CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS

CLUSTER ALGEBRAS AND CLUSTER CATEGORIES

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

cluster algebras and cluster categories

these are notes from introductory survey lectures given at the institute for studies in theoretical physics and mathematics (ipm), teheran, in 2008 and 2010. we present the definition and the fundamental properties of fomin-zelevinsky’s cluster algebras. then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of ...

Grassmannians and Cluster Algebras

Root games on Grassmannians

We recall the root game, introduced in [P], which gives a fairly powerful sufficient condition for non-vanishing of Schubert calculus on a generalised flag manifold G/B. We show that it gives a necessary and sufficient rule for nonvanishing of Schubert calculus on Grassmannians. In particular, a LittlewoodRichardson number is non-zero if and only if it is possible to win the corresponding root ...

AN INTRODUCTION TO HIGHER CLUSTER CATEGORIES

In this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. We focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of Fomin and Reading, and colored quiver mutation.