## Combinatorics of Coxeter GroupsIncludes a rich variety of exercises to accompany the exposition of Coxeter groups Coxeter groups have already been exposited from algebraic and geometric perspectives, but this book will be presenting the combinatorial aspects of Coxeter groups |

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### Contents

4 | |

to test understanding of the material and more difficult ones representing | 14 |

Bruhat order | 27 |

Weak order and reduced words | 65 |

Roots games and automata | 89 |

KazhdanLusztig and Rpolynomials | 131 |

KazhdanLusztig representations | 173 |

Combinatorial descriptions | 245 |

A2 Graphs posets and complexes | 299 |

A3 Permutations and tableaux | 307 |

A4 Enumeration and symmetric functions | 319 |

327 | |

352 | |

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### Common terms and phrases

affine Weyl groups algebra automaton bijection braid-moves Bruhat graph Bruhat order combinatorial complete notation compute Corollary coset Coxeter graph Coxeter group Coxeter matrix Coxeter system CW complex Deduce deﬁne deﬁnition denote descent set diagram dIef dihedral group DR(w edges elementary dual equivalent entries equation example Exercise exists fact Figure ﬁnite ﬁrst following hold G S5 G Sn graded poset group W Hence homomorphism i,j G implies induction integers invB isomorphic K-L graph Kazhdan-Lusztig polynomials labeled left cells Lemma length mapping Möbius function nodes normal form numbers game obtained order complex parabolic subgroups partial order partition permutation positive roots prove R-polynomials reduced decompositions reduced expression reduced words reﬂection ordering representation root poset s-relation Schubert polynomials sequence Show subset subword Suppose symmetric group tableau of shape Theorem weak order Weyl group window notation