Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T17:41:08.440Z Has data issue: false hasContentIssue false
  • English
  • Français

PyCox: computing with (finite) Coxeter groups and Iwahori–Hecke algebras

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 August 2012

Meinolf Geck
Meinolf Geck*
Affiliation:
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom (email: m.geck@abdn.ac.uk)

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce the computer algebra package PyCox, written entirely in the Python language. It implements a set of algorithms, in a spirit similar to the older CHEVIE system, for working with Coxeter groups and Hecke algebras. This includes a new variation of the traditional algorithm for computing Kazhdan–Lusztig cells and W-graphs, which works efficiently for all finite groups of rank ≤8 (except E8). We also discuss the computation of Lusztig’s leading coefficients of character values and distinguished involutions (which works for E8 as well). Our experiments suggest a re-definition of Lusztig’s ‘special’ representations which, conjecturally, should also apply to the unequal parameter case. Supplementary materials are available with this article.

MSC classification

Secondary: 20C40: Computational methods 20C08: Hecke algebras and their representations 20F55: Reflection and Coxeter groups
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 231 - 256
Copyright
Copyright © London Mathematical Society 2012

References

[1]Alvis, D., ‘The left cells of the Coxeter group of type H 4’, J. Algebra 107 (1987) 160–168 see also, http://mypage.iusb.edu/∼dalvis/h4data.CrossRefGoogle Scholar
[2]Casselman, B., ‘‘Verifying Kottwitz’ conjecture by computer’, Represent. Theory 4 (2000) 32–45.Google Scholar
[3]Chen, Y. and Shi, J.-Y., ‘Left cells in the Weyl group of type E 7’, Comm. Algebra 26 (1998) 3837–3852.CrossRefGoogle Scholar
[4]DuCloux, F., ‘The state of the art in the computation of Kazhdan–Lusztig polynomials’, Appl. Algebra Engrg. Comm. Comput. 7 (1996) 211–219.CrossRefGoogle Scholar
[5]DuCloux, F., ‘Coxeter: software for Kazhdan–Lusztig polynomials for Coxeter groups’; available at http://www.liegroups.org/coxeter/coxeter3/english.Google Scholar
[6]DuCloux, F., ‘Positivity results for the Hecke algebras of noncrystallographic finite Coxeter group’, J. Algebra 303 (2006) 731–741.Google Scholar
[7]Geck, M., ‘On the induction of Kazhdan–Lusztig cells’, Bull. Lond. Math. Soc. 35 (2003) 608–614.Google Scholar
[8]Geck, M., ‘Computing Kazhdan–Lusztig cells for unequal parameters’, J. Algebra 281 (2004) 342–365.CrossRefGoogle Scholar
[9]Geck, M., ‘Relative Kazhdan–Lusztig cells’, Represent. Theory 10 (2006) 481–524.CrossRefGoogle Scholar
[10]Geck, M., ‘Leading coefficients and cellular bases of Hecke algebras’, Proc. Edinb. Math. Soc. 52 (2009) 653–677.CrossRefGoogle Scholar
[11]Geck, M., ‘Some applications of CHEVIE to the theory of algebraic groups’, Carpathuian. Math. 27 (2011) 64–94.CrossRefGoogle Scholar
[12]Geck, M., ‘Kazhdan–Lusztig cells and the Frobenius–Schur indicator’, Preprint, 2011, arXiv:1110.5672.CrossRefGoogle Scholar
[13]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) 175–210, electronically available at http://www.math.rwth-aachen.de/∼CHEVIE.Google Scholar
[14]Geck, M. and Jacon, N., Representations of Hecke algebras at roots of unity, Algebra and Applications 15 (Springer, New York, 2011).CrossRefGoogle Scholar
[15]Geck, M. and Pfeiffer, G., Characters of finite Coxeter groups and Iwahori–Hecke algebras, London Mathematical Society Monographs, New Series 21 (Oxford University Press, Oxford, 2000).Google Scholar
[16]Howlett, R. B. and Yin, Y., ‘Inducing W-graphs’, Math. Z. 244 (2003) 415–431.Google Scholar
[17]Howlett, R. B. and Yin, Y., ‘Inducing W-graphs II’, Manuscripta Math. 115 (2004) 495–511.Google Scholar
[18]Howlett, R. B. and Yin, Y., ‘Computational construction of irreducible W-graphs for types E 6 and E 7’, J. Algebra 321 (2009) 2055–2067.Google Scholar
[19]Kazhdan, D. and Lusztig, G., ‘Representations of Coxeter groups and Hecke algebras’, Invent. Math. 53 (1979) 165–184.Google Scholar
[20]Kottwitz, R., ‘Involutions in Weyl groups’, Represent. Theory 4 (2000) 1–15.Google Scholar
[21]Lusztig, G., ‘A class of irreducible representations of a finite Weyl group’, Indag. Math. 41 (1979) 323–335.CrossRefGoogle Scholar
[22]Lusztig, G., ‘Left cells in Weyl groups’, Lie Group Representations, I, Lecture Notes in Mathematics 1024 (eds Herb, R. L. R. and Rosenberg, J.; Springer, New York, 1983) 99–111.Google Scholar
[23]Lusztig, G., Characters of reductive groups over a finite field, Annals of Mathematics Studies 107 (Princeton University Press, 1984).CrossRefGoogle Scholar
[24]Lusztig, G., ‘Cells in affine Weyl groups’, Algebraic groups and related topics, Advanced Studies in Pure Math. 6 (Kinokuniya and North–Holland, 1985) 255–287.CrossRefGoogle Scholar
[25]Lusztig, G., ‘Cells in affine Weyl groups II’, J. Algebra 109 (1987) 536–548.Google Scholar
[26]Lusztig, G., Leading coefficients of character values of Hecke algebras, Proceedings of Symposia in Pure Mathematics, 47 (American Mathematical Society, Providence, RI, 1987) 235–262.Google Scholar
[27]Lusztig, G., Hecke algebras with unequal parameters, CRM Monographs Series 18 (American Mathematical Society, Providence, RI, 2003).CrossRefGoogle Scholar
[28]Lusztig, G. and Vogan, D. A. Jr., ‘Hecke algebras and involutions in Weyl groups’, Bull. Inst. Math. Acad. Sin. (N.S.), to appear, arXiv:1109.4606.Google Scholar
[29]van Rossum, G.et al., Python Language Website, 1990–2012 (Python Software Foundation, see http://www.python.org/).Google Scholar
[30]Schönert, M.et al., GAP – Groups, Algorithms, and Programming, fourth ed. (Lehrstuhl D für Mathematik, RWTH Aachen, Germany, 1994).Google Scholar
[31]Stein, W. A.et al., Sage Mathematics Software (Version 4.7.1) (The Sage Development Team, 2011, see http://www.sagemath.org).Google Scholar
[32]Yin, Y., ‘An inversion formula for induced Kazhdan–Lusztig polynomials and duality of W-graphs’, Manuscripta Math. 121 (2006) 81–96.CrossRefGoogle Scholar
Supplementary material: File

Geck Supplementary Material

Supplementary Material

Download Geck Supplementary Material(File)
File 538.8 KB