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Invariant subspaces of the Lawrence-Krammer representation

Author: Claire Levaillant
Journal: Proc. Amer. Math. Soc. 140 (2012), 2599-2612
MSC (2010): Primary 20F36; Secondary 20C08
Posted: November 30, 2011
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Abstract: The Lawrence-Krammer representation was used in to show the linearity of the braid group. The problem had remained open for many years. The fact that the Lawrence-Krammer representation of the braid group is reducible for some complex values of its two parameters is now known, as well as the complete description of these values. It is also known that when the representation is reducible, the action on a proper invariant subspace is an Iwahori-Hecke algebra action. In this paper, we prove a theorem of classification for the invariant subspaces of the Lawrence-Krammer space. We classify the invariant subspaces in terms of Specht modules. We fully describe them in terms of dimension and spanning vectors in the Lawrence-Krammer space.

References

1. Stephen J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471–486 (electronic). MR 1815219 (2002a:20043), http://dx.doi.org/10.1090/S0894-0347-00-00361-1
2. Joan S. Birman and Hans Wenzl, Braids, link polynomials and a new algebra, Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273. MR 992598 (90g:57004), http://dx.doi.org/10.1090/S0002-9947-1989-0992598-X
3. Arjeh M. Cohen and David B. Wales, Linearity of Artin groups of finite type, Israel J. Math. 131 (2002), 101–123. MR 1942303 (2003j:20062), http://dx.doi.org/10.1007/BF02785852
4. Arjeh M. Cohen, Dié A. H. Gijsbers, and David B. Wales, BMW algebras of simply laced type, J. Algebra 286 (2005), no. 1, 107–153. MR 2124811 (2006f:16047), http://dx.doi.org/10.1016/j.jalgebra.2004.12.011
5. G. D. James, On the minimal dimensions of irreducible representations of symmetric groups, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 417–424. MR 720791 (86c:20018), http://dx.doi.org/10.1017/S0305004100000803
6. Daan Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131–156. MR 1888796 (2003c:20040), http://dx.doi.org/10.2307/3062152
7. R. J. Lawrence, Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1990), no. 1, 141–191. MR 1086755 (92d:16020)
8. C. Levaillant, Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$ , Ph.D. thesis, California Institute of Technology (2008), http://thesis.library.caltech.edu/2255/1/thesis.pdf
9. C. Levaillant, Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$ , arXiv:0901.3908 (arXiv version of the Ph.D. thesis).
10. Claire Levaillant, Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type 𝐴_{𝑛-1}, C. R. Math. Acad. Sci. Paris 347 (2009), no. 1-2, 15–20 (English, with English and French summaries). MR 2536742 (2010h:20078), http://dx.doi.org/10.1016/j.crma.2008.11.011
11. Claire Levaillant and David Wales, Parameters for which the Lawrence-Krammer representation is reducible, J. Algebra 323 (2010), no. 7, 1966–1982. MR 2594657 (2011d:20075), http://dx.doi.org/10.1016/j.jalgebra.2009.12.021
12. Ivan Marin, Sur les représentations de Krammer génériques, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1883–1925 (French, with English and French summaries). MR 2377890 (2009d:20083)
13. Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol. 15, American Mathematical Society, Providence, RI, 1999. MR 1711316 (2001g:20006)
14. John Atwell Moody, The Burau representation of the braid group 𝐵_{𝑛} is unfaithful for large 𝑛, Bull. Amer. Math. Soc. (N.S.) 25 (1991), no. 2, 379–384. MR 1098347 (92b:20041), http://dx.doi.org/10.1090/S0273-0979-1991-16080-5
15. H.R. Morton and A.J. Wasserman, A basis for the Birman-Wenzl algebra, preprint, 1989.
16. Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059 (89c:57007)
17. Hebing Rui and Mei Si, Blocks of Birman-Murakami-Wenzl algebras, Int. Math. Res. Not. IMRN 2 (2011), 452–486. MR 2764870, http://dx.doi.org/10.1093/imrn/rnq083
18. Matthew G. Zinno, On Krammer’s representation of the braid group, Math. Ann. 321 (2001), no. 1, 197–211. MR 1857374 (2002h:20055), http://dx.doi.org/10.1007/PL00004501

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