Specializations of the Lawrence Representations of the Braid Groups at Roots of Unity


Khanh Le

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The braid groups, Bn, were first defined by Emil Artin in 1925 and since then have come to play an important role in many areas of mathematics and physics. One of the longest open questions related to braid groups was answered in 2000 by Bigelow and Krammer, who proved that the braid groups are linear by exhibiting a faithful representation of Bn on the homology module of a certain configuration space over the ring Z[q^{+/- 1},t^{+/- 1}]. This representation, known as the Lawrence-Krammer-Bigelow (LKB) representation, is one of a family of homology representations, H(n,l), first discovered by Ruth Lawrence in 1990. In particular, H(n,2) is the LKB representation while H(n,1) is isomorphic to the famous Burau representation B(n).

In 2011, Jackson and Kerler proved that the Lawrence representations are irreducible over the quotient field Q(q,t). Moreover, they showed that when the parameters are specialized to tq = -1, the Lawrence representations admit a subrepresentation isomorphic to the Temperley-Lieb representation and that the the natural short exact sequence corresponding to this subrepresentation does not split for n >= 4.

In our study we investigate a different specialization of parameters. In particular, we show that when t is specialized at a primitive root of unity (t^l=1) the Lawrence representations admit a subrepresentation isomorphic to the Burau representation B(n). Furthermore, we prove that the corresponding short exact sequence does not split for n >= 3.

Jackson and Kerler also showed that the LKB representation H(n,2) is isomorphic to a highest weight representation W(n,2) constructed from the quantum algebra Uq(sl2). Ekenta and Jackson subsequently extended this argument for all l >= 2 by showing that H(n,l) is isomorphic to a diagonal subrepresentation of W(n,l). In our study we demonstrate that the converse is similarly true. Namely, we show that W(n,l) is isomorphic to a diagonal subrepresentation of H(n,l).

Faculty Mentor

Craig Jackson

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