The Lawrence–krammer Representation Is Unitary
نویسنده
چکیده
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.
منابع مشابه
On the image of the Lawrence– Krammer representation
A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negativedefinite Hermitian. Since unitary matrices diagonalize, the conjugacy class of a matrix in the un...
متن کاملA note on the image of the Lawrence - Krammer representation
A non-singular sesquilinear form is constructed that is preserved by the Lawrence-Krammer representation. It is shown that if the polynomial variables q and t of the Lawrence-Krammer representation are chosen to be appropriate algebraically independant unit complex numbers, then the form is negativedefinite Hermitian. Unitary matrices diagonalize, and it is shown that the eigenvalues of a Lawre...
متن کاملThe Density of Lawrence-krammer and Non-conjugate Braid Representations of Links
We use some Lie group theory and Budney’s unitarization of the Lawrence-Krammer representation, to prove that for generic parameters of definite form the image of the representation (also on certain types of subgroups) is dense in the unitary group. This implies that, except possibly for closures of full-twist braids, all links have infinitely many conjugacy classes of braid representations on ...
متن کاملInvariant Subspaces of the Lawrence–krammer Representation
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, t...
متن کاملar X iv : 0 90 1 . 38 56 v 1 [ m at h . R T ] 2 4 Ja n 20 09 Parameters for which the Lawrence - Krammer representation is reducible
We show that the representation, introduced by Lawrence and Krammer to show the linearity of Braid groups, is generically irreducible, but for that for some values of its two parameters when these are specialized to complex numbers , it becomes reducible. To do so, we construct a representation of degree n(n−1) 2 of the BMW algebra of type A n−1 inside the Lawrence-Krammer space. As a represent...
متن کامل