Sergey Fomin

Cluster algebras I: Foundations

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.

Cluster algebras II: Finite type classification

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent from its geometric origin. The combinatorial structure behind a cluster algebra of finite type is captured by its cluster complex. We identify this complex as the normal fan of a generalized associahedron introduced and studied in hep-th/0111053 and math.CO/0202004. Another essential combinatorial ingredient of our arguments is a new characterization of the Dynkin diagrams.

Double Bruhat cells and total positivity

We study intersections of opposite Bruhat cells in a semisimple complex Lie group, and associated totally nonnegative varieties.

The Yang-Baxter equation, symmetric functions, and Schubert polynomials

Abstract We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schutzenberger, and provide simplified proofs of their basic properties, along with various generalizations thereof. Our techniques make use of an explicit combinatorial interpretation of these polynomials in terms of configurations of labelled pseudo-lines.

Total positivity: Tests and parametrizations

An introduction to total positivity (TP), with the emphasis on efficient TP criteria and parametrizations of TP matrices. Intended for general mathematical audience.

Duality of Graded Graphs

AbstractA graph is said to be graded if its vertices are divided into levels numbered by integers, so that the endpoints of any edge lie on consecutive levels. Discrete modular lattices and rooted trees are among the typical examples. The following three types of problems are of interest to us:(1) path counting in graded graphs, and related combinatorial identities;(2) bijective proofs of these identities;(3) design and analysis of algorithms establishing corresponding bijections.This article is devoted to (1); its sequel [7] is concerned with the problems (2)–(3). A simplified treatment of some of these results can be found in [8].In this article, R.P. Stanleys [26, 27] linear-algebraic approach to (1) is extended to cover a wide range of graded graphs. The main idea is to consider pairs of graded graphs with a common set of vertices and common rank function. Such graphs are said to be dual if the associated linear operators satisfy a certain commutation relation (e.g., the “Heisenberg” one). The algebraic consequences of these relations are then interpreted as combinatorial identities. (This idea is also implicit in [27].)[7] contains applications to various examples of graded graphs, including the Young, Fibonacci, Young-Fibonacci and Pascal lattices, the graph of shifted shapes, the r-nary trees, the Schensted graph, the lattice of finite binary trees, etc. Many enumerative identities (both known and unknown) are obtained. Abstract of [7]. These identities can also be derived in a purely combinatorial way by generalizing the Robinson-Schensted correspondence to the class of graphs under consideration (cf. [5]). The same tools can be applied to permutation enumeration, including involution counting and rook polynomials for Ferrers boards. The bijective correspondences mentioned above are naturally constructed by Schensted-type algorithms. A general approach to these constructions is given. As particular cases we rederive the classical algorithm of Robinson, Schensted, and Knuth [20, 12, 21], the Sagan-Worley [17, 32] and Haiman [11] algorithms, the algorithm for the Young-Fibonacci graph [5, 15], and others. Several new applications are given.

Quantum Schubert polynomials

where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn. The corresponding cohomology classes oa, called Schubert classes, form an additive basis in H* (Fln 2) . To relate the two descriptions, one would like to determine which elements of 2[xl, ... , Xn]/In correspond to the Schubert classes under the isomorphism (1.1). This was first done in [2] (see also [8]) for a general case of an arbitrary complex semisimple Lie group. Later, Lascoux and Schiitzenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes oa called Schubert polynomials and denoted Gw. Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kahler algebraic manifold X, the (small) quantum cohomology ring QH* (X, 2), which is a certain deformation of the classical cohomology ring (see, e.g., [28, 19, 14] and references therein). The additive structure of QH* (X , 2) is essentially the same as that of ordinary cohomology. In particular, QH* (Fln , Z) is canonically isomorphic, as an abelian group, to the tensor product H* (Fln , 2) (0 Z[ql,..., qn-1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however

Quadratic Algebras, Dunkl Elements, and Schubert Calculus

We suggest a new combinatorial construction for the cohomology ring of the ag manifold. The degree 2 commutation relations satissed by the divided diierence operators corresponding to positive roots deene a quadratic associative algebra. In this algebra, the formal analogues of Dunkl operators generate a commuta-tive subring, which is shown to be canonically isomorphic to the cohomology of the ag manifold. This leads to yet another combinatorial version of the corresponding Schubert calculus. The paper contains numerous conjectures and open problems. We also discuss a generalization of the main construction to quantum cohomology. Contents

Generalized cluster complexes and Coxeter combinatorics

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 to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

Noncommutative Schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.