diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -104,7 +104,8 @@ combinatorial counterpart.
\section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families}
% TODO: Fix n >= 2
-% TODO: Does the analysis in here work for n = 1?
+% TODO: Does the analysis in here work for n = 1? We certainly need to adapt
+% the contitions of the following lemma in this situation
Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sp}_{2n}(K)\)
of diagonal matrices, as in Example~\ref{ex:cartan-of-sp}, and the basis
@@ -210,8 +211,13 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
\end{lemma}
\begin{definition}
- A \emph{\(\mathfrak{sl}_n\)-sequence} \(m\) is a \(n\)-tuple \(m \in K^n\)
- such that \(m_1 + \cdots + m_n = 0\).
+ A \emph{\(\mathfrak{sl}_n\)-sequence} \(m\) is a \(n\)-tuple \(m = (m_1,
+ \ldots, m_n) \in K^n\) such that \(m_1 + \cdots + m_n = 0\).
+\end{definition}
+
+\begin{definition}
+ A \(k\)-tuple \(m = (m_1, \ldots, m_k) \in K^k\) is called \emph{ordered} if
+ \(m_i - m_{i + 1}\) is a positive integer for all \(i < k\).
\end{definition}
% TODO: Revise the notation for this? I don't really like calling this
@@ -235,31 +241,32 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
\(\mathfrak{h}^*\) is given by the dot action and the action of \(W\) on the
space of \(\mathfrak{sl}_n\)-sequences is given my permuting coordinates. A
weight \(\lambda \in \mathfrak{h}^*\) satisfies the conditions of
- Lemma~\ref{thm:sl-bounded-weights} if, and only if the diferences between all
- but one consecutive coordinates of \(m(\lambda)\) are positive integers --
- i.e. there is some unique \(i < n\) such that \(m(\lambda)_i - m(\lambda)_{i
- + 1}\) is \emph{not} a positive integer.
+ Lemma~\ref{thm:sl-bounded-weights} if, and only if \(m(\lambda)\) is
+ \emph{not} ordered, but becomes ordered after removing one term.
\end{proposition}
% TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose
% union corresponds to condition (i)
\begin{definition}
We denote by \(\mathscr{B}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\)
- such that \(m_i - m_{i + 1}\) is a positive integer for all but a single \(i
- < n\). We also consider the subsets \(\mathscr{B}^+ = \{m \in \mathscr{B} :
- m_1 - m_2 \ \text{is \emph{not} a positive integer}\}\) and \(\mathscr{B}^- =
- \{m \in \mathscr{B} : m_{n-1} - m_n \ \text{is \emph{not} a positive
- integer}\}\).
+ which are \emph{not} ordered, but becomes ordered after removing one term. We
+ also consider the \emph{extremal} subsets \(\mathscr{B}^+ = \{m \in
+ \mathscr{B} : (m_2, m_3, \ldots, m_n) \ \text{is ordered}\}\) and
+ \(\mathscr{B}^- = \{m \in \mathscr{B} : (m_1, m_2, \ldots, m_{n - 1}) \
+ \text{is ordered}\}\).
\end{definition}
+% 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
% 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))
\begin{definition}
- Given \(m, m' \in \mathscr{B}\), say there is an arrow \(m \to m'\) if the
- unique \(i\) such that \(m_i - m_{i + 1}\) is not a positive integer is such
- that \(m' = \sigma_i \cdot m\).
+ Given \(m, m' \in \mathscr{B}\), say there is an arrow \(m \to m'\) if there
+ some \(i\) such that \(m_i - m_{i + 1}\) is \emph{not} a positive integer and
+ \(m' = \sigma_i \cdot m\).
\end{definition}
It should then be obvious that\dots
@@ -296,9 +303,9 @@ all \(i\) and \(j\).
following.
\begin{enumerate}
\item If \(m\) is regular and integral then there exists\footnote{Notice
- that in this case $m' \notin \mathscr{B}$, however.} a unique \(m' \in W
- \cdot m\) such that \(m_1' > m_2' > \cdots > m_n'\), in which case the
- connected component of \(m\) is given by
+ that in this case $m' \notin \mathscr{B}$, however.} a unique ordered
+ \(m' \in W \cdot m\), in which case the connected component of \(m\) is
+ given by
\[
\begin{tikzcd}[cramped, sep=small]
\sigma_1 \sigma_2 \cdots \sigma_i \cdot m' \rar &
@@ -320,9 +327,12 @@ 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?
+ % 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 \(m_1' > m_2' > \cdots > m_i' = m_{i + 1}' > \cdots >
- m_n'\), in which case the connected component of \(m\) is given by
+ and \(i\) such that \(m'_i = m'_{i + 1}\) and \((m_1', \cdots, m_i')\)
+ and \((m_{i + 1}', \ldots, m_n')\) are ordered, in which case the
+ connected component of \(m\) is given by
\[
\begin{tikzcd}[cramped, sep=small]
\sigma_1 \sigma_2 \cdots \sigma_{i-1} \cdot m' \rar &
@@ -344,8 +354,8 @@ all \(i\) and \(j\).
\(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{B}^-\).
\item If \(m\) is non-integral then there exists unique \(m' \in W \cdot
- m\) such that \(m_2' > m_3' > \cdots > m_n'\), in which case the
- connected component of \(m\) is given by
+ m\) with \(m' \in \mathscr{B}^+\), in which case the connected component
+ of \(m\) is given by
\[
\begin{tikzcd}[cramped]
m' \rar &
@@ -355,8 +365,7 @@ all \(i\) and \(j\).
\sigma_{n-1} \cdots \sigma_1 \cdot m' \lar &
\end{tikzcd}
\]
- with \(m' \in \mathscr{B}^+\) and \(\sigma_{n-1} \cdots \sigma_1 \cdot m'
- \in \mathscr{B}^-\).
+ with \(\sigma_{n-1} \cdots \sigma_1 \cdot m' \in \mathscr{B}^-\).
\end{enumerate}
\end{lemma}