Mathematics – Quantum Algebra
Scientific paper
2011-04-04
Mathematics
Quantum Algebra
This paper has been withdrawn because the author's claim of positivity may be false and needs to be investigated further
Scientific paper
In this paper we give a direct proof of the positivity conjecture for adapted quantum cluster variables. Moreover, our process allows one to explicitly compute formulas for all adapted cluster monomials and certain ordered products of adapted cluster monomials. In particular, we describe all cluster monomials in cluster algebras and quantum cluster algebras of rank 2. One may obtain similar formulas for all finite type cluster monomials. The above results are achieved by computing explicit set-theoretic decompositions of Grassmannians of subrepresentations in adapted valued quiver representations into a disjoint union of products of standard vector space Grassmannians. We actually prove a more general result which should be of independent interest: we compute these decompositions for arbitrary flags of subrepresentations in adapted valued quiver representations. This implies the existence of counting polynomials for the number of points in these sets over different finite fields. Using this we extend the results of \cite{rupel} to quantum cluster algebras $\Acal_q(Q,\bfd)$, where $q$ is an indeterminate.
No associations
LandOfFree
Positivity in Quantum Cluster Algebras and Flags of Valued Quiver Representations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Positivity in Quantum Cluster Algebras and Flags of Valued Quiver Representations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Positivity in Quantum Cluster Algebras and Flags of Valued Quiver Representations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-318506