18 research outputs found
Cohomology rings of toric varieties assigned to cluster quivers: the case of unioriented quivers of type A
The theory of cluster algebras of S. Fomin and A. Zelevinsky has assigned a
fan to each Dynkin diagram. Then A. Buan, R. Marsh, M. Reineke, I. Reiten and
G. Todorov have generalized this construction using arbitrary quivers on Dynkin
diagrams. In the special case of the unioriented quiver of type A, we describe
the cohomology ring of the toric variety associated to this fan. A natural base
is obtained and an explicit rule is given for the product of any two
generators.Comment: 12 page
Enumerative properties of generalized associahedra
Some enumerative aspects of the fans, called generalized associahedra,
introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras
are considered, in relation with a bicomplex and its two spectral sequences. A
precise enumerative relation with the lattices of generalized noncrossing
partitions is conjectured and some evidence is given.Comment: 15 page
Pre-Lie algebras and the rooted trees operad
A Pre-Lie algebra is a vector space L endowed with a bilinear product * : L
\times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all
x,y,z in L. We give an explicit combinatorial description in terms of rooted
trees of the operad associated to this type of algebras and prove that it is a
Koszul operad.Comment: 13 pages, uses xypic, typos corrected and more explicit description
of the free algebr
Cluster algebras as Hall algebras of quiver representations
Recent articles have shown the connection between representation theory of
quivers and the theory of cluster algebras. In this article, we prove that some
cluster algebras of type ADE can be recovered from the data of the
corresponding quiver representation category. This also provides some explicit
formulas for cluster variables.Comment: 17 pages ; 2 figures ; the title has changed ! some other minor
modification