## Kleshchev’s decomposition numbers for diagrammatic Cherednik algebras

HTML articles powered by AMS MathViewer

- by C. Bowman and L. Speyer PDF
- Trans. Amer. Math. Soc.
**370**(2018), 3551-3590 Request permission

## Abstract:

We construct a family of graded isomorphisms between certain subquotients of diagrammatic Cherednik algebras as the quantum characteristic, multicharge, level, degree, and weighting are allowed to vary; this provides new structural information even in the case of the classical $q$-Schur algebra. This also allows us to prove some of the first results concerning the (graded) decomposition numbers of these algebras over fields of arbitrary characteristic.## References

- Susumu Ariki,
*On the decomposition numbers of the Hecke algebra of $G(m,1,n)$*, J. Math. Kyoto Univ.**36**(1996), no. 4, 789–808. MR**1443748**, DOI 10.1215/kjm/1250518452 - Christopher Bowman,
*The many graded cellular bases of Hecke algebras*, arXiv:1702.06579, 2017, preprint. - Christopher Bowman, Anton Cox, and Liron Speyer,
*A family of graded decomposition numbers for diagrammatic Cherednik algebras*, Int. Math. Res. Not. IMRN**9**(2017), 2686–2734. MR**3658213**, DOI 10.1093/imrn/rnw101 - Joseph Chuang, Hyohe Miyachi, and Kai Meng Tan,
*Kleshchev’s decomposition numbers and branching coefficients in the Fock space*, Trans. Amer. Math. Soc.**360**(2008), no. 3, 1179–1191. MR**2357693**, DOI 10.1090/S0002-9947-07-04202-X - Joseph Chuang and Kai Meng Tan,
*Canonical bases for Fock spaces and tensor products*, Adv. Math.**302**(2016), 159–189. MR**3545928**, DOI 10.1016/j.aim.2016.07.008 - Richard Dipper, Gordon James, and Andrew Mathas,
*Cyclotomic $q$-Schur algebras*, Math. Z.**229**(1998), no. 3, 385–416. MR**1658581**, DOI 10.1007/PL00004665 - S. Donkin,
*The $q$-Schur algebra*, London Mathematical Society Lecture Note Series, vol. 253, Cambridge University Press, Cambridge, 1998. MR**1707336**, DOI 10.1017/CBO9780511600708 - Jun Hu and Andrew Mathas,
*Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type $A$*, Adv. Math.**225**(2010), no. 2, 598–642. MR**2671176**, DOI 10.1016/j.aim.2010.03.002 - Alexander Kleshchev,
*On decomposition numbers and branching coefficients for symmetric and special linear groups*, Proc. London Math. Soc. (3)**75**(1997), no. 3, 497–558. MR**1466660**, DOI 10.1112/S0024611597000427 - Alexander Kleshchev and David Nash,
*An interpretation of the Lascoux-Leclerc-Thibon algorithm and graded representation theory*, Comm. Algebra**38**(2010), no. 12, 4489–4500. MR**2764833**, DOI 10.1080/00927870903386536 - Ivan Losev,
*Proof of Varagnolo-Vasserot conjecture on cyclotomic categories $\mathcal {O}$*, Selecta Math. (N.S.)**22**(2016), no. 2, 631–668. MR**3477332**, DOI 10.1007/s00029-015-0209-7 - Raphaël Rouquier,
*$q$-Schur algebras and complex reflection groups*, Mosc. Math. J.**8**(2008), no. 1, 119–158, 184 (English, with English and Russian summaries). MR**2422270**, DOI 10.17323/1609-4514-2008-8-1-119-158 - Raphaël Rouquier, Peng Shan, Michela Varagnolo, and Eric Vasserot,
*Categorifications and cyclotomic rational double affine Hecke algebras*, Invent. Math.**204**(2016), no. 3, 671–786. MR**3502064**, DOI 10.1007/s00222-015-0623-7 - Kai Meng Tan and Wei Hao Teo,
*Sign sequences and decomposition numbers*, Trans. Amer. Math. Soc.**365**(2013), no. 12, 6385–6401. MR**3105756**, DOI 10.1090/S0002-9947-2013-05860-6 - Denis Uglov,
*Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan-Lusztig polynomials*, Physical combinatorics (Kyoto, 1999) Progr. Math., vol. 191, Birkhäuser Boston, Boston, MA, 2000, pp. 249–299. MR**1768086** - B. Webster,
*Weighted Khovanov–Lauda–Rouquier algebras*, arXiv:1209.2463v3, 2012, preprint. - Ben Webster,
*Rouquier’s conjecture and diagrammatic algebra*, Forum Math. Sigma**5**(2017), Paper No. e27, 71. MR**3732238**, DOI 10.1017/fms.2017.17

## Additional Information

**C. Bowman**- Affiliation: Department of Mathematics, City University London, Northampton Square, London, EC1V 0HB, United Kingdom
- Address at time of publication: School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, United Kingdom
- MR Author ID: 922280
- Email: c.d.bowman@kent.ac.uk
**L. Speyer**- Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
- Address at time of publication: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22904
- MR Author ID: 1076718
- Email: l.speyer@virginia.edu
- Received by editor(s): July 28, 2015
- Received by editor(s) in revised form: August 12, 2016
- Published electronically: December 20, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 3551-3590 - MSC (2010): Primary 05E10, 20C08, 20C30
- DOI: https://doi.org/10.1090/tran/7054
- MathSciNet review: 3766858