The important paper under review, which among other things solves a couple of conjectures due, respectively, to A. Berenstein et al. [Duke Math. J. 126, No. 1, 1–52 (2005; Zbl 1135.16013)] and to V. V. Fock and A. B. Goncharov [Ann. Sci. Éc. Norm. Supér. (4) 42, No. 6, 865–930 (2009; Zbl 1180.53081)], is concerned with a family of new varieties, that the authors call double Bott-Samelson cells, which are endowed with certain cluster structures.
Cluster varieties are so called because they are associated to cluster algebras, a notion introduced by S. Fomin and A. Zelevinsky in three important articles published in between 2002 and 2007, corresponding to the references [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017); Ann. Math. (2) 158, No. 3, 977–1018 (2003; Zbl 1057.52003); Compos. Math. 143, No. 1, 112–164 (2007; Zbl 1127.16023)] of the paper under review. They are suitable classes of commutative integral domains enriched with the action of distinguished subsets of finite cardinality, called clusters, which generate the algebra itself. Cluster structures already arise in more familiar cases of homogeneous spaces, such as Grassmannians. Indeed, the natural cells of a Grassmannian, attached to some complete flag, have cluster structures. Such structures are more generally studied in the case of double Bott-Samelson varieties.
But what are Bott-Samelson varieties and what does the adjective “double” stand for? Such varieties were introduced by R. Bott and H. Samelson [Am. J. Math. 80, 964–1029 (1961; Zbl 0101.39702)] to provide resolutions of singularities of (generalized) Schubert varieties and have many applications to geometric representation theory. They turn out to be natural generalizations of double Bruhat cells.
The Arianna thread to get to them is the following. If \(G\) is a complex connected linear algebraic group, \(T\) a maximal torus and \(B\) a Borel subgroup containing \(T\), it is known (see e.g. the classical book by T. A. Springer on algebraic groups) that to the root system \(R\) of \((G,B)\) one may associate a so-called generalised symmetrizable Cartan matrix. The entries of a Cartan matrix of a simple Lie algebra are normalized scalar product \(\frac{a_{ji}=2(r_{i},r_{j})}{(r_{j},r_{j})}\) of simple roots of the Lie algebra. In particular all the diagonal entries are equal to \(2\). To such matrix one may attach a braid group, whose elements satisfy the so called braid relations coming from topology, which saw several applications also in mathematical physics, for instance in the description of the Yang-BNaxter equation. To each simple root one may associate a double coset \(BwB\) and the the Bruhat decomposition is \(G=\coprod_{w\in W} BwB\). A double Bruhat cell is the intersection of two Bruhat cells with respect to two different elements of the Weil group (the group permuting the roots). Double Bott-Samelson cells are defined similarly to the double Bruhat cells, but the elements of the Weil group are replaced by braids of the Braid group associated to the generalised Cartan marix \(C\).
In the paper under review, the authors introduce four different versions of double Bott-Samelson cells associated to every pair of positive braids in the generalised braid group associated to \(C\). Their main and appealing result, among many others which are all incredibly interesting in their own, is the proof that certain decorated double Bott-Samelson cells are smooth affine varieties, whose coordinate rings are isomorphic to upper cluster algebras (Theorems 2.30, 3.45 and 3.46). These specific mathematical goals are achieved through an amazing blend of notions and techniques coming from different branches of mathematics, such as algebra, representation theory, algebraic geometry, combinatorics (especially employed in the master manipulations with Legendrian links and Legendrian Reidemeister moves).
The 89 pages of the paper are organized as follows. Besides a very detailed and comprehensive introduction, which clarifies what purposes the authors want to pursue, the exposition is divided into five sections (from 2 to 5) and three very useful appendices. Section 1.2 is crucial as it is related, and explains the connection, to the Donaldson-Thomas Transformation of Bott-Samelson Cells treated in detail in Section 4.
The three appendices are devoted respectively to basics on Kac-Peterson groups, to basics on cluster algebras (very important for non expert approaching the paper) and to computational codes for computing decorated configuration spaces.
It is not an easy paper, certainly not suited for a general audience but adressed to specialists in the subject, in that it is very technical. Clearly, this is is not indicated as a defect but rather to make the reader aware about this rather unavoidable feature, due to the deepness of the difficult questions the authors deal with.