Abstract
The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A n − 1 on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A n − 1, D n , E6, E7, E8). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.
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07 October 2019
Recently, Ivan Marin informed us of an error in the proof of Proposition 5.4 in the paper [2] entitled Brauer algebras of simply laced type. He also mentioned this observation in his preprint [3]. Here we give an improved argument. The result is exactly the same as for our incorrect argument and so other results using the proposition remain valid. We are grateful to Ivan Marin for pointing out the error.
07 October 2019
Recently, Ivan Marin informed us of an error in the proof of Proposition 5.4 in the paper [2] entitled Brauer algebras of simply laced type. He also mentioned this observation in his preprint [3]. Here we give an improved argument. The result is exactly the same as for our incorrect argument and so other results using the proposition remain valid. We are grateful to Ivan Marin for pointing out the error.
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Cohen, A.M., Frenk, B. & Wales, D.B. Brauer algebras of simply laced type. Isr. J. Math. 173, 335–365 (2009). https://doi.org/10.1007/s11856-009-0095-9
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DOI: https://doi.org/10.1007/s11856-009-0095-9