diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -147,11 +147,11 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
\end{proposition}
\begin{definition}
- We denote by \(\mathscr{Q}\) the set of the \(m \in (\sfrac{1}{2} +
+ We denote by \(\mathscr{B}\) the set of the \(m \in (\sfrac{1}{2} +
\mathbb{Z})^n\) such that \(m_1 > m_2 > \cdots > m_{n - 1} > \pm m_n\). We
- also consider the canonical partition \(\mathscr{Q} = \mathscr{Q}^+ \cup
- \mathscr{Q}^-\) where \(\mathscr{Q}^+ = \{ m \in \mathscr{Q} : m_n > 0 \}\)
- and \(\mathscr{Q}^- = \{ m \in \mathscr{Q} : m_n < 0\}\).
+ also consider the canonical partition \(\mathscr{B} = \mathscr{B}^+ \cup
+ \mathscr{B}^-\) where \(\mathscr{B}^+ = \{ m \in \mathscr{B} : m_n > 0 \}\)
+ and \(\mathscr{B}^- = \{ m \in \mathscr{B} : m_n < 0\}\).
\end{definition}
% TODO: Prove this
@@ -164,7 +164,7 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
\mExt(L(\mu))\) if, and only if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and
\(m(\lambda)_n = \pm m(\mu)_n\). In particular, the isomorphism classes of
semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-families are
- parameterized by \(\mathscr{Q}^+\).
+ parameterized by \(\mathscr{B}^+\).
\end{theorem}
\section{Coherent \(\mathfrak{sl}_n(K)\)-families}
@@ -234,16 +234,14 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
+ 1}\) is \emph{not} a positive integer.
\end{proposition}
-% TODO: Change the notation for 𝓟 ? The paper on affine vertex algebras calls
-% this graph 𝓑
-% TODO: Explain the significance of 𝓟 + and 𝓟 -: these are the subsets whose
+% TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose
% union corresponds to condition (i)
\begin{definition}
- We denote by \(\mathscr{P}\) the set of \(\mathfrak{sl}_n\)-sequences \(m\)
+ 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 \(\mathscr{P}^+ = \{m \in \mathscr{P} :
- m_1 - m_2 \ \text{is \emph{not} a positive integer}\}\) and \(\mathscr{P}^- =
- \{m \in \mathscr{P} : m_{n-1} - m_n \ \text{is \emph{not} a positive
+ < n\). We also consider the subsets \(\mathscr{B}^+ = \{m \in \mathscr{B} :
+ m_1 - m_2 \ \text{is \emph{not} a positive integer}\}\) and \(\mathscr{B}^- =
+ \{m \in \mathscr{B} : m_{n-1} - m_n \ \text{is \emph{not} a positive
integer}\}\).
\end{definition}
@@ -252,7 +250,7 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
% 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 \(m, m' \in \mathscr{P}\), say there is an arrow \(m \to m'\) if
+ 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 \(m' = \sigma_i \cdot m\).
\end{definition}
@@ -261,8 +259,8 @@ It should then be obvious that\dots
\begin{proposition}
Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that
- \(m(\lambda) \in \mathscr{P}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
- is such that \(m(\mu) \in \mathscr{P}\) and there is an arrow \(m(\lambda)
+ \(m(\lambda) \in \mathscr{B}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
+ is such that \(m(\mu) \in \mathscr{B}\) and there is an arrow \(m(\lambda)
\to m(\mu)\). Then \(L(\mu)\) is also bounded and \(\mExt(L(\mu)) \cong
\mExt(L(\lambda))\).
\end{proposition}
@@ -278,7 +276,7 @@ all \(i \ne j\) or \(m_i = m_j\) for some \(i \ne j\), respectively. Similarly,
\(\mathfrak{sl}_n\)-sequence \(m\) integral if \(m_i - m_j \in \mathbb{Z}\) for
all \(i\) and \(j\).
-% TODO: Add notes on what are the sets W ⋅m ∩ 𝓟 : the connected component of
+% TODO: Add notes on what are the sets W ⋅m ∩ 𝓑 : the connected component of
% a given element is contained in its orbit, but a given orbit may contain
% multiple connected components. When m is regular and integral then its orbit
% is the union of n connected components, but otherwise its orbit is precisely
@@ -287,11 +285,11 @@ all \(i\) and \(j\).
% theorem? Perhaps it's best to create another lemma for this
% TODO: Restate the notation for σ_i beforehand
\begin{proposition}
- The connected component of some \(m \in \mathscr{P}\) is given by the
+ The connected component of some \(m \in \mathscr{B}\) is given by the
following.
\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
+ 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
\[
@@ -312,8 +310,8 @@ all \(i\) and \(j\).
\end{tikzcd}
\]
for some unique \(i\), with \(\sigma_1 \cdots \sigma_i \cdot m' \in
- \mathscr{P}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in
- \mathscr{P}^-\).
+ \mathscr{B}^+\) and \(\sigma_{n-1} \cdots \sigma_i \cdot m' \in
+ \mathscr{B}^-\).
\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 >
@@ -335,8 +333,8 @@ all \(i\) and \(j\).
\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
- \(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{P}^-\).
+ with \(\sigma_1 \cdots \sigma_{i-1} \cdot m' \in \mathscr{B}^+\) and
+ \(\sigma_{n-1} \cdots \sigma_{i+1} \cdot m' \in \mathscr{B}^-\).
\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
@@ -350,16 +348,16 @@ all \(i\) and \(j\).
\sigma_{n-1} \cdots \sigma_1 \cdot m' \lar &
\end{tikzcd}
\]
- with \(m' \in \mathscr{P}^+\) and \(\sigma_{n-1} \cdots \sigma_1 \cdot m'
- \in \mathscr{P}^-\).
+ with \(m' \in \mathscr{B}^+\) and \(\sigma_{n-1} \cdots \sigma_1 \cdot m'
+ \in \mathscr{B}^-\).
\end{enumerate}
\end{proposition}
-% TODO: Add pictures of parts of the graph 𝓟 ?
+% TODO: Add pictures of parts of the graph 𝓑 ?
-% TODO: Notice that this gives us that if m(λ)∈ 𝓟 then L(λ) is bounded: for λ
-% ∈ 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: 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
@@ -367,11 +365,11 @@ all \(i\) and \(j\).
\begin{theorem}[Mathieu]
Given \(\lambda, \mu \notin P^+\) with \(L(\lambda)\) and \(L(\mu)\) bounded,
\(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and
- \(m(\mu)\) lie in the same connected component of \(\mathscr{P}\). In
+ \(m(\mu)\) lie in the same connected component of \(\mathscr{B}\). In
particular, the isomorphism classes of semisimple irreducible coherent
\(\mathfrak{sl}_n(K)\)-families are parameterized by the set
- \(\pi_0(\mathscr{P})\) of the connected components of \(\mathscr{P}\), as
- well as by \(\mathscr{P}^+\).
+ \(\pi_0(\mathscr{B})\) of the connected components of \(\mathscr{B}\), as
+ well as by \(\mathscr{B}^+\).
\end{theorem}
% TODO: Change this