Dual canonical bases, quantum shuffles and q-characters

By Bernard Leclerc. 2002

123 citations recorded by the source corpus.

Abstract.Rosso and Green have shown how to embed the positive part Uq() of a quantum enveloping algebra Uq() in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis B* of Uq() under this embedding Φ. This is motivated by the fact that when is of type Ar, the elements of Φ(B*) are q-analogues of irreducible characters of the affine Iwahori-Hecke algebras attached to the groups GL(m) over a p-adic field.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e5402bd48e82ec8cc463bcb9be5bfa392dbdbae8

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Citations in the corpus

A diagrammatic approach to categorification of quantum groups II (2008, unknown): “Given two functions f and g on sets Seq(ν) and Seq(ν′), respectively, with values in some commutative ring which contains Z[q, q−1], we define their (quantum) shuffle product f ∪∪ g (see [30] and references therein) as a function on Seq(ν + ν′) given by (f ∪∪ g)(k) = ∑”
A Quantum Cluster Algebra of Kronecker Type and the Dual Canonical Basis (2010, methodology): “The corresponding version for dual canonical bases is described in [28], [32], and [25]; in our example the characterization is Theorem 4.1.”
Imaginary vectors in the dual canonical basis of Uq(n) (2002, methodology): “Following Rosso [Ro3] and Lalonde and Ram [LR], it is shown in [ Le ] that to each total ordering of the set of simple roots corresponds a certain (dual) PBW basis, whose root vectors can be readily calculated using (1) by taking some iterated q-bracketings.”
The cohomological Hall algebra of a preprojective algebra (2014, methodology): “The shuffle description of U q (ĝ) for an affine Lie algebra ĝ was used to provide a combinatorial characterization of the dual canonical basis, and hence the shuffle formula became a basic calculation tool in q-characters of the KLR algebra [25, 28].”
Graded decomposition numbers for cyclotomic Hecke algebras (2009, methodology): “If L ∈ Rep(H) is irreducible then L ∈ Rep(H) is irreducible too.”