The Lawrence--Krammer representation is unitary

Won Taek Song

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The Lawrence--Krammer representation is unitary

Won Taek Song

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Abstract

We show that the Lawrence--Krammer representation is unitary. We explicitly present the non-singular matrix representing the sesquilinear pairing invariant under the action. We show that reversing the orientation of a braid is equivalent to the transposition of its Lawrence--Krammer matrix followed by a certain conjugation. As corollaries it is shown that the characteristic polynomial of the Lawrence--Krammer matrix is invariant under substitution of its variables with their inverses up to multiplication by units, and is not a complete conjugacy invariant for braids.

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Won Taek Song. 2002-02-21. The Lawrence--Krammer representation is unitary. https://arxiv.org/abs/math/0202214

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