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Champ: a Cherednik algebra Magma package

  • U. Thiel (a1)
Abstract

We present a computer algebra package based on Magma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded $G$ -module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.

Supplementary materials are available with this article.

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References
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1. Bellamy, G., ‘On singular Calogero–Moser spaces’, Bull. Lond. Math. Soc. 41 (2009) no. 2, 315326.
2. Bellamy, G. and Martino, M., ‘On the smoothness of centres of rational Cherednik algebras in positive characteristic’, Glasg. Math. J. 55 (2011) no. A, 2754.
3. Benard, M., ‘Schur indices and splitting fields of the unitary reflection groups’, J. Algebra 38 (1976) no. 2, 318342.
4. Bezem, M., Klop, J. W. and de Vrijer, R., Term rewriting systems (Cambridge University Press, 2003).
5. Bonnafé, C. and Rouquier, R., ‘Cellules de Calogero–Moser’, Preprint, 2013, arXiv:1302.2720.
6. Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997) no. 3–4, 235265.
7. Broué, M. and Kim, S., ‘Familles de caractères des algèbres de Hecke cyclotomiques’, Adv. Math. 172 (2002) no. 1, 53136.
8. Chlouveraki, M., Blocks and families for cyclotomic Hecke algebras , Lecture Notes in Mathematics 1981 (Springer, 2009).
9. Etingof, P. and Ginzburg, V., ‘Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism’, Invent. Math. 147 (2002) no. 2, 243348.
10. Finkelberg, M. and Ginzburg, V., ‘Calogero–Moser space and Kostka polynomials’, Adv. Math. 172 (2002) no. 1, 137150.
11. Geck, M., An introduction to algebraic geometry and algebraic groups , Oxford Graduate Texts in Mathematics 10 (Oxford University Press, 2003).
12. Geck, M., Hiss, G., Lübeck, F., Malle, G. and Pfeiffer, G., ‘CHEVIE — a system for computing and processing generic character tables’, Appl. Algebra Engrg. Comm. Comput. 7 (1996) no. 3, 175210; Pre-packaged GAP3 version by J. Michel (version from March 2012).
13. Geck, M. and Jacon, N., Representations of Hecke algebras at roots of unity , Algebra and Applications 15 (Springer, London, 2011).
14. Geck, M. and Pfeiffer, G., Characters of finite Coxeter groups and Iwahori–Hecke algebras , London Mathematical Society Monographs. New Series 21 (Oxford University Press, 2000).
15. Geck, M. and Rouquier, R., ‘Centers and simple modules for Iwahori–Hecke algebras’, Finite reductive groups (Luminy, 1994) , Progress in Mathematics 141 (Birkhäuser, Boston, MA, 1997) 251272.
16. Ginzburg, V., Guay, N., Opdam, E. and Rouquier, R., ‘On the category O for rational Cherednik algebras’, Invent. Math. 154 (2003) no. 3, 617651.
17. Gordon, I., ‘Baby Verma modules for rational Cherednik algebras’, Bull. Lond. Math. Soc. 35 (2003) 321336.
18. Görtz, U. and Wedhorn, T., Algebraic geometry I: schemes with examples and exercises , Advanced Lectures in Mathematics (Vieweg-Teubner, Wiesbaden, 2010).
19. Holmes, R. R. and Nakano, D. K., ‘Brauer-type reciprocity for a class of graded associative algebras’, J. Algebra 144 (1991) no. 1, 117126.
20. Holt, D. F., ‘The Meataxe as a tool in computational group theory’, The atlas of finite groups: ten years on (Birmingham, 1995) , London Mathematical Society Lecture Note Series 249 (Cambridge University Press, 1998) 7481.
21. Holt, D. F., Eick, B. and O’Brien, E. A., Handbook of computational group theory , Discrete Mathematics and its Applications (Chapman & Hall/CRC, Boca Raton, FL, 2005).
22. Holt, D. F. and Rees, S., ‘Testing modules for irreducibility’, J. Aust. Math. Soc. Ser. A 57 (1994) no. 1, 116.
23. Lux, K. and Pahlings, H., Representations of groups: a computational approach , Cambridge Studies in Advanced Mathematics 124 (Cambridge University Press, Cambridge, 2010).
24. Malle, G. and Rouquier, R., ‘Familles de caractères de groupes de réflexions complexes’, Represent. Theory 7 (2003) 610640 (electronic).
25. Martino, M., ‘The Calogero–Moser partition and Rouquier families for complex reflection groups’, J. Algebra 323 (2010) no. 1, 193205.
26. Martino, M., ‘Blocks of restricted rational Cherednik algebras for G(m, d, n)’, J. Algebra 397 (2014) 209224.
27. Parker, R. A., ‘The computer calculation of modular characters (the Meat-Axe)’, Computational group theory (Durham, 1982) (Academic Press, 1984) 267274.
28. Ram, A. and Shepler, A., ‘Classification of graded Hecke algebras for complex reflection groups’, Comment. Math. Helv. 78 (2003) no. 2, 308334.
29. Steel, A., ‘Construction of ordinary irreducible representations of finite groups’, PhD Thesis, University of Sydney, 2012.
30. Thiel, U., ‘A counter-example to Martino’s conjecture about generic Calogero–Moser families’, Algebr. Represent. Theory 17 (2014) no. 5, 13231348.
31. Thiel, U., ‘Decomposition matrices are generically trivial’, Preprint, 2014, arXiv:1402.5122.
32. Thiel, U., ‘On restricted rational Cherednik algebras’, Dissertation, TU Kaiserslautern, 2014.
33. Thiel, U., ‘CHAMP: a Cherednik algebra Magma package’, Preprint, 2015, arXiv:1403.6686.
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