By Fulton W.

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Mb , where b is the number of bounded indices in i. 5). • The generators of O(Lu,v ) are organized into b + 1 groups (“clusters”) numbered 0 to b. Group 0 corresponds to the toric chart Ui , group q is associated with the toric chart Uq,i . Cluster 0 contains M1 , . . , Mb , cluster q contains M1 , . . , Mq−1 , Mq , Mq+1 , . . , Mb . These clusters form a star-like graph with b + 1 labeled vertices and edges connecting vertex 0 with all other vertices (see Fig. 19). 5). 19. Star on 7 clusters An algebra with such a special structure of generators is an example of the so called upper cluster algebra.

For the generalization of the same approach to double Bruhat cells, see [SSVZ, Z, GSV1]. Double Bruhat cells for semisimple Lie groups are introduced in [FZ1], reduced double Bruhat cells and the inverse problem of restoring factorization parameters are studied in [BZ2]. 3 we follow [SSVZ, Z]. 11 is proved in [Z]. The construction of functions Mi in the SLn -case based on pseudoline arrangements is borrowed from [BFZ1, FZ1]. 14 are obtained in [FZ1]. Functions τi are introduced and studied in [SSV2].

Mm . 34 2. BASIC EXAMPLES: RINGS OF FUNCTIONS ON SCHUBERT VARIETIES We start with the irreducibility of Mn . Assume that Mn = P · Q, where P and Q are some regular functions on Lu,v . Restrict all functions to Ui . Since P and Q are regular functions on Ui , they are Laurent polynomials in M1 , . . , Mm . Equality Mn = P ·Q implies that both P and Q are in fact monomials. However, a monomial dm is regular on Lu,v if and only if all di corresponding to i-bounded i M1d1 ×· · · ×Mm dm must be a regular function are nonnegative.