AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Affine Hecke algebras and quantum symmetric pairs
About this Title
Zhaobing Fan, Chun-Ju Lai, Yiqiang Li, Li Luo and Weiqiang Wang
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 281, Number 1386
ISBNs: 978-1-4704-5626-9 (print); 978-1-4704-7319-8 (online)
DOI: https://doi.org/10.1090/memo/1386
Published electronically: January 3, 2023
Table of Contents
Chapters
- Acknowledgment
- Notations
- 1. Introduction
1. Affine Schur algebras
- 2. Affine Schur algebras via affine Hecke algebras
- 3. Multiplication formula for affine Hecke algebra
- 4. Multiplication formula for affine Schur algebra
- 5. Monomial and canonical bases for affine Schur algebra
2. Affine quantum symmetric pairs
- 6. Stabilization algebra $\dot {\mathbf {K}}^{\mathfrak {c}}_n$ arising from affine Schur algebras
- 7. The quantum symmetric pair $(\mathbf {K}_n, \mathbf {K}^{\mathfrak {c}}_n)$
- 8. Stabilization algebras arising from other Schur algebras
- A. Length formulas in symmetrized forms by Zhaobing Fan, Chun-Ju Lai, Yiqiang Li and Li Luo
Abstract
We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra ${\mathbf K}^{\mathfrak c}_n$. We show that ${\mathbf K}^{\mathfrak c}_n$ is a coideal subalgebra of quantum affine algebra ${\mathbf {U}}(\widehat {\mathfrak {gl}}_n)$, and $\big ({\mathbf {U}}(\widehat { \mathfrak {gl}}_n), {\mathbf K}^{\mathfrak c}_n)$ forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.\frenchspacing
- Anders Björner and Francesco Brenti, Affine permutations of type $A$, Electron. J. Combin. 3 (1996), no. 2, Research Paper 18, approx. 35. The Foata Festschrift. MR 1392503, DOI 10.37236/1276
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
- Robert Bédard, Cells for two Coxeter groups, Comm. Algebra 14 (1986), no. 7, 1253–1286. MR 842039, DOI 10.1080/00927878608823364
- A. A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\textrm {GL}_n$, Duke Math. J. 61 (1990), no. 2, 655–677. MR 1074310, DOI 10.1215/S0012-7094-90-06124-1
- Huanchen Bao, Jonathan Kujawa, Yiqiang Li, and Weiqiang Wang, Geometric Schur duality of classical type, Transform. Groups 23 (2018), no. 2, 329–389. MR 3805209, DOI 10.1007/s00031-017-9447-4
- Huanchen Bao and Weiqiang Wang, A new approach to Kazhdan-Lusztig theory of type $B$ via quantum symmetric pairs, Astérisque 402 (2018), vii+134 (English, with English and French summaries). MR 3864017
- Hongjia Chen, Nicolas Guay, and Xiaoguang Ma, Twisted Yangians, twisted quantum loop algebras and affine Hecke algebras of type $BC$, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2517–2574. MR 3165646, DOI 10.1090/S0002-9947-2014-05994-1
- Charles W. Curtis, On Lusztig’s isomorphism theorem for Hecke algebras, J. Algebra 92 (1985), no. 2, 348–365. MR 778453, DOI 10.1016/0021-8693(85)90125-5
- Bangming Deng, Jie Du, and Qiang Fu, A double Hall algebra approach to affine quantum Schur-Weyl theory, London Mathematical Society Lecture Note Series, vol. 401, Cambridge University Press, Cambridge, 2012. MR 3113018, DOI 10.1017/CBO9781139226660
- Bangming Deng, Jie Du, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, Mathematical Surveys and Monographs, vol. 150, American Mathematical Society, Providence, RI, 2008. MR 2457938, DOI 10.1090/surv/150
- Richard Dipper and Gordon James, The $q$-Schur algebra, Proc. London Math. Soc. (3) 59 (1989), no. 1, 23–50. MR 997250, DOI 10.1112/plms/s3-59.1.23
- Jie Du and Qiang Fu, The integral quantum loop algebra of $\mathfrak {gl}_n$, Int. Math. Res. Not. IMRN 20 (2019), 6179–6215. MR 4031235, DOI 10.1093/imrn/rnx300
- Jie Du and Qiang Fu, Quantum affine $\mathfrak {gl}_n$ via Hecke algebras, Adv. Math. 282 (2015), 23–46. MR 3374521, DOI 10.1016/j.aim.2015.06.007
- Jie Du, Kazhdan-Lusztig bases and isomorphism theorems for $q$-Schur algebras, Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989) Contemp. Math., vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 121–140. MR 1197832, DOI 10.1090/conm/139/1197832
- Henrik Eriksson and Kimmo Eriksson, Affine Weyl groups as infinite permutations, Electron. J. Combin. 5 (1998), Research Paper 18, 32. MR 1611984, DOI 10.37236/1356
- Zhaobing Fan and Yiqiang Li, Positivity of canonical bases under comultiplication, Int. Math. Res. Not. IMRN 9 (2021), 6871–6931. MR 4251292, DOI 10.1093/imrn/rnz047
- Zhaobing Fan and Yiqiang Li, Affine flag varieties and quantum symmetric pairs, II. Multiplication formula, J. Pure Appl. Algebra 223 (2019), no. 10, 4311–4347. MR 3958094, DOI 10.1016/j.jpaa.2019.01.011
- Zhaobing Fan, Chun-Ju Lai, Yiqiang Li, Li Luo, and Weiqiang Wang, Affine flag varieties and quantum symmetric pairs, Mem. Amer. Math. Soc. 265 (2020), no. 1285, v+123. MR 4080913, DOI 10.1090/memo/1285
- R. M. Green, Hyperoctahedral Schur algebras, J. Algebra 192 (1997), no. 1, 418–438. MR 1449968, DOI 10.1006/jabr.1996.6935
- R. M. Green, The affine $q$-Schur algebra, J. Algebra 215 (1999), no. 2, 379–411. MR 1686197, DOI 10.1006/jabr.1998.7753
- Victor Ginzburg and Éric Vasserot, Langlands reciprocity for affine quantum groups of type $A_n$, Internat. Math. Res. Notices 3 (1993), 67–85. MR 1208827, DOI 10.1155/S1073792893000078
- Jun Hu and Zhiqiang Li, On tensor spaces over Hecke algebras of type $B_n$, J. Algebra 304 (2006), no. 1, 602–611. MR 2256408, DOI 10.1016/j.jalgebra.2006.04.003
- Stefan Kolb, Quantum symmetric Kac-Moody pairs, Adv. Math. 267 (2014), 395–469. MR 3269184, DOI 10.1016/j.aim.2014.08.010
- David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165–184. MR 560412, DOI 10.1007/BF01390031
- Gail Letzter, Coideal subalgebras and quantum symmetric pairs, New directions in Hopf algebras, Math. Sci. Res. Inst. Publ., vol. 43, Cambridge Univ. Press, Cambridge, 2002, pp. 117–165. MR 1913438
- Chun-Ju Lai and Li Luo, An elementary construction of monomial bases of modified quantum affine $\mathfrak {gl}_n$, J. Lond. Math. Soc. (2) 96 (2017), no. 1, 15–27. MR 3687937, DOI 10.1112/jlms.12049
- George Lusztig, Some examples of square integrable representations of semisimple $p$-adic groups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 623–653. MR 694380, DOI 10.1090/S0002-9947-1983-0694380-4
- G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
- George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
- G. Lusztig, Aperiodicity in quantum affine $\mathfrak {g}\mathfrak {l}_n$, Asian J. Math. 3 (1999), no. 1, 147–177. Sir Michael Atiyah: a great mathematician of the twentieth century. MR 1701926, DOI 10.4310/AJM.1999.v3.n1.a7
- George Lusztig, Transfer maps for quantum affine $\mathfrak {s}\mathfrak {l}_n$, Representations and quantizations (Shanghai, 1998) China High. Educ. Press, Beijing, 2000, pp. 341–356. MR 1802182
- Yiqiang Li and Weiqiang Wang, Positivity vs negativity of canonical bases, Bull. Inst. Math. Acad. Sin. (N.S.) 13 (2018), no. 2, 143–198. MR 3792711
- Jian Yi Shi, Some results relating two presentations of certain affine Weyl groups, J. Algebra 163 (1994), no. 1, 235–257. MR 1257316, DOI 10.1006/jabr.1994.1015
- O. Schiffmann and E. Vasserot, Geometric construction of the global base of the quantum modified algebra of $\widehat {\mathfrak {gl}}_n$, Transform. Groups 5 (2000), no. 4, 351–360. MR 1800532, DOI 10.1007/BF01234797
- Marie-France Vignéras, Schur algebras of reductive $p$-adic groups. I, Duke Math. J. 116 (2003), no. 1, 35–75. MR 1950479, DOI 10.1215/S0012-7094-03-11612-9