RUS  ENG
Full version
JOURNALS // Moscow Mathematical Journal
Mosc. Math. J., 2008, Volume 8, Number 1, Pages 119–158 (Mi mmj7)
$q$-Schur algebras and complex reflection groups
R. Rouquier

References
1. S. Ariki, “On the semi-simplicity of the Hecke algebra of $(\mathbb Z/r\mathbb Z)\wr\mathfrak S_n$”, J. Algebra, 169:1 (1994), 216–225  crossref  mathscinet  zmath
2. S. Ariki, “On the decomposition numbers of the Hecke algebra of $G(m,1,n)$”, J. Math. Kyoto Univ., 36:4 (1996), 789–808  mathscinet  zmath
3. S. Ariki and K. Koike, “A Hecke algebra of $(\mathbb Z/r\mathbb Z)\wr\mathfrak S_n$ and construction of its irreducible representations”, Adv. Math., 106:2 (1994), 216–243  crossref  mathscinet  zmath
4. D. J. Benson, Representations and cohomology. I, Basic representation theory of finite groups and associative algebras, Cambridge Stud. Adv. Math., 30, Cambridge University Press, Cambridge, 1991  mathscinet  zmath
5. Y. Berest, P. Etingof, and V. Ginzburg, “Finite-dimensional representations of rational Cherednik algebras”, Int. Math. Res. Not., 2003, no. 19, 1053–1088  crossref  mathscinet  zmath
6. R. Bezrukavnikov and P. Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, preliminary version
7. M. Broué, G. Malle, and J. Michel, “Towards spetses. I”, Dedicated to the memory of Claude Chevalley, Transform. Groups, 4:2–3 (1999), 157–218  crossref  mathscinet  zmath
8. M. Broué, G. Malle, and R. Rouquier, “Complex reflection groups, braid groups, Hecke algebras”, J. Reine Angew. Math., 500 (1998), 127–190  mathscinet  zmath
9. M. Broué and J. Michel, “Sur certains éléments réguliers des groupes de Weyl et les variétés de Deligne–Lusztig associées”, Finite reductive groups (Luminy, 1994), Progr. Math., 141, Birkhäuser Boston, Boston, MA, 1997, 73–139  mathscinet  zmath
10. T. Chmutova and P. Etingof, “On some representations of the rational Cherednik algebra”, Represent. Theory, 7 (2003), 641–650  crossref  mathscinet  zmath
11. J. Chuang and H. Miyachi, Runner removal Morita equivalences, Preprint, 2007
12. J. Chuang and R. Rouquier, Calabi–Yau algebras and perverse Morita equivalences, in preparation
13. E. Cline, B. Parshall, and L. Scott, “Finite-dimensional algebras and highest weight categories”, J. Reine Angew. Math., 391 (1988), 85–99  mathscinet  zmath
14. E. Cline, B. Parshall, and L. Scott, “Integral and graded quasi-hereditary algebras. I”, J. Algebra, 131:1 (1990), 126–160  crossref  mathscinet  zmath
15. R. Dipper, G. James, and A. Mathas, “Cyclotomic $q$-Schur algebras”, Math. Z., 229:3 (1998), 385–416  crossref  mathscinet  zmath
16. R. Dipper, G. James, and A. Mathas, “The $(Q,q)$-Schur algebra”, Proc. London Math. Soc. (3), 77:2 (1998), 327–361  crossref  mathscinet  zmath
17. R. Dipper and A. Mathas, “Morita equivalences of Ariki–Koike algebras”, Math. Z., 240:3 (2002), 579–610  crossref  mathscinet  zmath
18. S. Donkin, “On tilting modules for algebraic groups”, Math. Z., 212:1 (1993), 39–60  crossref  mathscinet  zmath
19. S. Donkin, “On Schur algebras and related algebras. III. Integral representations”, Math. Proc. Cambridge Philos. Soc., 116:1 (1994), 37–55  crossref  mathscinet  zmath
20. S. Donkin, The $q$-Schur algebra, London Math. Soc. Lecture Note Ser., 253, Cambridge University Press, Cambridge, 1998  mathscinet  zmath
21. S. Donkin, “Tilting modules for algebraic groups and finite dimensional algebras”, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., 332, Cambridge University Press, Cambridge, 2007, 215–257  mathscinet
22. J. Du, B. Parshall, and L. Scott, “Cells and $q$-Schur algebras”, Transform. Groups, 3:1 (1998), 33–49  crossref  mathscinet  zmath
23. J. Du, B. Parshall, and L. Scott, “Quantum Weyl reciprocity and tilting modules”, Comm. Math. Phys., 195:2 (1998), 321–352  crossref  mathscinet  zmath  adsnasa
24. J. Du, B. Parshall, and L. Scott, “Stratifying endomorphism algebras associated to Hecke algebras”, J. Algebra, 203:1 (1998), 169–210  crossref  mathscinet  zmath
25. J. Du and L. Scott, “Lusztig conjectures, old and new. I”, J. Reine Angew. Math., 1994, 141–182  mathscinet  zmath
26. J. Du and L. Scott, “The $q$-Schur$^2$ algebra”, Trans. Amer. Math. Soc., 352:9 (2000), 4325–4353  crossref  mathscinet  zmath
27. C. F. Dunkl and E. M. Opdam, “Dunkl operators for complex reflection groups”, Proc. London Math. Soc. (3), 86:1 (2003), 70–108  crossref  mathscinet  zmath
28. P. Etingof and V. Ginzburg, “Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism”, Invent. Math., 147:2 (2002), 243–348  crossref  mathscinet  zmath  adsnasa
29. P. Etingof and E. Rains, “Central extensions of preprojective algebras, the quantum Heisenberg algebra, and 2-dimensional complex reflection groups”, J. Algebra, 299:2 (2006), 570–588  crossref  mathscinet  zmath
30. M. Geck, “Brauer trees of Hecke algebras”, Comm. Algebra, 20:10 (1992), 2937–2973  crossref  mathscinet  zmath
31. M. Geck, “Kazhdan–Lusztig cells and decomposition numbers”, Represent. Theory, 2 (1998), 264–277, electronic  crossref  mathscinet  zmath
32. M. Geck, “Kazhdan–Lusztig cells, $q$-Schur algebras and James' conjecture”, J. London Math. Soc. (2), 63:2 (2001), 336–352  crossref  mathscinet  zmath
33. M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, “CHEVIE–a system for computing and processing generic character tables”, Computational methods in Lie theory (Essen, 1994), Appl. Algebra Engrg. Comm. Comput., 7:3 (1996), 175–210  crossref  mathscinet  zmath
34. M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Math. Soc. Monogr. New Ser., 21, The Clarendon Press, Oxford University Press, New York, 2000  mathscinet
35. M. Geck and R. Rouquier, “Filtrations on projective modules for Iwahori–Hecke algebras”, Modular representation theory of finite groups (Charlottesville, VA, 1998), de Gruyter, Berlin, 2001, 211–221  mathscinet  zmath
36. V. Ginzburg, N. Guay, E. Opdam, and R. Rouquier, “On the category $\mathcal O$ for rational Cherednik algebras”, Invent. Math., 154:3 (2003), 617–651  mathscinet  zmath
37. I. Gordon, Quiver varieties, category $\mathcal O$ for rational Cherednik algebras, and Hecke algebras, arXiv:math/0703150v1[math.RT]
38. I. Gordon and J. T. Stafford, “Rational Cherednik algebras and Hilbert schemes”, Adv. Math., 198:1 (2005), 222–274  crossref  mathscinet  zmath
39. I. Gordon and J. T. Stafford, “Rational Cherednik algebras and Hilbert schemes. II. Representations and sheaves”, Duke Math. J., 132:1 (2006), 73–135  crossref  mathscinet  zmath
40. D. J. Hemmer and D. K. Nakano, “Specht filtrations for Hecke algebras of type A”, J. London Math. Soc. (2), 69:3 (2004), 623–638  crossref  mathscinet  zmath
41. N. Jacon, “On the parametrization of the simple modules for Ariki–Koike algebras at roots of unity”, J. Math. Kyoto Univ., 44:4 (2004), 729–767  mathscinet  zmath
42. N. Jacon, “Sur les représentations modulaires des algèbres de Hecke de type $D_n$”, J. Algebra, 274:2 (2004), 607–628  crossref  mathscinet  zmath
43. N. Jacon, “Crystal graphs of higher level $q$-deformed Fock spaces, Lusztig $a$-values and Ariki–Koike algebras”, Algebr. Represent. Theory, 10:6 (2007), 565–591  crossref  mathscinet  zmath
44. G. James and A. Kerber, The representation theory of the symmetric group, Addison-Wesley Publ. Co., Reading, MA, 1981  mathscinet
45. A. Lascoux, B. Leclerc, and J.-Y. Thibon, “Hecke algebras at roots of unity and crystal bases of quantum affine algebras”, Comm. Math. Phys., 181:1 (1996), 205–263  crossref  mathscinet  zmath  adsnasa
46. B. Leclerc and J.-Y. Thibon, “Canonical bases of $q$-deformed Fock spaces”, Int. Math. Res. Not., 1996, no. 9, 447–456  crossref  mathscinet  zmath
47. G. Lusztig, “Left cells in Weyl groups”, Lie group representations, I (College Park, Md., 1982/1983), Lecture Notes in Math., 1024, Springer, Berlin, 1983, 99–111  mathscinet
48. G. Malle, “On the rationality and fake degrees of characters of cyclotomic algebras”, J. Math. Sci. Univ. Tokyo, 6:4 (1999), 647–677  mathscinet  zmath
49. G. Malle and A. Mathas, “Symmetric cyclotomic Hecke algebras”, J. Algebra, 205:1 (1998), 275–293  crossref  mathscinet  zmath
50. A. Mathas, “Tilting modules for cyclotomic Schur algebras”, J. Reine Angew. Math., 562 (2003), 137–169  mathscinet  zmath
51. A. Mathas, “The representation theory of the Ariki–Koike and cyclotomic $q$-Schur algebras”, Representation theory of algebraic groups and quantum groups, Adv. Stud. Pure Math., 40, Math. Soc. Japan, Tokyo, 2004, 261–320  mathscinet  zmath
52. R. Rouquier, “Familles et blocs d'algèbres de Hecke”, C. R. Acad. Sci. Paris Sér. I Math., 329:12 (1999), 1037–1042  mathscinet  zmath
53. R. Rouquier, “Representations of rational Cherednik algebras”, Infinite-dimensional aspects of representation theory and applications, Contemp. Math., 392, Amer. Math. Soc., Providence, RI, 2005, 103–131  mathscinet
54. R. Rouquier, “Derived equivalences and finite dimensional algebras”, International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, 191–221  mathscinet  zmath
55. T. Suzuki, “Double affine Hecke algebras, conformal coinvariants and Kostka polynomials”, C. R. Math. Acad. Sci. Paris, 343:6 (2006), 383–386  mathscinet  zmath
56. D. Uglov, “Canonical bases of higher-level $q$-deformed Fock spaces and Kazhdan–Lusztig polynomials”, Physical combinatorics (Kyoto, 1999), Progr. Math., 191, Birkhäuser Boston, Boston, MA, 2000, 249–299  mathscinet  zmath
57. M. Varagnolo and E. Vasserot, “On the decomposition matrices of the quantized Schur algebra”, Duke Math. J., 100:2 (1999), 267–297  crossref  mathscinet  zmath
58. M. Varagnolo and E. Vasserot, “From double affine Hecke algebras to quantized affine Schur algebras”, Int. Math. Res. Not., 2004, no. 26, 1299–1333  crossref  mathscinet  zmath
59. M. Varagnolo and E. Vasserot, Finite dimensional representations of DAHA and affine Springer fibers: the spherical case, arXiv:0705.2691
60. X. Yvonne, “A conjecture for $q$-decomposition matrices of cyclotomic $v$-Schur algebras”, J. Algebra, 304:1 (2006), 419–456  crossref  mathscinet  zmath

© Steklov Math. Inst. of RAS, 2026