PBW bases and marginally large tableaux in type $D$

Salisbury, Ben; Schultze, Adam; Tingley, Peter

doi:10.4310/joc.2018.v9.n3.a6

Cited by 4 publications

(7 citation statements)

References 11 publications

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“…Remark 4.12. There is another reduced expression i P Rpw 0 q which gives a nice combinatorial description of B i [30]. But we do not know whether we may obtain results similar to the ones in this paper using this reduced expression.…”

Section: 3mentioning

confidence: 70%

Quantum nilpotent subalgebras of classical quantum groups and affine crystals

Jang

Kwon

2019

Journal of Combinatorial Theory, Series A

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We study the crystal of quantum nilpotent subalgebra of UqpDnq associated to a maximal Levi subalgebra of type An´1. We show that it has an affine crystal structure of type D p1q n isomorphic to a limit of perfect Kirillov-Reshetikhin crystal B n,s for s ě 1, and give a new polytope realization of B n,s . We show that an analogue of RSK correspondence for type D due to Burge is an isomorphism of affine crystals and give a generalization of Greene's formula for type D.

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Section: 3mentioning

confidence: 70%

Quantum nilpotent subalgebras of classical quantum groups and affine crystals

Jang

Kwon

2019

Journal of Combinatorial Theory, Series A

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“…For other types, recall that there is another realization of Bp8q of type ABCD and type G 2 in terms of marginally large tableaux [8]. Recently, a simple combinatorial description of the isomorphism from the crystal of marginally large tableaux of type BCD to that of i-Lusztig data is obtained in [34,35] for some i which is different from our choice. Using the insertion scheme of BCD types, one may describe an embedding of KN λ into the crystal of marginally large tableaux to have an embedding from KN λ into B i , but it seems more involved than in type A (see Remark 5.10).…”

Section: Introductionmentioning

confidence: 91%

Lusztig data of Kashiwara–Nakashima tableaux in types B and C

Kwon

2018

Journal of Algebra

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We provide an explicit combinatorial description of the embedding of the crystal of Kashiwara-Nakashima tableaux in types B and C into that of i-Lusztig data associated to a family of reduced expressions i of the longest element w0. The description of the embedding is simple and elementary using only the Schützenberger's jeu de taquin and RSK algorithm. A spinor model for classical crystals plays an important role as an intermediate object connecting Kashiwara-Nakashima tableaux and Lusztig data.2010 Mathematics Subject Classification. 17B37, 22E46, 05E10.(1.2)where r " xω λ , h n y, T λ is another tableau model of Bpω λ q introduced in [19,20], and V λ is the crystal of a parabolic Verma module with respect to a maximal Levi subalgebra l of type A n´1 in g [16,17].

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“…Here we review the crystal structure on Kostant partitions from [10]. As explained there, this is naturally identified with the crystal of PBW monomials as given in [1,9] (see also [12]) for the reduced expression…”

Section: Crystal Of Kostant Partitionsmentioning

confidence: 99%

“…The first is the marginally large tableaux construction of [4,5]. The second uses the Kostant partitions from [10], which are related to Lusztig's PBW bases [9] (see also [12]). In [3] and [11], isomorphisms between these two realizations are studied in types A n and D n , respectively.…”

Section: Introductionmentioning

confidence: 99%