Reflection groups, root systems, Semisimple Lie algebras
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Based on what we've been discussing so far, it makes sense to start with
the concept of Finite reflection groups and later link to complex semisimple
Lie algebras.

So I will begin with some material from Reflection Groups and Coxeter groups
by Humphreys.

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Let V be a real Euclidean space with a symmetric bilinear form (-,-).

Definition: A *reflection* is a linear operator s on V that sends a nonzero
vector \alpha to its negative and fixes the hyperplane orthogonal to
\alpha, namely:

	H_{\alpha}={v in V: (v,\alpha)=0}.

We write s=s_{\alpha} to be this reflection.

In class exer:

s_{\alpha}(\lambda)=\lambda- [2(\lambda,\alpha)/(\alpha,\alpha)]\alpha.

Proof: The formula works for \lambda=\alpha and \lambda in H_{\alpha}.
Hence it works for all v in H_{\alpha}\oplus R\alpha and thus for all v
in V. QED

In class exer: s_{\alpha} is an orthogonal transformation. That is, it
preserves the symmetric blinear form (-,-), i.e.,
(s_{\alpha}v,s_{\alpha}w)=(v,w).

Proof:

 (s_{\alpha}v, s_{\alpha}w)

=( v - 2(v,\alpha)\alpha,  w-2(w,\alpha)\alpha)
       ----------------    -------------------
       (\alpha,\alpha)     (\alpha,\alpha)

Now use bilinearity and expand (I did it) giving

=(v,w). QED

In class exer: Obviously s_{\alpha}^2=id

Hence, each reflection is an order 2 element of O(V).

Definition: A *finite reflection group* is a finite subgroup of
O(V) generated by reflections.

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Side note on Lie groups: You are aware of the Lie groups GL_n(C) and 
GL_n(R). Each are complex or real manifolds that are also groups (in 
what we do we work with the complex case). The multiplication and 
inverses are continuous functions. For the purposes of these notes we 
won't get into Lie groups per se that much. The summary of what we've 
talked about is that much of their data and classification can be 
extracted from their lie algebras. Lie algebras avoid topology (by 
definition) and their representation theory is equivalent to the 
representation theory of the Lie group.

That said I should introduce you to some other Lie groups

* A_{n-1} SL_n(C) = invertible nxn matrices with complex entries and  det 1.

* O(V) = orthogonal linear transformations of a vector space V. 
Now, there is some topology to discuss. Any orthogonal linear transformation
t has determinant 1, -1. In fact the set of transformations of det 1 and
those of det -1 are connected components of O(V), i.e., O(V) is a union of
two disjoint open sets (proof: look at the map det:O(V)->{1,-1}. This map
is continuous and in the relative topology on R, {1}, {-1} are open in 
{1,-1}. Hence their inverse images are open.

In Lie theory often one adds the condition of wanting the 
connected case. Why? Because then the Lie group is "determined by the Lie
algebra". Why? Remember that the idea is the Lie algebra is the tangent
space at the identity. The example is GL_n(C) has Lie algebra gl_n(C)=
n x n matrices (no condition on det). Given a matrix M in gl_n(C) one
can create a matrix in GL_n(C) be matrix exponentiation. We then argued
that the identity isn't special and we can continuously move around
GL_n(C) to recover the behavior around any other point. In our O(V)
case, one of the components is disconnected from the identity so the
above argument breaks down. This is not a proof of the quoted statement,
but a sense of what the issue is.

Thus:

* B_{n}, D_{n} "Special Orthogonal group" SO(V) = orthogonal linear 
trans of a vector space, and of det 1. This is connected. This is type B_n
if V is 2n+1 dimensional and D_n if it is 2n+2 dimensional.

* C_n "Symplectic Group" Sp(V): let < , > be a *skew* symmetric bilinear 
form and look at linear transformations t that preserve < , >. Here
such matrices ALWAYS have det 1 and Sp(V) is connected to begin with.
Here V must be even dimensional.

The A,B,C,D as usual refer to their "type" in the Cartan-Killing classification
of complex Simple Lie groups. 

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Side note 2: Complex reflection groups -- we assumed V is real -- what 
about if it's complex? Yes, that's a thing too. See the Shephard-Todd 
theorem. These arise in invariant theory. The question is if a group G 
acts on a vector space V, when is it true that the invariant ring C[V]^G 
is a POLYNOMIAL ring? Answer: these guys, and they contain the Coxeter 
groups introduced by Zhuo Zhang. So even if you think you're happy with 
Coxeter groups -- there's always these guys to play with! (Ex: is there
a KL theory for complex reflection groups??)
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Examples of finite reflection groups

I_2(m), m>=3: This "type" is the Dihedral group D_m. (This does NOT generally
come from a Lie group! That is, it is not crystallographic, i.e., it is
not a Weyl group).

D_m is the dihedral group of order 2m consisting of orthogonal transformations
that preserve an m sided polygon centered at the origin. We usually think
of this as generated by a rotation of angle 2\pi/m and m diagonal flips.

As stated this is not, by definition, a reflection group. However you CAN
achieve it as such by using the diagonal reflections and additional
reflections, e.g., in the case of a square, the reflections through the
x and y axis plus the diagonal reflections. The \alphas are the normals
to the reflecting hyperplanes.

IN FACT, for a square you can do it with 2 reflections: the reflection
through the x-axis (normal is (0,1)) and the y=x line (normal is 
(1,-1)/sqrt(2)). Draw.

Ex: A_{n-1}. This is the symmetric group S_n. Let \alpha=e_i-e_j.
Check that s_{\alpha}(e_i)=e_j. Thus s_{\alpha} acts on V=R^n.
If we write w as a product of s_{\alpha} (say with a reduced word)
then the operator t=s_{\alpha^1}...s_{\alpha^L} acts by sending e_i
to w(e_i). What is this? (It's called a representation of S_n!)

Here we use \alphas to be e_i-e_j 1\leq i<j<=n.

Notice that this reflection group actually fixes e_1+e_2+...+e_n and
leaves stable the orthogonal complement consisting of 
V':={(x1,x2,...,xn) in R^n: x1+...+xn=0}.
Thus we really think of S_n acting on V' rather than V. This is why
"A_{n-1}" rather than "A_n".

Ex: B_n=C_n. This is the "Hyperoctohedral group". Let V=R^n. Have the 
same reflections as in A_{n-1} AND also the reflections that send e_i-> 
-e_i. The group W is S_n semi-direct product the group of sign changes 
(in class exer: work the meaning out -- S_n normalizes the sign change 
group so W=Z_2^n x| S_n) and |W|=2^n n!. 
Here we use \alphas to be
e_i i=1,...,n and e_i-e_j 1\leq i<j<=n.

Ex: D_n. This time we only do sign changes with an even number of sign 
changes, together with the permutations.

Here we use \alphas to be
e_i+e_i, e_i-e_j 1\leq i<j<=n.

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The alphas in each case will be the "roots". What properties do these roots
have?

Prop: If t in O(V) and \alpha is any nonzero
vector in V then ts_{\alpha}t^{-1}=s_{t\alpha}. Thus, if w in W
then s_{w\alpha} belongs to W whenever s_{\alpha} does.

Proof: ts_{\alpha}^{-1} sends t\alpha to its negative. Now check it
preserves H_{t\alpha}. QED

Exer. If alpha=e_{i}-e_{i+1} and w is some permutation then ws_{\alpha}w^{-1}
is ...?

The upshot of the proposition is that W sends reflections to reflections.
BUT we want reflections to be defined by roots. This leads to the following
definition

Definition: A root system \Phi is a finite collection of nonzero vectors
in V such that

R1. \Phi intersect R\alpha = {\alpha,-\alpha}
R2. s_{\alpha}\Phi =\Phi

The elements of \Phi are called roots.

Why "roots"?  This is precisely because of the proof of the classification of
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complex semisimple Lie algebras. See "Cartan decomposition of Lie algebra".
The \alpha's there are the roots. We'll see this more later.

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Thing to think about: A finite reflection group W can be realized as
reflections for many different choices of roots. For example, instead of
e_i-e_j in the S_n case, we could have used e_j-e_i. Or, since 
simple transpositions work, we could get away with a much smaller set of
reflections: e_i -e_{i+1}.

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To capture this problem:

Definition: A *simple system* of a root system \Phi is a subcollection 
of vectors, the *simple root*, that is a basis of the R-span of \Phi in 
V and such that every \alpha in \Phi is a linear combo of the simples 
WITH ALL COEFFICIENTS OF THE SAME SIGN.

Thm: simple systems exist!

Definition: the *positive roots* are those which can be expressed in 
terms of the simples with nonnegative coeffs. Similarly, the *negative 
roots* are those which can be expressed in terms of the simples with 
nonpositive coeffs.

Notation: the simple roots are usually denoted \Delta, the positive roots
\Phi^+ and the negative roots \Phi^-

Definition: The rank of a root system is the size of the simple system =
dimension of the R-span of the root system.

Thm: W is generated by the simple reflections (=reflections corresponding
to simple roots). 

Thm: W is a Coxeter group. 

Suppose {s_\beta: \beta in \Delta} are the simple reflections. 

Definition: m_{\alpha \beta} is the order of s_i s_j, i.e., the minimum 
positive integer m_{alpha,beta} such that 
(s_{alpha}s_{beta})^{m(alpha,beta)}=id.

Definition: A *Coxeter graph* has as vertices the simple roots
and an edge between alpha and beta whenever m(alpha,beta)>=3. If 
m(\alpha,beta)>=4 we mark the value on the edge, otherwise it is assumed
to be 3. No edge means the value is 2.

Note: The Coxeter graph is more or less synonymous with the Dynkin diagram 
(which instead uses double edges if m(alpha,beta)=4, triple edges if 
m(alpha,beta)=5 etc. The Dynkin diagram only applies to the 
"crystallographic" case.

Definition: A Coxeter system (W,S) where W is a Coxeter group with simple
reflections S is *irreducible* if its Coxeter graph is irreducible.

Associated to a Coxeter graph is a matrix ... "A" that is r x r
with

a(s,s'):=-cos(\pi/m(s,s'))

Since this matrix is symmetric, it defines a bilinear form

x^t A y.

Def: 
A is *positive definite* if x^t A x >0 for all x \neq 0.
A is *positive semidefinite* if x^t A x >= 0 for all x \neq 0.
A is *positive type* if either positive semidef or def.

Call \Gamma positive if A is positive, etc.

Notice that if \Gamma comes from a finite reflection group then A
is positive definite b/c it is computing the standard Euclidean inner
product relative to the simple system.

Idea of Classification of finite reflection groups: Classify all positive
matrices and show they actually correspond to a Coxeter group.

Idea of Classification of semisimple Lie algebras: Prove the Cartan
"root" decomposition of any such Lie algebra, notice the roots form a root
system then observe that there are only so many possibilities by the argument
above.

Note: The A matrix is close to the "Cartan matrix" below. The "Cartan matrix"
C isn't necessarily symmetric but C=DA where A is the matrix above and
D is diagonal.

Thm: The length of w in W is the number of positive roots w sends negative.

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Crystallographic condition: A root system \Phi is crystallographic if

R3. 2(\alpha,\beta)/(\beta,\beta) \in Integers

In this case we call W generated by s_{\alpha} a *Weyl group*.
(Indeed, if a root system comes from a Cartan decomposition of a 
complex simple Lie algebra, which in turn comes from a complex
simple Lie group by taking tangent space at the identity, which is how
you classify simple Lie groups.)

R3 => all roots are Z-linear combo of \Delta and the Z-span of
\Delta is W-stable.

coroots, weights
----------------

Two other notions that come up a lot are coroots and weights.

Definition: If \alpha is a root then 

\alpha^{\vee}:=2\alpha/(\alpha,\alpha) is its *coroot*.

The set of coroots is also a root system in V which is the "dual" root
system (sometimes the "Langlands dual" root system). Example, B_n vs C_n.

Definition: The *root lattice* of \Phi, denote L(\Phi) is the Z-span of
\Phi in V. Similarly onle define the *coroot lattice*.

In our discussion of irreps of GL_n(C) we mentioned the root-system way
of indexing them.

Definition: The *weight lattice* of \Phi is

\Gamma(\Phi)
   ={\lambda in V| (\lambda,\alpha^{\vee}) in Z for all alpha  in \Phi}

Definition: A weight is *dominant (integral)* if
(\lambda,\alpha^{\vee})>=0 (and integer) for all \alpha.

Exer: In type A_{n-1} there is a bijection between dominant integral
weights and partitions of n with at most n rows!

In general: dominant integral weights index irreps of a complex simple
Lie algebra g.

[Definition: The *coweight lattice* of \Phi is

\Gamma(\Phi^{\vee})
   ={\lambda in V| (\lambda,\alpha) in Z for all alpha  in \Phi}]

Definition: A vector \omega_\alpha (\alpha\in \Delta) is a *fundamental 
weight* if (\omega_{\alpha},\beta)=Kronecker-delta(\alpha,\beta).
In particular, fundamental weights are dominant integral.

Exer: For type A_{n-1} the fundamentals correspond to the Alt reps (I think)
which correspond to (1,1,1,...1,0,0,0).


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Below, I am following notes of Zelevinsky

https://arxiv.org/abs/math/0112062

Another reference is

Humphreys, "Introduction to Lie algebras and representation theory"

Yet another is

Fulton-Harris, "Representation theory"

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Recall that a Lie algebra is a vector space V over a field k
with a Lie bracket [-,-]:V x V->V that is 

bilinear
skew-symmetric, i.e., [x,x]=0
satisfies the Jacobi identity
[[a,b],c]+[[b,c],a]+[[c,a],b]]=0

Definition 1.1: A *Cartan matrix* of rank r is an r x r integer matrix
A=(a_{ij}) such that

(1) a_{ii}=2 for all i
(2) a_{ij}<=0 for i \neq j and a_{ij}=0 => a_{ji}=0
(3) for every non-empty subset I \subset [1,r]:={1,2,...,r}, the principal
minor with rows and columns I is positive.

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Cartan-Killing classification: every Cartan matrix can be transformed by
a simultaneous permutation of row and columns into a direct sum of matrices
of 

four "classical types" 

A_r (r\geq 1)
B_r (r\geq 2)
C_r (r\geq 3)
D_r (r\geq 4)

and five "exceptional types"

E_6
E_7
E_8
F_4
G_2

Above, the subscript indicates the rank.

Main example: type A_r. Here a_{ij}=-1 if |i-j|=1 and a_{ij}=0 
if |i-j|>2. For instance

A_4 is the 4x4 matrix

2 -1 0   0 
-1 2 -1  0
0  -1 2 -1
0   0 -1 2

Example (rank 2 cases): 

A_2 

2  -1
-1  2

B_2

2  -2
-1  2

G_2

2  -1
-3  2

A_1 x A_1

2 0 
0 2

(The last one isn't an "irreducible" case, it is a direct sum of two A_1
Cartan matrices.

(Often people have conjectures about Lie algebras and give evidence
in the rank 2 case.)

A complete list of all matrices is found on page 59 of Humphreys.

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Definition 1.2: The *complex semisimple Lie algebra* g=g(A) associated 
to a Cartan matrix A is generated by 3r Chevalley generators

e_i, \alpha_i^{\vee} and f_i   for i=1,2,...,r

(that is, take the FREE LIE ALGEBRA with that basis where we force the
skew-symmetry and the jacobi identity) AND THEN IN ADDITION

satisfying these Serre relations

[\alpha_i^{\vee}, \alpha_i^{\vee}] = 0

[\alpha_i^{\vee}, e_j] = a_{ij}e_j

[\alpha_i^{\vee}, f_j] = -a_{ij}f_j

[e_i, f_j] = \delta_{ij} \alpha_i^{\vee} [Kronecker delta]

AND

(ad e_i)^{1-a_{ij}}(e_j) = (ad f_i)^{1-a_{ij}}(f_j)=0 = 0 (for i\neq j)

For the last line recall that for any Lie algebra V, if x in V then
(ad x) in End(V) so (ad x)^t means [x...[x,[x,-]] (t many compositions).

----

(Quick Definition (Free Lie algebra): The usual universal definition giving
a functor from the category of sets to the category of Lie algebras.

https://en.wikipedia.org/wiki/Free_Lie_algebra)

Let X be a set, A be a Lie algebra
       i
   X ----> L
   |     /
 f |    /
   |   / g
   v  /
   A<- 

Here f is a set theoretic map, g is UNIQUE Lie algebra homomorphism
(recall a Lie algebra homomorphism MEANS f([x,y]_L)=[f(x),f(y)]_A.

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