2009
DOI: 10.1093/imrn/rnp014
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A Quiver Construction of Symmetric Crystals

Abstract: In the papers [EK1], [EK2] and [EK3] with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type B. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type B are described by using the lower global basis of symmetric crystals of V θ (λ). In the present paper, we prove the existence of crystal bases and global bases of V θ (0) fo… Show more

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Cited by 8 publications
(12 citation statements)
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“…Returning to Hall modules, define operators E i , F i , T i ∈ End R (M Q ) as follows (see also [8]). Put…”
Section: Hall Modules From Quivers With Involutionmentioning
confidence: 99%
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“…Returning to Hall modules, define operators E i , F i , T i ∈ End R (M Q ) as follows (see also [8]). Put…”
Section: Hall Modules From Quivers With Involutionmentioning
confidence: 99%
“…The right-hand side of this equation is equal to E(U ⊕ V ), proving that E descends to K(A). Finally, it is straightforward to verify equation (7) using equation (8).…”
Section: Introductionmentioning
confidence: 99%
See 1 more Smart Citation
“…Similarly, the simplicial stack version of R • (C) is relative 2-Segal over S • (C) and recovers the perverse sheaf theoretic [10], motivic and cohomological [47] Hall algebra representations which appear in the representation theory of quantum enveloping algebras and orientifold Donaldson-Thomas theory. ⊳…”
Section: Remarksmentioning
confidence: 80%
“…The input for the R • -construction is a proto-exact category with duality which satisfies a reduction assumption. In the case of exact categories the R • -construction categorifies the Hall algebra representations of [43], [10], [46], [47] while for the proto-exact category Rep F1 (Q) of representations of a quiver over F 1 we obtain new modules over Szczesny's combinatorial Hall algebras [41]. The latter modules will be the subject of future work.…”
Section: Introductionmentioning
confidence: 99%