The papers below are listed in (roughly) reverse chronological
order. All of the files are PostScript.
If you have trouble downloading, or would prefer a hardcopy, please
email me at: fomin@umich.edu
Alternatively, you may look up my papers on the Mathematics
ArXiv,
Google Scholar,
or
MathSciNet (requires subscription).
For most of the papers listed below,
I would be glad to email the pdf file of the
final published version.
This is a preliminary draft of Chapters 13 of our forthcoming
textbook.
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
SL_{3}invariants of collections of
vectors, covectors, and matrices,
using combinatorics of webs on marked surfaces with boundary.
We explore the notion of subtractionfree arithmetic circuit
complexity, and compare it to other complexity measures.
We describe and explore cluster structures in classical rings of
SL_{3} invariants, and relate them to the web bases introduced
by G.Kuperberg.
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
type,
using generalized decorated Teichmüller spaces.
The cluster variables are interpreted as suitably renormalized lambda
lengths of tagged arcs.
Also available (but not recommendedread 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
coefficients.
We study a family of simplicial complexes
which generalize cluster complexes of finite type.
This leads to combinatorial algorithms for determining
Coxetertheoretic invariants.
Lecture notes for CDM2003.
An introduction to cluster algebras in the historical order
of development.
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 offdiagonal 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 CartanKilling 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
algebras.
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.

Fibered
quadratic Hopf algebras related to Schubert calculus (with C. Procesi,
8 pages)
Journal of
Algebra
230 (2000), 174183.
We introduce and study certain quadratic Hopf algebras related to Schubert
calculus of the flag manifold.

Mixed Bruhat
operators and YangBaxter equations for Weyl groups (with
F. Brenti and A. Postnikov, 19 pages)
International
Mathematics Research Notices 1999:8, 419441.
We study a family of operators acting in the span of a Weyl group which
provide a solution of the YangBaxter equations of corresponding type.

Double
Bruhat cells and total positivity (with A. Zelevinsky,
45 pages, with color figures); the grayscale
version also available
Journal of the AMS 12
(1999), 335380.
We study intersections of opposite Bruhat cells in a semisimple complex
Lie group, and associated totally nonnegative varieties.
Cf. P. Littelmann's
Séminaire
Bourbaki talk

Knuth
equivalence, jeu de taquin, and the LittlewoodRichardson rule
(30 pages)
Appendix 1 to Chapter 7 in: R. P. Stanley,
Enumerative
Combinatorics, vol. 2, Cambridge University Press, 1999.

Quadratic
algebras, Dunkl elements, and Schubert calculus
(with A. N. Kirillov,
34 pages)
Advances in Geometry, Progress in Mathematics 172 (1999),
147182.
We suggest a new combinatorial construction for the cohomology ring
(ordinary or quantum) of the flag manifold.

Inverting
the Frobenius map (with J.Y. Thibon,
10 pages)
St.
Petersburg Mathematical Journal 10 (1999), 545552.
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
Frobenius image.

Quantum
Schubert Polynomials (with S. Gelfand and A. Postnikov)
Journal of the AMS 10
(1997), 565596.
We compute the GromovWitten invariants of the flag manifold and derive
the quantum Monk's formula.

Sparse
Interpolation of Symmetric Polynomials (with A. Barvinok)
Advances in Applied
Mathematics 18 (1997), 271285.
We develop efficient algorithms for computing expansions of symmetric
polynomials into Schur functions.

Reduced
words and plane partitions (with A. N. Kirillov,
10 pages)
Journal
of Algebraic Combinatorics 6 (1997), 311319.
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.

Parametrizations
of canonical bases and totally positive matrices
(with A. Berenstein
and A. Zelevinsky)
Advances in
Mathematics 122
(1996), 49149.
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.

Noncommutative
Schur functions and their applications (with C. Greene, 21 pages)
Discrete
Mathematics 193
(1998), 179200.
Reprinted in the Discrete
Math Anniversary Volume, 306 (2006), 10801096.
We develop a theory of Schur functions in noncommuting variables.

Balanced
labellings and Schubert polynomials (with C. Greene,
V. Reiner,
and M. Shimozono)
European Journal
of Combinatorics 18 (1997), 373389.
We introduce and study balanced labellings of diagrams representing
the inversions in a permutation.

Combinatorial
B_{n}analogues of Schubert polynomials (with A. N. Kirillov)
Transactions of the AMS 348
(1996), 35913620.
Three flavors of Schubert polynomials of types B and C are constructed
and studied.

Grothendieck
polynomials and the YangBaxter equation (with A. N. Kirillov)
Proc. 6th Intern. Conf. on Formal Power Series and Algebraic Combinatorics,
DIMACS, 1994, 183190.
A new development of the theory of Grothendieck polynomials based on
an exponential solution of the YangBaxter equation in the degenerate Hecke
algebra is given.

Universal
exponential solution of the YangBaxter equation (with A. N. Kirillov)
Letters
in Mathematical Physics 37 (1996), 273284.

On the
number of rim hook tableaux (with N. Lulov), Zapiski Nauchn.
Sem. POMI 223 (1995), 219226.
More papers by Sergey
Fomin (available upon request):

The YangBaxter equation, symmetric functions, and Schubert
polynomials
(with
A. N. Kirillov),
Discrete
Mathematics 153 (1996), 123143.

Dual graphs and Schensted correspondences,
Series formelles et
combinatoire algebrique, P. Leroux and C. Reutenauer, Ed., Montreal,
LACIM, UQAM, 1992, 221236.