Séminaires : Séminaire d'Algèbre

The theory of Newton-Okounkov bodies is a generalization of 
that of Newton polytopes for toric varieties. One of the ingredients for 
the definition of a Newton-Okounkov body is a valuation on the function 
field of a given projective variety. In this talk, we consider Newton-
Okounkov bodies of Schubert varieties defined from specific valuations 
which generalize extended g-vectors in cluster theory. We show that they 
provide polytopes unimodularly equivalent to string polytopes and 
Nakashima-Zelevinsky polytopes, both of which are well-known polytopes 
in representation theory. Indeed, this framework allows us to connect 
string polytopes with Nakashima-Zelevinsky polytopes by tropicalized 
cluster mutations. 
This talk is based on a joint work with Naoki Fujita.