An Introduction to the Brauer and BMW Algebras1 (with an emphasis on the canonical basis and cells of BMW3) Summary: An Introduction to the Brauer and BMW Algebras1 (with an emphasis on the canonical basis and cells of BMW3) Geordie Williamson 1 This document was produced as part of a vacation scholarship at the University of New South Wales. Thanks to the University of New South Wales and especially to Jie Du who supervised the project. Thanks also to Steve Ward for proof reading. Contents 1 The Brauer and BMW Algebras 2 1.1 The Brauer Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Tangles and the BMW Monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 The Monoid Algebras of Bn and BMWn . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Canonical Bases 16 2.1 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Applying the Main Theorem to the BMW Algebra . . . . . . . . . . . . . . . . . . . 18 2.3 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3 Results for BMW2 and BMW3 22 3.1 BMW2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.2 BMW3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Collections: Mathematics