Chapter 2 considers in detail two basic sources of cluster-like structures in rings
of functions related to Schubert varieties: the homogeneous coordinate ring of the
Grassmannian G2(m) of 2-dimensional planes and the ring of regular functions on
a double Bruhat cell. The first of the two rings is studied in Section 2.1. We show
that Pl¨ ucker coordinates of G2(m) can be organized into a finite number of groups
of the same cardinality (clusters), covering the set of all Pl¨ ucker coordinates. Every
cluster corresponds to a triangulation of a convex m-gon. The system of clusters
has a natural graph structure, so that adjacent clusters differ exactly by two Pl¨ucker
coordinates corresponding to a pair of crossing diagonals (and contained together
in exactly one short Pl¨ ucker relation). Moreover, this graph is a 1-skeleton of the
Stasheff polytope of m-gon triangulations. We proceed to show that if one fixes
an arbitrary cluster, any other Pl¨ ucker coordinate can be expressed as a rational
function in the Pl¨ ucker coordinates entering this cluster. We prove that this rational
function is a Laurent polynomial and find a geometric meaning for its numerator
and denominator, see Proposition 2.1.
Section 2.2 starts with the formulation of Arnold’s problem: find the number of
connected components in the variety of real complete flags intersecting transversally
a given pair of flags. We reformulate this problem as a problem of enumerating con-
nected components in the intersection of two real open Schubert cells, and proceed
to a more general problem of enumerating connected components of a real double
Bruhat cell. It is proved that components in question are in a bijection with the
orbits of a group generated by symplectic transvections in a vector space over the
field F2, see Theorem 2.10. As one of the main ingredients of the proof, we provide
a complete description of the ring of regular functions on the double Bruhat cell.
It turns out that generators of this ring can be grouped into clusters, and that they
satisfy Pl¨ ucker-type exchange relations.
In Chapter 3 we introduce cluster algebras and prove two fundamental results
about them. Section 3.1 contains basic definitions and examples. In this book we
mainly concentrate on cluster algebras of geometric type, so the discussion in this
Section is restricted to such algebras, and the case of general coefficients is only
mentioned in Remark 3.13. We define basic notions of cluster and stable variables,
seeds, exchange relations, exchange matrices and their mutations, exchange graphs,
and provide extensive examples. The famous Laurent phenomenon is treated in Sec-
tion 3.2. We prove both the general statement (Theorem 3.14) and its sharpening
for cluster algebras of geometric type (Proposition 3.20). The second fundamental
result, the classification of cluster algebras of finite type, is discussed in Section 3.3.
We state the result (Theorem 3.26) as an equivalence of three conditions, and
provide complete proofs for two of the three implications. The third implication is
discussed only briefly, since a complete proof would require exploring intricate com-
binatorial properties of root systems for different Cartan–Killing types, which goes
beyond the scope of this book. In Section 3.4 we discuss relations between cluster
algebras and rings of regular functions, see Proposition 3.37. Finally, Section 3.5
contains a list of conjectures, some of which are treated in subsequent chapters.
Chapter 4 is central to the book. In Section 4.1 we introduce the notion of
Poisson brackets compatible with a cluster algebra structure and provide a complete
characterization of such brackets for cluster algebras with the extended exchange
matrix of full rank, see Theorem 4.5. In this context, mutations of the exchange
matrix are explained as transformations of the coefficient matrix of the compatible
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