Def: If F is a root system we write W = <sa : a in F> (the group generated by the reflections sa (b) = b - <b, a>a) and call W the Weyl group of F.
Note: W is a finite subgroup of Sym F, as every element of W is a permutation of the finitely many roots.
Lemma: Suppose T is a transformation such that all eigenvalues of T are 1 and Tk = 1 for some k > 0, then T = 1.
Proof: Consider the minimal polynomial of T, p(T). Then p(T) | (T - 1)n as all the roots of p(T) are equal to 1. Also p(T) | Tk - 1. This implies p(T) = T - 1 which means that T must be the identity. n
Lemma: Suppose w in GL(E) leaves invariant. Then
1) wsaw-1
= sw(a)
for all a in F.
2) <w(a), w(b)>
= <a, b> for all
a, b in F.
Proof: Let wsaw-1 = t. Then t(w(a)) = wsa(a) = w(-a) = -w(a) and t(w(Pa)) = wsa(Pa) = w(Pa). Thus t maps w(a) to -w(a) and t fixes w(Pa). (To be filled in soon, hopefully)
Def: Suppose F is a root system in
Euclidean space E. A subset D of F
is called a base (or a set of simple roots) if
1) D is a basis of E
2) Every root g in F
can be written uniquely in the form g = Sa
in D ka
a where the kas
are either all in {0, 1, 2, ...} or {0, -1, -2, ...}.
Def: A vector g in E is called regular if (g, a) is nonzero for all a in F (Pg does not intersect F).
Def: Fix g regular. F+(g) = {a in F : (g, a) > 0} and F-(g) = {a in F : (g, a) < 0}.
Def: A root f in F+(g) is called decomposable if f = f1 + f2 for some f1, f2 in F+(g). Otherwise f is called indecomposable.
Lemma: Suppose v1, v2,
..., vk satisfy
1) (vi, vj) < or
= 0 for i not equal to j.
2) There is some g such that (g,
vi) > or = 0 for all i.
Then {v1, v2, ..., vk}
is linearly independent.
Theorem: Let D = D(g)
be the indecomposable roots in F+(g),
then D(g) is a base.
Moreover, every base is of the form D(g)
for some g.
4/1/99
Note: Last time we showed for regular g in E we can construct a base D(g) = {indecomposable roots a in the set (a, g) > 0}. It remains to show that every base arises in this fashion.
Proposition: (Linear Algebra) Suppose V is a vector space over the field R and l1*, l2*, ... , lk* are linearly independent in V*. Let H1, H2, ... , Hk be the open half spaces in V. Hi = {v in V : (li*, v) > 0}. Then the intersection of these Hi is nonempty.
Note: Consider the map g |~> D(g). What are all the gs so that D(g) stays the same? We now look to split E into components which generate the same D(g).
Def: The root hyperplanes Pa for a in F divide E into several connected components. We call these components open Weyl chambers.
Lemma: (A) If a is in F+ and not in D then there is b in D such that a - b is in F+.
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