lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -1,7 +1,7 @@
 \chapter{Classification of Coherent Families}
 
-% TODOO: Is this decomposition unique??
-% TODO: Prove this
+% TODOOO: Is this decomposition unique??
+% TODOO: Prove this
 \begin{proposition}
   Suppose \(\mathfrak{g} = \mathfrak{s}_1 \oplus \cdots \oplus \mathfrak{s}_r\)
   and let \(\mathcal{M}\) be a semisimple irreducible coherent
@@ -84,7 +84,8 @@ combinatorial counterpart.
 
 % TODO: Note we will prove that central characters are also invariants of
 % coherent families
-% TODO: Prove this
+
+% TODOO: Prove this
 \begin{proposition}[Mathieu]\label{thm:coherent-family-has-uniq-central-char}
   Suppose \(\lambda, \mu \in \mathfrak{h}^*\) are such that \(L(\lambda)\) and
   \(L(\mu)\) are both bounded and \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\).
@@ -92,7 +93,8 @@ combinatorial counterpart.
 \end{proposition}
 
 % TODO: Remark that the probability of σ_β ∙ λ ∈ P+ is slight: there precisely
-% one eleement in the orbit of λ which is dominant integral
+% one element in the orbit of λ which is dominant integral, so the odds are
+% 1/|W ∙ λ|
 \begin{proposition}\label{thm:lemma6.1}
   Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that.
   \(L(\lambda)\) is bounded and \(\lambda(H_\beta) \notin \mathbb{N}\). Then
@@ -102,7 +104,7 @@ combinatorial counterpart.
   \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\).
 \end{proposition}
 
-% TODOO: Treat the case of sl(2) here?
+% TODOO: Treat the case of sl(2) here
 
 \section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families}
 
@@ -119,6 +121,9 @@ the \(i\)-th entry of the diagonal of a given matrix, as described in
 Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 \cdots + \sfrac{1}{2} \beta_n\).
 
+% TODO: Add some comments on the proof of this: verifying that these conditions
+% are necessary is a purely combinatorial affair, while checking that these are
+% sufficient involves some analysis envolving the Shane-Weil module
 \begin{lemma}\label{thm:sp-bounded-weights}
   Then \(L(\lambda)\) is bounded if, and only if
   \begin{enumerate}
@@ -244,10 +249,8 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 % TODO: Add some comments on the proof of this: while the proof that these
 % conditions are necessary is a purely combinatorial affair, the proof of the
 % fact that conditions (ii) and (iii) imply L(λ) is bounded requires some
-% results on the connected components of of the graph ℘ (which we will only
+% results on the connected components of of the graph 𝓑  (which we will only
 % state later down the line)
-% TODO: Change the notation for A(λ)? I hate giving objets genereic names, but
-% there really isn't any reasonable name for it
 \begin{lemma}\label{thm:sl-bounded-weights}
   Let \(\lambda \notin P^+\) and \(A(\lambda) = \{ i : \lambda(H_{\beta_i})\
   \text{is \emph{not} a non-negative integer}\}\). Then \(L(\lambda)\) is
@@ -275,8 +278,6 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 
 % TODO: Revise the notation for this? I don't really like calling this
 % bijection m
-% TODO: Note that this prove is similar to the previous one, and that the
-% equivariance of the map follows from the nature of the isomorphism W ≅ S_n
 \begin{proposition}
   The map
   \begin{align*}
@@ -299,9 +300,13 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   \emph{not} ordered, but becomes ordered after removing one term.
 \end{proposition}
 
-% TODO: Note the normalization constant 2n is choosen because
+% TODOO: Note that this prove is similar to the previous one, and that the
+% equivariance of the map follows from the nature of the isomorphism W ≅ S_n
+% TODO: The normalization constant 2n is choosen because
 % λ(H_β) = 2n κ(λ, β) and m(λ) is thus uniquely characterized by the fact that
-% (λ + ρ)(H_β_i) = m(λ)_i - m(λ)_i+1
+% (λ + ρ)(H_β_i) = m(λ)_i - m(λ)_i+1. This comes in when verifying the
+% equivalence between the conditions of the previous lemma and those described
+% in the theorem
 
 % TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose
 % union corresponds to condition (i)
@@ -314,10 +319,12 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   \text{is ordered}\}\).
 \end{definition}
 
+% TODO: Add a picture of parts of 𝓑  for n = 3 in here
+
 % TODO: Explain that for each m ∈ 𝓑  there is a unique i such that so that
 % m_i - m_i+1 is not a positive integer. For m ∈ 𝓑 + this is i = 1, while for
 % m ∈ 𝓑 - this is i = n-1
-% TODO: Explain the intuition behind defining the arrows like so: the point is
+% TODOO: Explain the intuition behind defining the arrows like so: the point is
 % that if there is an arrow m(λ) → m(μ) then μ = σ_i ∙ λ for some i, which
 % implies L(μ) is contained in 𝓔𝔁𝓽(L(λ)) - so that L(μ) is also bounded and
 % 𝓔𝔁𝓽(L(λ)) ≅ 𝓔𝔁𝓽(L(μ))
@@ -378,7 +385,7 @@ all \(i\) and \(j\).
       \mathscr{B}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in
       \mathscr{B}^-\).
 
-    % TODOO: What happens when i = 1?? Do we need to suppose i > 1?
+    % TODOOO: What happens when i = 1?? Do we need to suppose i > 1?
     % TODO: For instance, consider m = (1, 1, -2)
     \item If \(m\) is singular then there exists unique \(m' \in W \cdot m\)
       and \(i\) such that \(m'_i = m'_{i + 1}\) and \((m_1', \cdots, m_{i-1}',
@@ -421,13 +428,9 @@ all \(i\) and \(j\).
   \end{enumerate}
 \end{lemma}
 
-% TODO: Add pictures of parts of the graph 𝓑 ?
-
 % TODO: Notice that this gives us that if m(λ)∈ 𝓑  then L(λ) is bounded: for λ
 % ∈ 𝓑 + ∪ 𝓑 - we stablish this by hand, and for the general case it suffices to
 % notice that there is always some path μ → ... → λ with μ ∈ 𝓑 + ∪ 𝓑 -
-% TODO: Perhaps this could be incorporated into the discussion of the lemma
-% that characterizes the weights of sl(n) whose L is bounded
 
 \begin{theorem}[Mathieu]
   Given \(\lambda, \mu \notin P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded,