Papers by Sergey Fomin (1990-1999)

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.

• Lecture notes on quantum cohomology of the flag manifold  (9 pages)

Publications de l'Institut Mathematique (Beograd)(N.S.) 66 (80) (1999), 91-100.

An exposition of some recent developments related to the object in the title.
 

• Mixed Bruhat operators and Yang-Baxter equations for Weyl groups (with F. Brenti and A. Postnikov, 19 pages)

International 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.
 

• 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), 335-380.

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 Littlewood-Richardson 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), 147-182.

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), 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 Frobenius image.


• Quantum Schubert Polynomials (with S. Gelfand and A. Postnikov)

Journal of the AMS 10 (1997), 565-596.

We compute the Gromov-Witten 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), 271-285.

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), 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.


• Parametrizations of canonical bases and totally positive matrices (with A. Berenstein and A. Zelevinsky)

Advances in Mathematics 122 (1996), 49-149.

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), 179-200. Reprinted in the Discrete Math Anniversary Volume, 306 (2006), 1080-1096.

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), 373-389.

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), 3591-3620.

Three flavors of Schubert polynomials of types B and C are constructed and studied.
 

• Grothendieck 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.
 

• Universal exponential solution of the Yang-Baxter equation (with A. N. Kirillov)

Letters in Mathematical Physics 37 (1996), 273-284.
 
• On the number of rim hook tableaux (with N. Lulov), Zapiski Nauchn. Sem. POMI 223 (1995), 219-226.

• Rim hook lattices (with D. Stanton, 9 pages, figures not included)

St. Petersburg Mathematical Journal 9 (1998), 1007-1016.
 
• Generalizations of the Greene-Kleitman theorem (2 pages), Discrete Mathematics 157 (1996), 376-377.

More papers by Sergey Fomin (available upon request):

• Schur operators and Knuth correspondences, Journal of Combinatorial Theory, Ser.A 72 (1995), 277-292.

• Duality of graded graphs, Journal of Algebraic Combinatorics 3 (1994), 357-404.

• Schensted algorithms for dual graded graphs, Journal of Algebraic Combinatorics 4 (1995), 5-45.

• Schubert polynomials and the nilCoxeter algebra (with R. P. Stanley), Advances in Mathematics 103 (1994), 196-207.

• A Littlewood-Richardson miscellany (with C. Greene), European Journal of Combinatorics 14 (1993), 191-212.

• The Yang-Baxter equation, symmetric functions, and Schubert polynomials (with A. N. Kirillov), Discrete 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.

• Fourier-Walsh expansions with random coefficients (with P. P. Kargaev), Theory of Probability and Its Applications 35 (1990), 258-268.