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Automorphisms of the Bruhat order on Coxeter Groups

Automorphisms of the Bruhat order on Coxeter Groups AUTOMORPHISMS OF THE BRUHAT ORDER ON COXETER GROUPS WILLIAM C. WATERHOUSE Every Coxeter group (W,S) carries a natural partial order, the (strong) Bruhat order. In this paper I shall show that the Bruhat order is very rigid: apart from the dihedral cases, all order automorphisms come from the group automorphisms preserving S and from group inversion. This result was recently proved by Kung and Sutherland [8] for Coxeter groups of type A (symmetric groups), and Sutherland in his thesis [10] established it for the two other classical finite types. Their proofs use specific representations of the groups as (signed) permutations, and they point out the desirability of a unified argument. The one I supply here, though following roughly the outline of [8], covers not only the classical and exceptional groups but also all the infinite Coxeter groups. 1. Summary review I shall begin by summarizing some basic facts about Coxeter groups and the Bruhat order, drawing on Chapter 1 of Hiller [7] and Section 1 of Chapter 4 of Bourbaki [5]. This section is designed to include everything needed for the rest of the paper. A Coxeter group on a finite set of generators S is a group W http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Automorphisms of the Bruhat order on Coxeter Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/21.3.243
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Abstract

AUTOMORPHISMS OF THE BRUHAT ORDER ON COXETER GROUPS WILLIAM C. WATERHOUSE Every Coxeter group (W,S) carries a natural partial order, the (strong) Bruhat order. In this paper I shall show that the Bruhat order is very rigid: apart from the dihedral cases, all order automorphisms come from the group automorphisms preserving S and from group inversion. This result was recently proved by Kung and Sutherland [8] for Coxeter groups of type A (symmetric groups), and Sutherland in his thesis [10] established it for the two other classical finite types. Their proofs use specific representations of the groups as (signed) permutations, and they point out the desirability of a unified argument. The one I supply here, though following roughly the outline of [8], covers not only the classical and exceptional groups but also all the infinite Coxeter groups. 1. Summary review I shall begin by summarizing some basic facts about Coxeter groups and the Bruhat order, drawing on Chapter 1 of Hiller [7] and Section 1 of Chapter 4 of Bourbaki [5]. This section is designed to include everything needed for the rest of the paper. A Coxeter group on a finite set of generators S is a group W

Journal

Bulletin of the London Mathematical Society – Wiley

Published: May 1, 1989

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