Invariant subspaces of the Lawrence–Krammer representation

Levaillant, Claire

doi:10.1090/S0002-9939-2011-11107-9

Invariant subspaces of the Lawrence–Krammer representation
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by Claire Levaillant

Proc. Amer. Math. Soc. 140 (2012), 2599-2612

DOI: https://doi.org/10.1090/S0002-9939-2011-11107-9

Published electronically: November 30, 2011

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Abstract:

The Lawrence–Krammer representation was used in $2000$ 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

Stephen J. Bigelow, Braid groups are linear, J. Amer. Math. Soc. 14 (2001), no. 2, 471–486. MR 1815219, DOI 10.1090/S0894-0347-00-00361-1
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, DOI 10.1090/S0002-9947-1989-0992598-X
Arjeh M. Cohen and David B. Wales, Linearity of Artin groups of finite type, Israel J. Math. 131 (2002), 101–123. MR 1942303, DOI 10.1007/BF02785852
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, DOI 10.1016/j.jalgebra.2004.12.011
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, DOI 10.1017/S0305004100000803
Daan Krammer, Braid groups are linear, Ann. of Math. (2) 155 (2002), no. 1, 131–156. MR 1888796, DOI 10.2307/3062152
R. J. Lawrence, Homological representations of the Hecke algebra, Comm. Math. Phys. 135 (1990), no. 1, 141–191. MR 1086755
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
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).
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, DOI 10.1016/j.crma.2008.11.011
Claire Levaillant and David Wales, Parameters for which the Lawrence-Krammer representation is reducible, J. Algebra 323 (2010), no. 7, 1966–1982. MR 2594657, DOI 10.1016/j.jalgebra.2009.12.021
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
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, DOI 10.1090/ulect/015
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, DOI 10.1090/S0273-0979-1991-16080-5
H.R. Morton and A.J. Wasserman, A basis for the Birman-Wenzl algebra, preprint, 1989.
Jun Murakami, The Kauffman polynomial of links and representation theory, Osaka J. Math. 24 (1987), no. 4, 745–758. MR 927059
Hebing Rui and Mei Si, Blocks of Birman-Murakami-Wenzl algebras, Int. Math. Res. Not. IMRN 2 (2011), 452–486. MR 2764870, DOI 10.1093/imrn/rnq083
Matthew G. Zinno, On Krammer’s representation of the braid group, Math. Ann. 321 (2001), no. 1, 197–211. MR 1857374, DOI 10.1007/PL00004501