Preface
Cluster algebras introduced by Fomin and Zelevinsky in [FZ2] are commu-
tative rings with unit and no zero divisors equipped with a distinguished family
of generators (cluster variables) grouped in overlapping subsets (clusters) of the
same cardinality (the rank of the cluster algebra) connected by exchange relations.
Among these algebras one finds coordinate rings of many algebraic varieties that
play a prominent role in representation theory, invariant theory, the study of total
positivity, etc. For instance, homogeneous coordinate rings of Grassmannians, coor-
dinate rings of simply-connected connected semisimple groups, Schubert varieties,
and other related varieties carry (possibly after a small adjustment) a cluster alge-
bra structure. A prototypic example of exchange relations is given by the famous
Pl¨ ucker relations, and precursors of the cluster algebra theory one can observe in
the Ptolemy theorem expressing product of diagonals of inscribed quadrilateral in
terms of side lengths and in the Gauss formulae describing Pentagrama Myrificum.
Cluster algebras were introduced in an attempt to create an algebraic and
combinatorial framework for the study dual canonical bases and total positivity in
semisimple groups. The notion of canonical bases introduced by Lusztig for quan-
tized enveloping algebras plays an important role in the representation theory of
such algebras. One of the approaches to the description of canonical bases uti-
lizes dual objects called dual canonical bases. Namely, elements of the enveloping
algebra are considered as differential operators on the space of functions on the
corresponding group. Therefore, the space of functions is considered as the dual
object, whereas the pairing between a differential operator D and a function F is
defined in the standard way as the value of DF at the unity. Elements of dual
canonical bases possess special positivity properties. For instance, they are regular
positive-valued functions on the so-called totally positive subvarieties in reductive
Lie groups, first studied by Lusztig. In the case of GLn the notion of total posi-
tivity coincides with the classical one, first introduced by Gantmakher and Krein:
a matrix is totally positive if all of its minors are positive. Certain finite collec-
tions of elements of dual canonical bases form distinguished coordinate charts on
totally positive varieties. The positivity property of the elements of dual canonical
bases and explicit expressions for these collections and transformations between
them were among the sources of inspiration for designing cluster algebra transfor-
mation mechanism. Transitions between distinguished charts can be accomplished
via sequences of relatively simple positivity preserving transformations that served
as a model for an abstract definition of a cluster transformation. Cluster alge-
bra transformations construct new distinguished elements of the cluster algebra
from the initial collection of elements. In many concrete situations all constructed
distinguished elements have certain stronger positivity properties called Laurent
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