lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: da5c06bd3e86cf0ae7adea2fdf6d0b42993042e3
Parent: 9b7c99d6ec1c7fa8b4cb4101909ced13e73ce4e1
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 29 Jun 2023 13:45:13 +0000

Added some proofs to the last chapter

Added proofs for the theorems which describe the classification of coherent families for sp(2n) and sl(n)

Also fixed a typo: for Mathieus theorems to work we needed to noralize m(λ) by a factor of 2/n

Also fixed a typo in the description of the isomorphism W ≅ Sₙ⋉ (ℤ/2)ⁿ

Diffstat

2 files changed, 129 insertions, 22 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	142	124	18
Modified	sections/fin-dim-simple.tex	9	5	4

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -85,7 +85,7 @@ combinatorial counterpart.
 % TODO: Note we will prove that central characters are also invariants of
 % coherent families
 % TODO: Prove this
-\begin{proposition}[Mathieu]
+\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 \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\).
   Then \(\chi_\lambda = \chi_\mu\).
@@ -93,7 +93,7 @@ combinatorial counterpart.
 
 % TODO: Note that if σ_β ∙ λ is not dominant integral then L(σ_β ∙ λ) is
 % infinite-dimensional and 𝓔𝔁𝓽(L(σ_β ∙ λ)) ≅  𝓔𝔁𝓽(L(λ))
-\begin{proposition}
+\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
   \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\).
@@ -128,8 +128,6 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 % TODO: Note that we need a better set of parameters to the space of weights
 % such that L(λ) is bounded
 
-% TODO: Prove this. The only part worth proving is the fact this is
-% W-equivariant, and this is clear from the isomorphism W ≅ S_n ⋉ (ℤ/2)^n
 % TODO: Revise the notation for this? I don't really like calling this
 % bijection m
 \begin{proposition}
@@ -154,6 +152,32 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   m(\lambda)_n\).
 \end{proposition}
 
+\begin{proof}
+  The fact \(m : \mathfrak{h}^* \to K^n\) is a bijection is clear from the fact
+  that \(\{\epsilon_1, \ldots, \epsilon_n\}\) is an orthonormal basis for
+  \(\mathfrak{h}^*\). Veryfying that \(L(\lambda)\) is bounded if, and only if
+  \(m(\lambda)_1 > m(\lambda)_2 > \cdots > m(\lambda)_{n - 1} > \pm
+  m(\lambda)_n\) is also a simple combinatorial affair.
+
+  The only part of the statement worth proving is the fact that \(m\) is an
+  equivariant map, which is equivalent to showing the map
+  \begin{align*}
+    \mathfrak{h}^* & \to K^n \\
+    \lambda &
+    \mapsto (\kappa(\epsilon_1, \lambda), \ldots, \kappa(\epsilon_1, \lambda))
+  \end{align*}
+  is equivariant with respect to the natural action of \(W\) on
+  \(\mathfrak{h}^*\). But this also clear from the isomorphism \(W \cong S_n
+  \ltimes (\mathbb{Z}/2\mathbb{Z})^n\), as described in
+  Example~\ref{ex:sp-weyl-group}: \((\sigma_i, (\bar 0, \ldots, \bar 0)) =
+  \sigma_{\beta_i}\) permutes \(\epsilon_i\) and \(\epsilon_{i + 1}\) for \(i <
+  n\) and \((1, (\bar 0, \ldots, \bar 0, \bar 1) = \sigma_{\beta_n}\) flips the
+  sign of \(\epsilon_n\). Hence \(m(\sigma_{\beta_i} \cdot \epsilon_j) =
+  \sigma_{\beta_i} \cdot m(\epsilon_j)\) for all \(i\) and \(j\). Since \(W\)
+  is generated by the \(\sigma_{\beta_i}\), this implies that the required map
+  is equivariant.
+\end{proof}
+
 \begin{definition}
   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
@@ -162,10 +186,6 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   and \(\mathscr{B}^- = \{ m \in \mathscr{B} : m_n < 0\}\).
 \end{definition}
 
-% TODO: Prove this
-% Given the Harish-Chandra theorem and the previous proposition, all its left
-% is to show that of m(λ)_n = - m(μ)_n then indeed Ext(L(λ)) = Ext(L(μ)). I
-% think this is cavered in Lemma 6.1 of Mathieu
 \begin{theorem}[Mathieu]
   Given \(\lambda\) and \(\mu\) satisfying the conditions of
   Lemma~\ref{thm:sp-bounded-weights}, \(\mExt(L(\lambda)) \cong
@@ -175,6 +195,36 @@ Example~\ref{ex:sp-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
   parameterized by \(\mathscr{B}^+\).
 \end{theorem}
 
+\begin{proof}
+  Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\),
+  so that \(m(\lambda), m(\mu) \in \mathscr{B}\).
+
+  Suppose \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). By
+  Proposition~\ref{thm:coherent-family-has-uniq-central-char}, \(\chi_\lambda =
+  \chi_\mu\). It thus follows from the Harish-Chandra Theorem that \(\mu \in W
+  \bullet \lambda\). Since \(m\) is equivariant, \(m(\mu) \in W \cdot
+  m(\lambda)\). But the only elements in of \(\mathscr{B}\) in \(W \cdot
+  m(\lambda)\) are \(m(\lambda)\) and \((m(\lambda)_1, m(\lambda)_2, \ldots,
+  m(\lambda)_{n-1}, - m(\lambda)_n)\).
+
+  Conversely, if \(m(\lambda)_i = m(\mu)_i\) for \(i < n\) and \(m(\mu)_n = -
+  m(\lambda)_n\) then \(m(\mu) = \sigma_{\beta_n} \cdot m(\lambda)\) and \(\mu
+  = \sigma_n \bullet \lambda\). Since \(m(\lambda) \in \mathscr{B}\),
+  \(\lambda(H_{\beta_n}) \in \sfrac{1}{2} + \mathbb{Z}\) and thus
+  \(\lambda(H_{\beta_n}) \notin \mathbb{N}\). Hence by
+  Proposition~\ref{thm:lemma6.1} \(L(\mu) \subset \mExt(L(\lambda))\) and
+  \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\).
+
+  For each semisimple irreducible coherent \(\mathfrak{sp}_{2n}(K)\)-family
+  \(\mathcal{M}\) there is some \(m = m(\lambda) \in \mathscr{B}\) such that
+  \(\mathcal{M} = \mExt(L(\lambda))\). The only other \(m' \in \mathscr{B}\)
+  which generates the same coherent family as \(m\) is \(m' = \sigma_{\beta_n}
+  \cdot m\). Since \(m\) and \(m'\) lie in different elements of the partition
+  \(\mathscr{B} = \mathscr{B}^+ \cup \mathscr{B}^-\), the is a unique \(m'' =
+  m(\nu) \in \mathscr{B}^+\) -- either \(m\) or \(m'\) -- such that
+  \(\mathcal{M} \cong \mExt(L(\nu))\).
+\end{proof}
+
 \section{Coherent \(\mathfrak{sl}_n(K)\)-families}
 
 % TODO: Fix n >= 3
@@ -231,6 +281,7 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
         \to \{ \mathfrak{sl}_n\textrm{\normalfont-sequences} \} \\
         \lambda &
         \mapsto
+        \frac{2}{n}
         (
           \kappa(\epsilon_1, \lambda + \rho),
           \ldots,
@@ -245,6 +296,10 @@ 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 2/n is choosen because
+% λ(H_β) = 2/n 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
+
 % TODO: Explain the significance of 𝓑 + and 𝓑 -: these are the subsets whose
 % union corresponds to condition (i)
 \begin{definition}
@@ -271,7 +326,7 @@ Example~\ref{ex:sl-canonical-basis}. Also fix \(\rho = \sfrac{1}{2} \beta_1 +
 
 It should then be obvious that\dots
 
-\begin{proposition}
+\begin{proposition}\label{thm:arrow-implies-ext-eq}
   Let \(\lambda \notin P^+\) be such that \(L(\lambda)\) is bounded -- so that
   \(m(\lambda) \in \mathscr{B}\) -- and suppose that \(\mu \in \mathfrak{h}^*\)
   is such that \(m(\mu) \in \mathscr{B}\) and there is an arrow \(m(\lambda)
@@ -290,15 +345,8 @@ 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 ∩ 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
-% its connected component (see Lemma 8.3)
-% TODO: Perhaps this could be incorporated into the proof of the following
-% theorem? Perhaps it's best to create another lemma for this
 % TODO: Restate the notation for σ_i beforehand
-\begin{lemma}
+\begin{lemma}\label{thm:connected-components-B-graph}
   The connected component of some \(m \in \mathscr{B}\) is given by the
   following.
   \begin{enumerate}
@@ -377,7 +425,6 @@ all \(i\) and \(j\).
 % TODO: Perhaps this could be incorporated into the discussion of the lemma
 % that characterizes the weights of sl(n) whose L is bounded
 
-% TODO: Prove this
 \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
@@ -388,6 +435,65 @@ all \(i\) and \(j\).
   well as by \(\mathscr{B}^+\).
 \end{theorem}
 
+\begin{proof}
+  Let \(\lambda, \mu \notin P^+\) be such that \(L(\lambda)\) and \(L(\mu)\),
+  so that \(m(\lambda), m(\mu) \in \mathscr{B}\).
+
+  It is clear from Proposition~\ref{thm:arrow-implies-ext-eq} that if
+  \(m(\lambda)\) and \(m(\mu)\) lie in the same connected component of
+  \(\mathscr{B}\) then \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\). On the other
+  hand, if \(\mExt(L(\lambda)) \cong \mExt(L(\mu))\) then \(\chi_\lambda =
+  \chi_\mu\) and thus \(\mu \in W \bullet \lambda\). We now investigate which
+  elements of \(W \bullet \lambda\) satisfy the conditions of
+  Lemma~\ref{thm:sl-bounded-weights}. To do so,  we describe the set
+  \(\mathscr{B} \cap W \cdot m(\lambda)\).
+
+  % Great migué
+  If \(\lambda\) is regular and integral then the only permutations of
+  \(m(\lambda)\) which lie in \(\mathscr{B}\) are \(\sigma_k \sigma_{k+1}
+  \cdots \sigma_i \cdot m'\) for \(k \le i\) and \(\sigma_k \sigma_{k-1} \cdots
+  \sigma_i \cdot m'\) for \(k \ge i\), where \(m'\) is the unique ordered
+  element of \(W \cdot m(\lambda)\). Hence by
+  Lemma~\ref{thm:connected-components-B-graph} \(\mathscr{B} \cap W \cdot
+  m(\lambda)\) is the union of the connected components of the \(\sigma_i \cdot
+  m'\) for \(i \le n\). On the other hand, if \(\lambda\) is singular or
+  non-integral then the only permutations of \(m(\lambda)\) which lie in
+  \(\mathscr{B}\) are the ones from the connected component of \(m(\lambda)\)
+  in \(\mathscr{B}\), so that \(\mathscr{B} \cap W \cdot m(\lambda)\) is
+  exactly the connected component of \(m(\lambda)\).
+
+  In both cases, we can see that if \(B(\lambda)\) is the set of the \(m' =
+  m(\mu) \in \mathscr{B}\) such that \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\)
+  then \(B(\lambda) \subset \mathscr{B} \cap W \cdot m(\lambda)\) is contain in
+  a union of connected components of \(\mathscr{B}\) -- including that of
+  \(m(\lambda)\) itself. We now claim that \(B(\lambda)\) is exactly the
+  connected component of \(m(\lambda)\). This is already clear when \(\lambda\)
+  is singular or non-integral, so we may assume that \(\lambda\) is regular and
+  integral, in which case every other \(\mu \in W \bullet \lambda\) is regular
+  and integral.
+
+  % Great migué
+  In this situation, \(m(\mu) \in \mathscr{B}^+\) implies \(\mu(H_{\beta_1}) =
+  m(\mu)_1 - m(\mu)_2 \in \mathbb{Z}\) is negative. But it follows from
+  Proposition~\ref{thm:lemma6.1} that for each \(\beta \in \Sigma\) there is at
+  most one \(\mu \notin P^+\) with \(\mExt(L(\mu)) \cong \mExt(L(\lambda))\)
+  such that \(\mu(H_\beta)\) is a negative integer -- see Lemma~6.5 of
+  \cite{mathieu}. Hence there is at most one \(m' \in \mathscr{B}^+ \cap W
+  \cdot m(\lambda)\). Since every connected component of \(\mathscr{B}\) meets
+  \(\mathscr{B}^+\) -- see Lemma~\ref{thm:connected-components-B-graph} -- this
+  implies \(B(\lambda)\) is precisely the connected component of
+  \(m(\lambda)\).
+
+  Another way of stating this is to say that \(\mExt(L(\lambda)) \cong
+  \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and \(m(\mu)\) lie in the same
+  connected component. There is thus a one-to-one correspondance between
+  \(\pi_0(\mathscr{B})\) and the isomorphism classes of semisimple irreducible
+  coherent \(\mathfrak{sl}_n(K)\)-families. Since every connected component of
+  \(\mathscr{B}\) meets \(\mathscr{B}^+\) precisely once -- again, see
+  Lemma~\ref{thm:connected-components-B-graph} -- we also get that such
+  isomorphism classes are parameterized by \(\mathscr{B}^+\).
+\end{proof}
+
 % TODO: Change this
 % I don't really think these notes bring us to this conclusion
 % If anything, these notes really illustrate the incredible vastness of the

diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex
@@ -687,7 +687,7 @@ This already allow us to compute some examples of Weyl groups.
   \end{align*}
 \end{example}
 
-\begin{example}
+\begin{example}\label{ex:sp-weyl-group}
   Suppose \(\mathfrak{g} = \mathfrak{sp}_{2n}(K)\) and \(\mathfrak{h} \subset
   \mathfrak{g}\) is as in Example~\ref{ex:cartan-of-sp}. Let \(\epsilon_1,
   \ldots, \epsilon_n \in \mathfrak{h}^*\) be as in
@@ -697,10 +697,11 @@ This already allow us to compute some examples of Weyl groups.
   \(\epsilon_{i+1}\) for \(i < n\) and \(\sigma_{\beta_n}\) switches the sign
   of \(\epsilon_n\). This translates to a canonical isomorphism
   \begin{align*}
-                   W & \isoto  S_n \ltimes (\mathbb{Z}/2\mathbb{Z})^n        \\
-    \sigma_{\beta_i} & \mapsto (\sigma_i, \bar 0) = ((i \; i\!+\!1), \bar 0) \\
-    \sigma_{\beta_n} & \mapsto (1, \bar 1)
+                   W & \isoto  S_n \ltimes (\mathbb{Z}/2\mathbb{Z})^n \\
+    \sigma_{\beta_i} & \mapsto (\sigma_i, (\bar 0, \ldots, \bar 0))   \\
+    \sigma_{\beta_n} & \mapsto (1, (\bar 0, \ldots, \bar 0, \bar 1)),
   \end{align*}
+  where \(\sigma_i = (i \ i\!+\!1)\) are the canonical transpositions.
 \end{example}
 
 If we conjugate some \(\sigma \in W\) by the isomorphism \(\mathfrak{h}^*