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

Published electronically:
November 30, 2011

MathSciNet review:
2910748

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**Stephen J. Bigelow,*Braid groups are linear*, J. Amer. Math. Soc.**14**(2001), no. 2, 471–486 (electronic). MR**1815219**, 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**, 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**, 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**, 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**, 10.1017/S0305004100000803**6.**Daan Krammer,*Braid groups are linear*, Ann. of Math. (2)**155**(2002), no. 1, 131–156. MR**1888796**, 10.2307/3062152**7.**R. J. Lawrence,*Homological representations of the Hecke algebra*, Comm. Math. Phys.**135**(1990), no. 1, 141–191. MR**1086755****8.**C. Levaillant,*Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type*, 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*, 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**, 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**, 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****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****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**, 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****17.**Hebing Rui and Mei Si,*Blocks of Birman-Murakami-Wenzl algebras*, Int. Math. Res. Not. IMRN**2**(2011), 452–486. MR**2764870**, 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**, 10.1007/PL00004501

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
20F36,
20C08

Retrieve articles in all journals with MSC (2010): 20F36, 20C08

Additional Information

**Claire Levaillant**

Affiliation:
Department of Mathematics, Caltech, Pasadena, California 91125

Address at time of publication:
Department of Mathematics, University of California Santa Barbara, Santa Barbara, California 93106

Email:
cl@caltech.edu, claire@math.ucsb.edu

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

Keywords:
Braid groups,
representation theory

Received by editor(s):
July 31, 2010

Received by editor(s) in revised form:
January 28, 2011, February 15, 2011, and March 4, 2011

Published electronically:
November 30, 2011

Communicated by:
Birge Huisgen-Zimmermann

Article copyright:
© Copyright 2011
American Mathematical Society