By Louis Auslander

Complaints of the yank Mathematical Society

Vol. sixteen, No. 6 (Dec., 1965), pp. 1230-1236

Published by way of: American Mathematical Society

DOI: 10.2307/2035904

Stable URL: http://www.jstor.org/stable/2035904

Page count number: 7

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V))a ^€ ADefine S = {sa}aeA- Then, (W,S) is a Coxeter system. The rank 2 root systems Type Ai x -a2 ot2 - a i - a2 -a2 Type G2 Type B2 = 0L2 —2ai — a2 ai + a2 —ai — a 2 a2 — —3ari— The action of W on $ gives another interpretation of the length function [Bki, Chap. 3 Let w € W. The cardinality of $~ n w($+) is the length The set $ v defines a root system in V* (the root system inverse or dual to $). There is an isomorphism of groups ~ sending 3 a on sav. Through this isomorphism, M^($) operates on V*.

A general Z-category should be thought of as a 'ring with several objects' [25]. e. we have Hom^O,^) = 0 = Hom^(X,0) for all X) and such that all pairs of 41 42 B. e. an object X n ^ endowed with morphisms px : X]\Y —¥ X and py : X Y[ Y —> Y such that the map ) x ILomc(U,Y) , h H> (Px h,py h) is bijective. In other words, the pair of maps (PX,PY) is universal among all pairs of morphisms (/,#) from an object U to X and F, respectively. Universal properties of this type are most conveniently expressed in the language of representable functors: Recall that a contravariant functor F defined on a category C with values in the category of sets is representable if there is an object Z £ C and an isomorphism of functors Note that this determines the object Z uniquely up to canonical isomorphism.

This means that the set {ind(Tu,)~1Tw;-i}tuevv is the dual basis of {Tw}w^w with respect to r. , the morphism : h^(hf^ r(hh')) is an isomorphism. Together with the fact that % is a deformation of ZW, this explains the structure of % over an algebraic closure K of the field of fractions of O (Tits' deformation theorem) [Bki, Chap. 7 The algebra % ®o K is semi-simple and isomorphic to KW. 8 Assume W is a finite Weyl group. Then, the algebra QW is isomorphic to a direct product of matrix algebras over Q and the algebra H ®o Q(\/^)s€5 i 5 isomorphic to a direct product of matrix algebras over The theorem above generalizes to finite Coxeter groups : if W is a finite reflection group over i f c R , then KW is isomorphic to a product of matrix algebras over K and H ®o K(y/

### An Account of the Theory of Crystallographic Groups by Louis Auslander

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