Brauer algebras of simply laced type |
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Authors: | Arjeh M Cohen Bart Frenk David B Wales |
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Institution: | (1) Department of Mathematics, East China Normal University, Shanghai, 200062, China |
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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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