TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE <font>D</font><sub>n</sub>

Cohen, Arjeh M.; Gijsbers, Dah Dié; Wales, DB David

doi:10.1142/s0218216509007063

Cited by 11 publications

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“…In view of Corollary 5.5, these triples may be thought of as the abstract Brauer diagrams for any M ∈ ADE. For M = D n , there is a diagrammatic description of BrM(D n ) in [7].…”

Section: Irreducibility Of Representations and Lower Bounding The Dimmentioning

confidence: 99%

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Brauer algebras of simply laced type

2009

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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 M , called the Brauer algebra of type M , and study its structure in the cases where M is a Coxeter graph of simply laced spherical type (so its connected components are of type A n−1 , Dn, E 6 , E 7 , E 8 ). We determine the representations and find the dimension. The algebra is semisimple and contains the group algebra of the Coxeter group of type M as a subalgebra. It is a ring homomorphic image of the Birman-MurakamiWenzl algebra of type M ; 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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“…In view of Corollary 5.5, these triples may be thought of as the abstract Brauer diagrams for any M ∈ ADE. For M = D n , there is a diagrammatic description of BrM(D n ) in [7].…”

Section: Irreducibility Of Representations and Lower Bounding The Dimmentioning

confidence: 99%

“…− (2 n−1 + 1)n! E 6 1, 440, 585 E 7 139, 613, 625 E 8 53, 328, 069, 225 Table 2. Brauer algebra dimensions…”

Section: Introductionmentioning

confidence: 99%

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Brauer algebras of simply laced type

2009

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“…This relation will be extensively used in the present paper. The other relations are referred to by the authors in [6] as the first pole-related self-intersection relation, the second pole-related self-intersection relation and the first closed pole loop relation respectively. For these, we refer the reader to the diagrams (v), (vi) and (vii) of [6].…”

Section: )mentioning

confidence: 99%

“…It is called Ξ + in [6] and it has many interesting properties. One of them is that it commutes with another twist around the pole, as shown on Fig.…”

Section: Construction Of the Representationmentioning

confidence: 99%

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n

Levaillant

2012

J. Knot Theory Ramifications

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Abstract. Using knot theory, we construct a linear representation of the CGW algebra of type D n . This representation has degree n 2 − n, the number of positive roots of a root system of type D n . We show that the representation is generically irreducible, but that when the parameters of the algebra are related in a certain way, it becomes reducible. As a representation of the Artin group of type D n , this representation is equivalent to the faithful linear representation of Cohen-Wales. We give a reducibility criterion for this representation as well as a conjecture on the semisimplicity of the CGW algebra of type D n . Our proof is computer-assisted using Mathematica.

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The Birman-Murakami-Wenzl algebras of type E n

Cohen

Wales

2011

Transformation Groups

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The Birman-Murakami-Wenzl algebras (BMW algebras) of type En for n = 6, 7, 8 are shown to be semisimple and free over the integral domain Z[δ ±1 , l ±1 , m]/(m(1− δ) − (l − l −1 )) of ranks 1, 440, 585; 139, 613, 625; and 53, 328, 069, 225. We also show they are cellular over suitable rings. The Brauer algebra of type En is a homomorphic ring image and is also semisimple and free of the same rank as an algebra over the ring Z[δ ±1 ]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. The generalized Temperley-Lieb algebra of type En turns out to be a subalgebra of the BMW algebra of the same type. So, the BMW algebras of type En share many structural properties with the classical ones (of type An) and those of type Dn.

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TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE D_n

Cited by 11 publications

References 10 publications

Brauer algebras of simply laced type

Brauer algebras of simply laced type

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n

The Birman-Murakami-Wenzl algebras of type E n

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TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE Dn

Cited by 11 publications

References 10 publications

Brauer algebras of simply laced type

Brauer algebras of simply laced type

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

The Birman-Murakami-Wenzl algebras of type E n

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Product

Resources

About

TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE D_n

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE D_n