Geometric Schur duality of classical type

Bao, Huanchen; Kujawa, Jonathan R.; Li, Yiqiang; Wang, Weiqiang

doi:10.48550/arxiv.1404.4000

Cited by 17 publications

(35 citation statements)

References 22 publications

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“…Recall (cf. [BW13,BKLW] and the references therein) there is a quantum coideal algebra U  (gl n ) which can be embedded in U(gl n ), and (U(gl n ), U  (gl n )) form a quantum symmetric pair in the sense of Letzter. For our purpose here, its modified version U (gl n ) is more directly relevant; we recall its presentation below from [BKLW,§4.4] to fix some notation.…”

Section: 3mentioning

confidence: 99%

“…Denote |A| = 2d 0 +1, and d = d 0 +pn/2. We have the following commutative diagram [BKLW,(4.8)], we have…”

Section: Negativity Of Stably Canonical Basis For U (Glmentioning

confidence: 99%

“…As in [BW13,BKLW], the different behaviors in the cases for n odd and even force us to carry out the studies of the two cases separately in this paper. The case of odd n, indicated by the superscript , is easier and done first, while the remaining case is indicated by the superscript ı.…”

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confidence: 99%

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Positivity VS Negativity of Canonical Bases

Li¹,

Wang²

2018

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

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We provide examples for negativity of structure constants of the stably canonical basis of modified quantum gl n and an analogous basis of modified quantum coideal algebra of gl n . In contrast, we construct the canonical basis of the modified quantum coideal algebra of sln, establish the positivity of its structure constants, the positivity with respect to a geometric bilinear form as well as the positivity of its action on the tensor powers of the natural representation. The matrix coefficients of the transfer map on these Schur algebras with respect to the canonical bases are shown to be positive. Formulas for canonical basis of the iSchur algebra of rank one are obtained.

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Section: 3mentioning

confidence: 99%

“…Denote |A| = 2d 0 +1, and d = d 0 +pn/2. We have the following commutative diagram [BKLW,(4.8)], we have…”

Section: Negativity Of Stably Canonical Basis For U (Glmentioning

confidence: 99%

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Positivity VS Negativity of Canonical Bases

Li¹,

Wang²

2018

Bulletin of the Inst. of Math. Academia Sinica(N.S.)

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“…By definition, U ı = U ı ς is a coideal subalgebra of U depending on parameters ς = (ς i ) i∈I (subject to some compatibility conditions) and will be referred to as an ıquantum group in this paper. As suggested in [BW18a], most of the fundamental constructions in the theory of quantum groups should admit generalizations in the setting of ıquantum groups; see [BW18a,BK19,BW18b] for generalizations of (quasi) R-matrix and canonical bases, and also see [BKLW18] (and [Li18]) for a geometric realization and [BSWW18] for KLR type categorification of a class of (modified) U ı .…”

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confidence: 99%

Hall algebras and quantum symmetric pairs I: foundations

Lu¹,

Wang²

2019

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A quantum symmetric pair consists of a quantum group U and its coideal subalgebra U ı ς with parameters ς (called an ıquantum group). We initiate a Hall algebra approach for the categorification of ıquantum groups. A universal ıquantum group U ı is introduced and U ı ς is recovered by a central reduction of U ı . The modified Ringel-Hall algebras of the first author and Peng, which are closely related to semi-derived Hall algebras of Gorsky and motivated by Bridgeland's work, are extended to the setting of 1-Gorenstein algebras, as shown in Appendix A by the first author. A new class of 1-Gorenstein algebras (called ıquiver algebras) arising from acyclic quivers with involutions is introduced. The modified Ringel-Hall algebras for the Dynkin ıquiver algebras are shown to be isomorphic to the universal quasi-split ıquantum groups of finite type, and a reduced version provides a categorification of U ı ς . Monomial bases and PBW bases for these Hall algebras and ıquantum groups are constructed. In the special case of quivers of diagonal type, our construction reduces to a reformulation of Bridgeland's Hall algebra realization of quantum groups. MING LU AND WEIQIANG WANG 3.3. Λ ı -modules with finite projective dimensions 3.4. Projective Λ ı -modules 3.5. Singularity category of Gorenstein algebras 3.6. Singularity category for ıquivers 4. Hall algebras for ıquivers 4.1. Euler forms 4.2. Modified Ringel-Hall algebras for Λ ı 4.3. A Hall basis 4.4. Hall algebras for ıquivers 4.5. Hall algebras for ısubquivers 5. Monomial bases and PBW bases of Hall algebras for ıquivers 5.1. Monomial basis of H(kQ) 5.2. Monomial bases for Hall algebras of ıquivers 5.3. PBW bases for Hall algebras of ıquivers Part 2. ıHall Algebras and ıQuantum Groups 6. Quantum symmetric pairs and ıquantum groups 6.1. Quantum groups 6.2. The ıquantum groups U ı and U ı 6.3. ıQuantum groups of type ADE 6.4. Presentation of U ı 7. Hall algebras for ıquivers and ıquantum groups 7.1. Computations for rank 2 ıquivers, I 7.2. Computations for rank 2 ıquivers, II 7.3. The homomorphism ψ 7.4. ıQuantum groups via ıHall algebras 8. Bridgeland's theorem revisited 8.1. A category equivalence 8.2. Quantum group as an ıquantum group 8.3. Bridgeland's theorem reformulated 9. Generic Hall algebras for Dynkin ıquivers 9.1. Hall polynomials 9.2. Generic Hall algebras, I 9.3. Generic Hall algebras, II Appendix A. Modified Ringel-Hall algebras of 1-Gorenstein algebras by Ming Lu A.1. Hall algebras A.2. Definition of modified Ringel-Hall algebras A.3. 1-Gorenstein algebras A.4. Semi-derived Hall algebras A.5. Tilting invariance References

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“…There has been a completely different geometric construction of the modified quantum groups U (of type A) and their canonical bases with positivity [BLM90,Lus93], which is compatible with the Khovanov-Lauda-Rouquier (KLR) categorification [KL10,R08]. There has also been a geometric construction of the modified ıquantum groups Uı (of type AIII) and their ıcanonical bases with positivity [BKLW18,LiW18,BW18b], which is again compatible with a KLR type categorification [BSWW18]. It is natural to ask if there is any connection between these constructions for modified quantum groups and canonical bases (respectively, for ıquantum groups and ıcanonical bases) and the geometric constructions of the Drinfeld doubles and dual canonical bases in [Qin16,SS16] (respectively, in this paper).…”

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Hall algebras and quantum symmetric pairs III: Quiver varieties

Lu¹,

Wang²

2019

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The ıquiver algebras were introduced recently by the authors to provide a Hall algebra realization of universal ıquantum groups, which is a generalization of Bridgeland's Hall algebra construction for (Drinfeld doubles of) quantum groups; here an ıquantum group and a corresponding Drinfeld-Jimbo quantum group form a quantum symmetric pair. In this paper, the Dynkin ıquiver algebras are shown to arise as new examples of singular Nakajima-Keller-Scherotzke categories. Then we provide a geometric construction of the universal ıquantum groups and their "dual canonical bases" with positivity, via the quantum Grothendieck rings of Nakajima-Keller-Scherotzke quiver varieties, generalizing Qin's geometric realization of quantum groups of type ADE.2010 Mathematics Subject Classification. Primary 17B37, 18E30. Key words and phrases. Hall algebras, ıquantum groups, quantum symmetric pairs, NKS categories and quiver varieties. MING LU AND WEIQIANG WANG 4.1. Strongly l-dominant pairs (v, w i ) 23 4.2. Strongly l-dominant pairs (v, w ij ) 26 4.3. Strongly l-dominant pairs (v, w ijk ) 30 5. Computation in quantum Grothendieck rings for ıquivers 32 5.1. A bilinear form 33 5.2. Computation for rank 2 ıquivers, I 35 5.3. Computation for rank 2 ıquivers, II 37 5.4. Computation for rank 2 ıquivers, III 39 6. Geometric realization of ıquantum groups 41 6.1. Quantum groups 42 6.2. Theorems of Hernandez-Leclerc and Qin 42 6.3. The ıquantum groups 44 6.4. Decomposition of R ı 45 6.5. Filtered algebra R ı 49 6.6. Generators for R ı 51 6.7. Realization of ıquantum groups 53 References 54

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Geometric Schur duality of classical type

Cited by 17 publications

References 22 publications

Positivity VS Negativity of Canonical Bases

Positivity VS Negativity of Canonical Bases

Hall algebras and quantum symmetric pairs I: foundations

Hall algebras and quantum symmetric pairs III: Quiver varieties

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