Quantum cluster algebras of type <i>A</i>
 and the dual canonical basis

Lampe, Philipp

doi:10.1112/plms/pds098

Cited by 12 publications

(8 citation statements)

References 52 publications

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“…It was recently proved in [9] that cluster monomials in A Q are always linearly independent over Z. Moreover, cluster monomials play a prominent role in the construction of Z-linear bases of A Q which are of interest with respect to the study of dual canonical bases of quantum groups; see, for instance, [20,22,24,26,27,33].…”

Section: Cluster Algebrasmentioning

confidence: 99%

Atomic bases of cluster algebras of types A and Ã

Dupont

Thomas

2013

Proceedings of the London Mathematical Society

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Section: Cluster Algebrasmentioning

confidence: 99%

Atomic bases of cluster algebras of types A and Ã

Dupont

Thomas

2013

Proceedings of the London Mathematical Society

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“…Grabowski and Launois [GL] have shown that the quantum coordinate rings of the Grassmannians Gr(2, n) (n ≥ 2), Gr(3, 6), Gr(3, 7), and Gr(3, 8) have a quantum cluster algebra structure. Lampe [La1,La2] has proved two particular instances of Theorem 1.1, namely when g has type A n or A (1) 1 and w = c 2 is the square of a Coxeter element. Recently, the existence of a quantum cluster structure on every algebra A q (n(w)) was conjectured by Kimura [Ki,Conj.1.1].…”

Section: Introductionmentioning

confidence: 99%

“…In fact, it is not too difficult to show that κ(Y T ) always satisfies one of the two characteristic properties of B * (see below §12.3). But unfortunately, the second property remains elusive, although Lampe [La1,La2] has proved it for all cluster variables in the two special cases mentioned above.…”

Section: Introductionmentioning

confidence: 99%

Cluster structures on quantum coordinate rings

Geiß

Leclerc

Schröer

2012

Sel. Math. New Ser.

127

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We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a q-analogue of a T -system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.

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“…In the hopes of gaining a combinatorial grasp of the structure of the canonical basis they provided the definition of a recursively, combinatorially defined algebra with a deep conjecture in mind: certain "cluster monomials" appearing in this construction should be identifiable with (dual) canonical basis vectors. This motivating conjecture has been established so far in a very restricted set of cases by Lampe [20,21] and Kimura-Qin [16]. One aim of this note is to shed some additional light on this "dual canonical basis conjecture".…”

Section: Introductionmentioning

confidence: 76%

The Feigin Tetrahedron

Rupel¹

2015

SIGMA

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Abstract. 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 skewsymmetrizable 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. We sketch a proof in the symmetric case giving an alternative to the proof of Kimura-Qin that all non-initial cluster variables in an acyclic skew-symmetric quantum cluster algebra are contained in the dual canonical basis. With these results in mind we interpret the cluster mutations directly in terms of the representation theory of the KLR algebra.

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Quantum cluster algebras of type A and the dual canonical basis

Cited by 12 publications

References 52 publications

Atomic bases of cluster algebras of types A and Ã

Atomic bases of cluster algebras of types A and Ã

Cluster structures on quantum coordinate rings

The Feigin Tetrahedron

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