diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -162,17 +162,17 @@ We can find an orthonormal basis \(\{\epsilon_1, \ldots, \epsilon_n\}\) for
with \(m_1 > m_2 > \cdots > m_n > 0\).
\end{theorem}
-% TODOO: Change the notation in here to work with sl(n) (n >= 3) instead of
-% sl(n + 1) (n >= 2)
-\section{Coherent \(\mathfrak{sl}_{n + 1}(K)\)-families}
+\section{Coherent \(\mathfrak{sl}_n(K)\)-families}
+
+% TODO: Fix n >= 3
% TODOO: Add notes about this basis beforehand
-Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in \mathfrak{sl}_{n
-+ 1}(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_{n + 1}(K)\) as in
+Take the standard Cartan subalgebra \(\mathfrak{h} = \{ X \in
+\mathfrak{sl}_n(K) : X \ \text{is diagonal}\}\) of \(\mathfrak{sl}_n(K)\) as in
Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
-\ldots, \epsilon_{n + 1} \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the
+\ldots, \epsilon_n \in \mathfrak{h}^*\) such that \(\epsilon_i(H)\) is the
\(i\)-th entry of the diagonal of \(H\). Let \(\Sigma = \{ \beta_1, \ldots,
-\beta_n \}\) be the basis for \(\Delta\) given by \(\beta_i = \epsilon_i -
+\beta_{n-1} \}\) be the basis for \(\Delta\) given by \(\beta_i = \epsilon_i -
\epsilon_{i + 1}\).
% TODO: Add some comments on the proof of this: while the proof that these
@@ -184,52 +184,50 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
% there really isn't any reasonable name for it
\begin{lemma}\label{thm:sl-bounded-weights}
Let \(\lambda \notin P^+\) and \(A(\lambda) = \{ i : \lambda(H_{\beta_i})\
- \text{is not a non-negative integer}\}\). Then \(L(\lambda)\) is bounded if,
- and only if one of the following assertions holds.
+ \text{is \emph{not} a non-negative integer}\}\). Then \(L(\lambda)\) is
+ bounded if, and only if one of the following assertions holds.
\begin{enumerate}
- \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n\}\).
- \item \(A(\lambda) = \{i\}\) for some \(1 < i < n\) and \((\lambda +
+ \item \(A(\lambda) = \{1\}\) or \(A(\lambda) = \{n - 1\}\).
+ \item \(A(\lambda) = \{i\}\) for some \(1 < i < n - 1\) and \((\lambda +
\rho)(H_{\beta_{i - 1}} + H_{\beta_i})\) or \((\lambda +
\rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive integer.
- \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n\) and
+ \item \(A(\lambda) = \{i, i + 1\}\) for some \(1 \le i < n - 1\) and
\((\lambda + \rho)(H_{\beta_i} + H_{\beta_{i + 1}})\) is a positive
integer.
\end{enumerate}
\end{lemma}
-% TODO: Change the notation: these should be called "sl_n+1-sequences", not
-% "sl_n+1(K)-sequences"
\begin{definition}
- A \emph{\(\mathfrak{sl}_{n + 1}(K)\)-sequence} \(m\) is a \(n + 1\)-tuple \(m
- \in K^{n + 1}\) such that \(m_1 + \cdots + m_{n + 1} = 0\).
+ A \emph{\(\mathfrak{sl}_n\)-sequence} \(m\) is a \(n\)-tuple \(m \in K^n\)
+ such that \(m_1 + \cdots + m_n = 0\).
\end{definition}
% 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+1
+% equivariance of the map follows from the nature of the isomorphism W ≅ S_n
% TODOO: Describe this isomorphism beforehand
\begin{proposition}
The map
\begin{align*}
- m : \mathfrak{h}^* & \to K^{n + 1} \\
+ m : \mathfrak{h}^* & \to K^n \\
\lambda &
\mapsto
(
\kappa(\epsilon_1, \lambda + \rho),
\cdots,
- \kappa(\epsilon_{n + 1}, \lambda + \rho)
+ \kappa(\epsilon_n, \lambda + \rho)
)
\end{align*}
- is \(W\)-equivariant bijection onto the space of all \(\mathfrak{sl}_{n +
- 1}(K)\)-sequences, where the action \(W \cong S_{n + 1}\) on
+ is \(W\)-equivariant bijection onto the space of all
+ \(\mathfrak{sl}_n\)-sequences, where the action \(W \cong S_n\) on
\(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on
\(K^n\) 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 \(i \le n\) such that
- \(m(\lambda)_i - m(\lambda)_{i + 1}\) is not a positive integer.
+ 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.
\end{proposition}
% TODO: Change the notation for 𝓟 ? The paper on affine vertex algebras calls
@@ -239,11 +237,11 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
% 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}
- Denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_{n + 1}(K)\)-sequences
- \(m\) such that \(m_i - m_{i + 1}\) is a positive integer for all but a
- single \(i \le n\). Given \(m, m' \in \mathscr{P}\), 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 \(m' = \sigma_i \cdot m\).
+ Denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\) such
+ that \(m_i - m_{i + 1}\) is a positive integer for all but a single \(i <
+ n\). Given \(m, m' \in \mathscr{P}\), 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 \(m' = \sigma_i \cdot m\).
\end{definition}
It should then be obvious that\dots
@@ -257,21 +255,21 @@ It should then be obvious that\dots
\end{proposition}
\begin{definition}
- Let \(\mathscr{P}^+ = \{m \in \mathscr{P} : m_1 - m_2 \ \text{is not a
- positive integer}\}\) and \(\mathscr{P}^- = \{m \in \mathscr{P} : m_n - m_{n
- + 1} \ \text{is not a positive integer}\}\).
+ Let \(\mathscr{P}^+ = \{m \in \mathscr{P} : m_1 - m_2 \ \text{is \emph{not} a
+ positive integer}\}\) and \(\mathscr{P}^- = \{m \in \mathscr{P} : m_{n-1} -
+ m_n \ \text{is \emph{not} a positive integer}\}\).
\end{definition}
\begin{definition}
- A \(\mathfrak{sl}_{n + 1}(K)\)-sequence \(m\) is called \emph{integral} if
- \(m_i - m_j \in \mathbb{Z}\) for all \(i\) and \(j\).
+ A \(\mathfrak{sl}_n\)-sequence \(m\) is called \emph{integral} if \(m_i - m_j
+ \in \mathbb{Z}\) for all \(i\) and \(j\).
\end{definition}
% TODOO: Discuss the notion of a regular weight beforehand
\begin{definition}
- A \(\mathfrak{sl}_{n + 1}(K)\)-sequence \(m\) is called \emph{regular} if
- \(m_i \ne m_j\) for all \(i \ne j\). A sequence is called \emph{singular} if
- it is not regular.
+ A \(\mathfrak{sl}_n\)-sequence \(m\) is called \emph{regular} if \(m_i \ne
+ m_j\) for all \(i \ne j\). A sequence is called \emph{singular} if it is not
+ regular.
\end{definition}
% TODO: Add notes on what are the sets W . m ∩ 𝓟 : the connected component of
@@ -288,8 +286,8 @@ It should then be obvious that\dots
\begin{enumerate}
\item If \(m\) is regular and integral then there exists\footnote{Notice
that in this case $m' \notin \mathscr{P}$, however.} a unique \(m' \in W
- \cdot m\) such that \(m_1' > m_2' > \cdots > m_{n + 1}'\), in which case
- the connected component of \(m\) is given by
+ \cdot m\) such that \(m_1' > m_2' > \cdots > m_n'\), in which case the
+ connected component of \(m\) is given by
\[
\begin{tikzcd}[cramped, sep=small]
\sigma_1 \sigma_2 \cdots \sigma_i \cdot m' \rar &
@@ -304,16 +302,16 @@ It should then be obvious that\dots
\sigma_i \cdot m' &
\sigma_{i+1} \sigma_i \cdot m' \lar &
\cdots \lar &
- \sigma_n \cdots \sigma_i \cdot m' \lar &
+ \sigma_{n-1} \cdots \sigma_i \cdot m' \lar &
\end{tikzcd}
\]
for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in
- \mathscr{P}^+\) and \(\sigma_n \cdots \sigma_i \cdot m' \in
+ \mathscr{P}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in
\mathscr{P}^-\).
\item If \(m\) is singular then there exists unique \(m' \in W \cdot m\)
and \(i\) such that \(m_1' > m_2' > \cdots > m_i' = m_{i + 1}' > \cdots >
- m_{n + 1}'\), in which case the connected component of \(m\) is given by
+ m_n'\), in which case the connected component of \(m\) is given by
\[
\begin{tikzcd}[cramped, sep=small]
\sigma_1 \sigma_2 \cdots \sigma_{i-1} \cdot m' \rar &
@@ -328,26 +326,26 @@ It should then be obvious that\dots
m' &
\sigma_{i+1} \cdot m' \lar &
\cdots \lar &
- \sigma_n \cdots \sigma_{i+1} \cdot m' \lar &
+ \sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \lar &
\end{tikzcd}
\]
with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathscr{P}^+\) and
- \(\sigma_n \cdots \sigma_{i+1} \cdot m' \in \mathscr{P}^-\).
+ \(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{P}^-\).
\item If \(m\) is non-integral then there exists unique \(m' \in W \cdot
- m\) such that \(m_2' > m_3' > \cdots > m_{n + 1}'\), in which case the
+ m\) such that \(m_2' > m_3' > \cdots > m_n'\), in which case the
connected component of \(m\) is given by
\[
\begin{tikzcd}[cramped]
- m' \rar &
- \sigma_1 \cdot m' \rar \lar &
- \sigma_2 \sigma_1 \cdot m' \rar \lar &
- \cdots \rar \lar &
- \sigma_n \cdots \sigma_1 \cdot m' \lar &
+ m' \rar &
+ \sigma_1 \cdot m' \rar \lar &
+ \sigma_2 \sigma_1 \cdot m' \rar \lar &
+ \cdots \rar \lar &
+ \sigma_{n-1} \cdots \sigma_1 \cdot m' \lar &
\end{tikzcd}
\]
- with \(m' \in \mathscr{P}^+\) and \(\sigma_n \cdots \sigma_1 \cdot m' \in
- \mathscr{P}^-\).
+ with \(m' \in \mathscr{P}^+\) and \(\sigma_{n-1} \cdots \sigma_1 \cdot m'
+ \in \mathscr{P}^-\).
\end{enumerate}
\end{proposition}
@@ -357,7 +355,7 @@ It should then be obvious that\dots
% ∈ 𝓟 + ∪ 𝓟 - we stablish this by hand, and for the general case it suffices to
% notice that there is always some path μ → ... → λ with μ ∈ 𝓟 + ∪ 𝓟 -
% TODO: Perhaps this could be incorporated into the discussion of the lemma
-% that characterizes the weights of sl(n + 1) whose L is bounded
+% that characterizes the weights of sl(n) whose L is bounded
% TODO: Prove this
\begin{theorem}[Mathieu]
@@ -365,7 +363,7 @@ It should then be obvious that\dots
\(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and
\(m(\mu)\) lie in the same connected component of \(\mathscr{P}\). In
particular, the isomorphism classes of semisimple irreducible coherent
- \(\mathfrak{sl}_{n + 1}(K)\)-families are parameterized by the set
+ \(\mathfrak{sl}_n(K)\)-families are parameterized by the set
\(\pi_0(\mathscr{P})\) of the connected components of \(\mathscr{P}\).
\end{theorem}