The Dipper-Du conjecture revisited

Norton, Emily

doi:10.1090/ert/581

The Dipper-Du conjecture revisited
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by Emily Norton

Represent. Theory 25 (2021), 748-759

DOI: https://doi.org/10.1090/ert/581

Published electronically: September 3, 2021

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Abstract:

We consider vertices, a notion originating in local representation theory of finite groups, for the category $\mathcal {O}$ of a rational Cherednik algebra and prove the analogue of the Dipper-Du Conjecture for Hecke algebras of symmetric groups in that setting. As a corollary we obtain a new proof of the Dipper-Du Conjecture over $\mathbb {C}$.

References

J. L. Alperin, Local representation theory, Cambridge Studies in Advanced Mathematics, vol. 11, Cambridge University Press, Cambridge, 1986. Modular representations as an introduction to the local representation theory of finite groups. MR 860771, DOI 10.1017/CBO9780511623592
Yuri Berest, Pavel Etingof, and Victor Ginzburg, Finite-dimensional representations of rational Cherednik algebras, Int. Math. Res. Not. 19 (2003), 1053–1088. MR 1961261, DOI 10.1155/S1073792803210205
Roman Bezrukavnikov and Pavel Etingof, Parabolic induction and restriction functors for rational Cherednik algebras, Selecta Math. (N.S.) 14 (2009), no. 3-4, 397–425. MR 2511190, DOI 10.1007/s00029-009-0507-z
Michel Broué, Higman’s criterion revisited, Michigan Math. J. 58 (2009), no. 1, 125–179. MR 2526081, DOI 10.1307/mmj/1242071686
E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85–99. MR 961165
Richard Dipper and Jie Du, Trivial and alternating source modules of Hecke algebras of type $A$, Proc. London Math. Soc. (3) 66 (1993), no. 3, 479–506. MR 1207545, DOI 10.1112/plms/s3-66.3.479
Jie Du, The Green correspondence for the representations of Hecke algebras of type $A_{r-1}$, Trans. Amer. Math. Soc. 329 (1992), no. 1, 273–287. MR 1022164, DOI 10.1090/S0002-9947-1992-1022164-1
Olivier Dudas and Jean Michel, Lectures on finite reductive groups and their representations, (2015), http://webusers.imj-prg.fr/~jean.michel/papiers/lectures_beijing_2015.pdf
Richard Dipper and Gordon James, Blocks and idempotents of Hecke algebras of general linear groups, Proc. London Math. Soc. (3) 54 (1987), no. 1, 57–82. MR 872250, DOI 10.1112/plms/s3-54.1.57
Meinolf Geck and Nicolas Jacon, Representations of Hecke algebras at roots of unity, Algebra and Applications, vol. 15, Springer-Verlag London, Ltd., London, 2011. MR 2799052, DOI 10.1007/978-0-85729-716-7
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphaël Rouquier, On the category $\scr O$ for rational Cherednik algebras, Invent. Math. 154 (2003), no. 3, 617–651. MR 2018786, DOI 10.1007/s00222-003-0313-8
Stephen Griffeth and Daniel Juteau, $W$-exponentials, Schur elements, and the support of the spherical representation of the rational Cherednik algebra, Ann. Sci. Éc. Norm. Supér. (4) 53 (2020), no. 5, 1335–1362 (English, with English and French summaries). MR 4174848, DOI 10.24033/asens.2448
Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
Toshiro Kuwabara, Hyohe Miyachi and Kentaro Wada. On the Mackey formula for cyclotomic Hecke algebras and categories $\mathcal {O}$ of rational Cherednik algebras. arXiv:1801.03761, 2018.
Markus Linckelmann, The block theory of finite group algebras. Vol. I, London Mathematical Society Student Texts, vol. 91, Cambridge University Press, Cambridge, 2018. MR 3821516
Ivan Losev, On isomorphisms of certain functors for Cherednik algebras, Represent. Theory 17 (2013), 247–262. MR 3054265, DOI 10.1090/S1088-4165-2013-00437-5
Ivan Losev, Finite-dimensional quotients of Hecke algebras, Algebra Number Theory 9 (2015), no. 2, 493–502. MR 3320850, DOI 10.2140/ant.2015.9.493
Ivan Losev and Seth Shelley-Abrahamson, On refined filtration by supports for rational Cherednik categories $\mathcal {O}$, Selecta Math. (N.S.) 24 (2018), no. 2, 1729–1804. MR 3782434, DOI 10.1007/s00029-018-0390-6
Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
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
Peng Shan, Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 147–182 (English, with English and French summaries). MR 2760196, DOI 10.24033/asens.2141
P. Shan and E. Vasserot, Heisenberg algebras and rational double affine Hecke algebras, J. Amer. Math. Soc. 25 (2012), no. 4, 959–1031. MR 2947944, DOI 10.1090/S0894-0347-2012-00738-3
James R. Whitley, Vertices for Iwahori-Hecke algebras and the Dipper-Du conjecture, Proc. Lond. Math. Soc. (3) 119 (2019), no. 2, 379–408. MR 3959048, DOI 10.1112/plms.12230
Stewart Wilcox, Supports of representations of the rational Cherednik algebra of type A, Adv. Math. 314 (2017), 426–492. MR 3658722, DOI 10.1016/j.aim.2017.04.031