A decorated surface S is an oriented surface, with or without boundary, and a finite set {s 1,..., s n} of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over [special characters omitted].

We define Stasheff polytopes in the spaces of tropical points of cluster A -varieties. We study the supports of products of elements of canonical bases for cluster A -varieties. We prove that, for the cluster A -variety of type A , such supports are Stasheff polytopes.

Let G be a connected almost simple algebraic group with a Dynkin automorphism . Let G be the connected almost simple algebraic group associated with G and . We prove that the dimension of the tensor invariant space of G is equal to the trace of on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does G . As a consequence, we show that the spin group Spin (2 n+ 1) has saturation factor 2, which strengthens the results of BelkaleKumar [1] and Sam [28] in the case of type B n.

Abstract Kontsevich and Soibelman defined DonaldsonThomas invariants of a 3d CalabiYau category equipped with a stability condition [41]. Any cluster variety gives rise to a family of such categories. Their DT invariants are encapsulated in a single formal automorphism of the cluster variety, called the DT-transformation. Given a stability condition, the DT-transformation allows to recover the DT-invariants. Let S be an oriented surface with punctures and a finite number of special points on the boundary, considered modulo isotopy. It gives rise to a moduli space X P G L m, S, closely related to the moduli space of P G L m-local systems on S, which carries a canonical cluster Poisson variety structure [13]. For each puncture of S, there is a birational Weyl group action on the space X P G L m, S. We prove that it is given by cluster Poisson transformations. We prove a similar result for the involution of X P G L m, S

Given a complex analytic function f on a Whitney stratified complex analytic variety of complex dimension n, whose real part Re (f) is Morse, we prove the existence of a stratified gradient-like vector field for Re (f) such that the unstable set of a critical point p on a stratum S of complex dimension s has real dimension m (p)+ ns as was conjectured by Goresky and MacPherson.