Mathematics 601
Homework (due Nov 26)
A. Hulpke
5) Let G be a Coxeter group.
a) Show that there can be at most one element T ∈ G such that ∆+ T = ∆− .
b) If T ∈ G such that ∆+ T =/ ∆− , show that there is r i ∈ Π such that −r i ∈/ T(∆+ ). Conclude that
n(S i T) = n(T) + 1.
c) Conclude that G has a unique element with maximal length.
d) Determine this element for the Group of type A3 .
6) Let G be a Coxeter group with fundamental reflections {s i }ni=1 . An element x ∈ G is called a
Coxeter element if there exists a permutation π of {1, . . . , n} such that
x = s π(1) ⋅ s π(2) ⋅ ⋯ ⋅ s π(n)
Show that all Coxeter elements of G are conjugate in G.
7) Let G be a Coxeter group. We consider elements of G as words in the fundamental reflections
{s i }ni=1 . A word w ∈ G is called reduced if it has exactly l(w) factors. If w = s i1 ⋯s i k is reduced, then
w ′ is a subword of w if w ′ can be obtained from w by ommitting factors.
a) Show that if w ′ is a subword of w, then this holds for any other reduced words representing the
same elements. Thus “subword” defines a relation ≤ on W.
b) Show that ≤ is reflexive, antisymmetric and transitive. We call ≤ the Bruhat-Ordering on W.
8) Let α∶ (−є, є) → GLn (R) be a differentiable curve. Show that
d(det α(t))
dt
= traceα ′ (0).
∣
t=0
9)
For an n × n matrix A over a field, show that det exp(A) = exp(traceA).
10) Let A be a simple Lie Algebra. For x, y ∈ A define (x, y) = trace(adx ⋅ ady). (This is called the
Killing form.) Show that (⋅, ⋅) is a scalar product.
11) Let G = SO n (R) = {A ∈ GLn (R) ∣ det(A) = 1, AT A = I} be the special orthogonal group of
dimension n. Show that the associated Lie-Algebra TI (G) is the set of all skew-symmetric matrices.