28 research outputs found
On Quantum Analogue of The Caldero-Chapoton Formula
Let be any invertible valued quiver without oriented cycles. We study
connections between the category of valued representations of and
expansions of cluster variables in terms of the initial cluster in quantum
cluster algebras. We show that an analogue of the Caldero-Chapoton formula
holds for all quantum cluster algebras of finite type and for any cluster
variable in an almost acyclic cluster
Greedy bases in rank 2 generalized cluster algebras
In this note we extend the notion of greedy bases developed by Lee, Li, and
Zelevinsky to rank two generalized cluster algebras, i.e. binomial exchange
relations are replaced by polynomial exchange relations. In the process we give
a combinatorial construction in terms of a refined notion of compatible pairs
on a maximal Dyck path.Comment: 25 page
Quantum Cluster Characters
Let \FF be a finite field and (Q,\bfd) an acyclic valued quiver with
associated exchange matrix . We follow Hubery's approach \cite{hub1}
to prove our main conjecture of \cite{rupel}: the quantum cluster character
gives a bijection from the isoclasses of indecomposable rigid valued
representations of to the set of non-initial quantum cluster variables for
the quantum cluster algebra \cA_{|\FF|}(\tilde{B},\Lambda). As a corollary we
find that, for any rigid valued representation of , all Grassmannians of
subrepresentations Gr_\bfe^V have counting polynomials.Comment: material reorganized, some proofs rewritte
The Feigin Tetrahedron
The first goal of this note is to extend the well-known Feigin homomorphisms
taking quantum groups to quantum polynomial algebras. More precisely, we define
generalized Feigin homomorphisms from a quantum shuffle algebra to quantum
polynomial algebras which extend the classical Feigin homomorphisms along the
embedding of the quantum group into said quantum shuffle algebra. In a recent
work of Berenstein and the author, analogous extensions of Feigin homomorphisms
from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial
algebras were defined. To relate these constructions, we establish a
homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel
algebra to the quantum shuffle algebra which relates the generalized Feigin
homomorphisms. These constructions can be compactly described by a commuting
tetrahedron of maps beginning with the quantum group and terminating in a
quantum polynomial algebra. The second goal in this project is to better
understand the dual canonical basis conjecture for skew-symmetrizable quantum
cluster algebras. In the symmetrizable types it is known that dual canonical
basis elements need not have positive multiplicative structure constants, while
this is still suspected to hold for skew-symmetrizable quantum cluster
algebras. We propose an alternate conjecture for the symmetrizable types: the
cluster monomials should correspond to irreducible characters of a KLR algebra.
Indeed, the main conjecture of this note would establish this "KLR conjecture"
for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture
that the images of rigid representations under the quantum shuffle character
give irreducible characters for KLR algebras
Rank Two Non-Commutative Laurent Phenomenon and Pseudo-Positivity
We study polynomial generalizations of the Kontsevich automorphisms acting on
the skew-field of formal rational expressions in two non-commuting variables.
Our main result is the Laurentness and pseudo-positivity of iterations of these
automorphisms. The resulting expressions are described combinatorially using a
generalization of the combinatorics of compatible pairs in a maximal Dyck path
developed by Lee, Li, and Zelevinsky. By specializing to quasi-commuting
variables we obtain pseudo-positive expressions for rank 2 quantum generalized
cluster variables. In the binomial case when all internal exchange coefficients
are zero, this quantum specialization provides a positive combinatorial
construction of counting polynomials for Grassmannians of submodules in
exceptional representations of valued quivers with two vertices.Comment: Version 2: 31 pages, several examples added to improve readabilit
Some Consequences of Categorification
Several conjectures on acyclic skew-symmetrizable cluster algebras are proven
as direct consequences of their categorification via valued quivers. These
include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella
related to -vectors, -vectors, and -polynomials