps related
to the canonical basis of the quantum group of type A; (ii) formulas for
the factorizations of a square matrix into elementary Jacobi matrices;
(iii) a family of new total positivity criteria.
Schur functions and their applications (with C. Greene, 21 pages)
Reprinted in the Discrete
Math Anniversary Volume, 306 (2006), 1080-1096.
We develop a theory of Schur functions in noncommuting variables.
labellings and Schubert polynomials (with C. Greene,
and M. Shimozono)
of Combinatorics 18 (1997), 373-389.
We introduce and study balanced labellings of diagrams representing
the inversions in a permutation.
Bn-analogues of Schubert polynomials (with A. N. Kirillov)
Transactions of the AMS 348
Three flavors of Schubert polynomials of types B and C are constructed
polynomials and the Yang-Baxter equation (with A. N. Kirillov)
Proc. 6th Intern. Conf. on Formal Power Series and Algebraic Combinatorics,
DIMACS, 1994, 183-190.
A new development of the theory of Grothendieck polynomials based on
an exponential solution of the Yang-Baxter equation in the degenerate Hecke
algebra is given.
exponential solution of the Yang-Baxter equation (with A. N. Kirillov)
in Mathematical Physics 37 (1996), 273-284.
number of rim hook tableaux (with N. Lulov), Zapiski Nauchn.
Sem. POMI 223 (1995), 219-226.
More papers by Sergey
Fomin (available upon request):
The Yang-Baxter equation, symmetric functions, and Schubert
A. N. Kirillov),
Mathematics 153 (1996), 123-143.
Dual graphs and Schensted correspondences,
Series formelles et
combinatoire algebrique, P. Leroux and C. Reutenauer, Ed., Montreal,
LACIM, UQAM, 1992, 221-236.