In 2003, Gijsbers, Wales, and I formulated conjectures about the structure of the BMW algebras of type Dn (n > 3), and En (n = 6, 7, 8). Subsequently, we proved these conjectures for Dn (n > 3). Recently, Wales and I worked on En leading to proofs of freeness and cellularity and to a verification of the conjectured dimensions of the BMW algebras of the title. In this talk I will describe the algebras, the related Brauer and Temperley-Lieb algebras, the connections with root systems. If time allows, I will attempt a sketch of proof of the announced results.
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