A representation $\grep : G \to \GL(\s V)$ is said to be \defnsty{completely reducible} if it is isomorphic to a finite direct sum of irreducible representations, i.e., there exists a finite set of irreducible representations $\grep_{i} : G \to \GL(\s V_{i})$ indexed by $i \in [r]$ such that $\s V$ decomposes as a vector space into a finite direct sum,
\begin{equation}
\s V = \s V_{1} \oplus \s V_{2} \cdots \oplus \s V_{r},
\end{equation}
and furthermore $\grep(g)$ decomposes, for each $g$, as an operator
\begin{equation}
\grep(g) = \grep_{1}(g) \oplus \cdots \oplus \grep_{r}(g).
\end{equation}
The group $G$ itself is said to be \defnsty{completely reducible} if every finite-dimensional representation of $G$ is completely reducible.
\end{defn}

The following is \cite[Prop. 4.27]{hall2015lie}.
\begin{lem}
\label{lem:unitary_reps_completely_reducible}
Let $\grep : K \to \U(\s V)$ be a finite-dimensional unitary representation of $K$ on a Hilbert space $\s V$ with inner product $\braket{\cdot, \cdot}$. Then $\grep$ is completely reducible.
\end{lem}
\begin{proof}
First, we prove that any reducible, unitary representation $\grep : K \to \s U(\s V)$ of $K$ on the Hilbert space $\s V$ \textit{splits} into the orthogonal direct sum of unitary representations.
Let $\s W \subset \s V$ be a proper invariant subspace of $\s V$.
The orthogonal complement $\s W^{\perp}$ of $\s W$ (evaluated with respect to the inner product $\braket{\cdot, \cdot}$ on $\s V$), leads to the decomposition $\s V = \s W \oplus \s W^{\perp}$.
That $\s W^{\perp}$ is also an invariant subspace follows from the unitarity of $\grep(k)$ for all $k \in K$.
The invariance of $\s W$ means $\grep(k)w \in \s W$ for all $w \in \s W$ and $k \in K$ (including $k^{-1}$).
Therefore, for generic $w' \in \s W^{\perp}$ and $w \in \s W$,
\begin{equation}
0 = \braket{w', \grep(k^{-1}) w} = \braket{w', \grep(k)^{*} w} = \braket{\grep(k)w',  w},
\end{equation}
which means $\grep(k)w' \in \s W^{\perp}$ and thus $\s W^{\perp}$ is also invariant.

Second, if the representation on $\s W$ (or $\s W^{\perp}$) is reducible, then the above argument can be applied to recursively split the subrepresentation on $\s W$ (or $\s W^{\perp}$).
Since $\s V$ is finite-dimensional and one-dimensional representations are irreducible by definition, this recursive process much terminate, producing a orthogonal direct-sum decomposition of $\s V$ into irreducible invariant subspaces.
\end{proof}

The following is \cite[Thm 4.28]{hall2015lie} as is sometimes known as \textit{Weyl's unitary trick}.
\begin{thm}
\label{thm:compact_completely_reducible}
If $K$ is compact matrix Lie group, then every finite-dimensional representation of $K$ is completely reducible.
\end{thm}
\begin{proof}
Let $\grep : K \to \GL(\s V)$ be a representation of $K$ on a finite-dimensional Hilbert space $\s V$ with inner product $\braket{\cdot, \cdot}$.
If $\grep$ is a unitary representation, i.e. $\grep(k)^{*} = \grep(k^{-1})$ (with respect to $\braket{\cdot, \cdot}$) for all $k \in K$, then \cref{lem:unitary_reps_completely_reducible} implies $\grep$ is completely reducible.
Thus the only remaining case to consider is when $\grep$ is not a unitary representation.
The trick here is to show that whenever $\grep(k)$ is not unitary with respect to the inner product $\braket{\cdot, \cdot}$, one can define a new inner product, $\braket{\cdot, \cdot}_{K}$ on $\s V$ such that $\grep(k)$ is unitary.
This new inner product is obtained by \textit{twirling} the old inner product, namely for $v, w \in \s V$,
\begin{equation}
\braket{v, w}_{G} \coloneqq \int_{G} \braket{\grep(k)v, \grep(k)w} \diff \mu(k),
\end{equation}
where $\diff \mu$ is the unique up to scaling left and right $K$-invariant Haar measure on $K$ (compactness of $K$ ensures this existence and invariance of this measure, as well as the convergence of the integral).
That $\grep(k)$ is a unitary operator with respect to $\braket{\cdot, \cdot}_{K}$ follows from the invariance of the Haar measure $\diff \mu$ which implies $\braket{\grep(k)v, \grep(k)w}_{K} = \braket{v, w}_{K}$ for all $v, w \in \s V$ which means $\grep(k)^{*} \grep(k) = \ident_{\s V}$ is the identity on $\s V$ and thus $\grep(k)^{-1} = \grep(k^{-1}) = \grep(k)^{*}$ is unitary for all $g \in G$.
By \cref{lem:unitary_reps_completely_reducible}, again, $\grep$ is completely reducible.
\end{proof}

\subsection{Complexification \& semisimplicity}
\label{sec:complexification}

At this stage, we turn our attention to a particular class of Lie algebras called \textit{semisimple Lie algebras} which are obtained from the Lie algebras of compact Lie groups by a process known as \textit{complexification}.
Our interest in semisimple Lie algebras stems from the fact that, much like \cref{thm:compact_completely_reducible}, their finite-dimensional representations are completely reducible~\cite[Thm. 10.9]{hall2015lie}.

Semisimple Lie algebras include many of the familiar examples of including $\mathfrak{sl}(d, \mathbb C)$, $\mathfrak{so}(d, \mathbb C)$, and $\mathfrak{sp}(d, \mathbb C)$.
Our main goal will be to understand the decomposition of reductive Lie algebras and establish some consistent notation along the way.

To begin, we consider the complexification of a real Lie algebras, $\mathfrak k$, which is formed by extending the field of real numbers, $\mathbb R$, to the field of complex numbers, $\mathbb C$, in the usual way.
\begin{defn}
Let $\mathfrak k$ be a  Lie algebra with Lie bracket $[\cdot, \cdot]$.
The \defnsty{complexification} of $\mathfrak k$, expressed as $\mathfrak k_{\mathbb C}$ or $\mathfrak k \oplus i \mathfrak k$, is the Lie algebra of formal linear combinations,
\begin{equation}
X = X_1 + i X_2,
\end{equation}
where $X_1, X_2 \in \mathfrak k$ and where $iX \coloneqq - X_2 + i X_1$.
The Lie bracket is extended linearly for $X, Y \in \mathfrak k_{\mathbb C}$ to
\begin{equation}
[X, Y] = ([X_1, Y_1] - [X_2, Y_2]) + i ([X_1, Y_2] + [X_2, Y_1]).
\end{equation}
\end{defn}
\begin{exam}
\label{exam:complexification}
The complexification of the Lie algebra of real $d \times d$ matrices is isomorphic to the Lie algebra of complex $d \times d$ matrices,
\begin{equation}
\mathfrak{gl}(d,\mathbb R)_{\mathbb C} \cong \mathfrak{gl}(d, \mathbb C).
\end{equation}
Similarly, the complexification of the Lie algebra of skew-Hermitian $d \times d$ matrices (which is a real Lie algebra) is also isomorphic to the Lie algebra of complex $d\times d$ matrices,
\begin{equation}
\mathfrak{u}(d)_{\mathbb C} \cong \mathfrak{gl}(d, \mathbb C).
\end{equation}
This second example follows because every $d \times d$ complex matrix $M \in \mathfrak{gl}(d, \mathbb C)$ is expressible as
\begin{equation}
M = \left(\frac{M-M^{*}}{2}\right) + i \left(\frac{M+M^{*}}{2i}\right)
\end{equation}
where $M^{*}$ is the conjugate transpose of the matrix $M$ and where \textit{both} the first and second terms in parentheses are skew-Hermitian matrices, and thus elements of $\mathfrak{u}(d)_{\mathbb C}$.
For more examples of complexifications, see \cite[Eq. 3.17]{hall2015lie}.
\end{exam}

\begin{rem}
\label{rem:complexification_convention}
When working with Lie algebras and their complexifications, it can quickly become quite challenging to keep track of all of the different relationships between the various subspaces.
It is therefore essential to establish some consistent notational conventions.
Our notational conventions are fairly standard and strongly mirrors those of Ref.~\cite{botero2021large}.
As a starting point we consider a real Lie algebra $\mathfrak k$ and its complexification
\begin{equation}
\mathfrak g = \mathfrak k_{\mathbb C} = \mathfrak k \oplus i \mathfrak k.
\end{equation}
Acting on the complexified Lie algebra $\mathfrak g = \mathfrak k_{\mathbb C}$, there exists a map, called the \defnsty{local Cartan involution},
\begin{equation}
\theta : \mathfrak g \to \mathfrak g,
\end{equation}
which sends each $X = X_1 + i X_2 \in \mathfrak g$ (where $X_1, X_2 \in \mathfrak k$) to
\begin{equation}
\theta(X) = \theta(X_1 + i X_2) = X_1 - i X_2.
\end{equation}
In this way, $\mathfrak k$ can be identified with the $+1$ eigenspace of $\theta$, while $i\mathfrak k$ is identified with its $-1$ eigenspace.
In addition, we can define a \defnsty{local $*$-involution}, $* : \mathfrak g \to \mathfrak g$, on $\mathfrak g$ as the map sending $X \in \mathfrak g$ to
\begin{equation}
\label{eq:star_involution}
X^{*} = -\theta(X) = -X_1 + i X_2.
\end{equation}
In this way, if $\mathfrak k = \mathfrak u(d)$ is the Lie algebra of $d \times d$ skew-Hermitian matrices, then $i\mathfrak k = i \mathfrak u(d)$ is the Lie algebra of $d \times d$ Hermitian matrices (with negated Lie bracket), and then the $*$-involution coincides with the complex conjugate of a complex matrix in the sense that
\begin{align}
\begin{split}
X \in \mathfrak k &\iff X^{*} = - \theta(X) = - X, \\
X \in i\mathfrak k &\iff X^{*} = - \theta(X) = + X.
\end{split}
\end{align}
Finally, it is sometimes useful to give the factor $i\mathfrak k$ its own dedicated symbol, namely
\begin{equation}
\mathfrak p \coloneqq i \mathfrak k,
\end{equation}
and thus we have $\mathfrak g = \mathfrak k_{\mathbb C} = \mathfrak k \oplus i\mathfrak k = \mathfrak k \oplus \mathfrak p$.
\end{rem}
\begin{rem}
In addition to the notational conventions from \cref{rem:complexification_convention} for dealing with the complexification, $\mathfrak g = \mathfrak k_{\mathbb C} = \mathfrak k \oplus i\mathfrak k$, of a real Lie algebra $\mathfrak k$, it will also be important, for the sake of consistency, to establish a notational convention for dealing with the subalgebras of $\mathfrak k$ and their complexification. In particular, the maximal abelian subalgebra (or Cartan subalgebra) of $\mathfrak k$, will always be denoted by $\mathfrak t$, in which case we have the relationship
\begin{equation}
\mathfrak t \subseteq \mathfrak k.
\end{equation}
The complexification of $\mathfrak t$, will be denoted by
\begin{equation}
\mathfrak h = \mathfrak t_{\mathbb C} = \mathfrak t \oplus i \mathfrak t.
\end{equation}
Evidently, just as $\mathfrak t$ is a subalgebra of $\mathfrak k$, the complexification of $\mathfrak t$ is a subalgebra of the complexification of $\mathfrak k$,
\begin{equation}
\mathfrak h = \mathfrak t \oplus i \mathfrak t \subseteq \mathfrak k \oplus i \mathfrak k = \mathfrak g.
\end{equation}
Furthermore, $\mathfrak h$ is the maximal abelian subalgebra of $\mathfrak g$~\cite[Prop. 7.11]{hall2015lie}.
Finally, in addition to sometimes letting $\mathfrak p \coloneqq i\mathfrak k$, we sometimes let $\mathfrak a \coloneqq i \mathfrak t$, in which case $\mathfrak a \subseteq \mathfrak p$.
\end{rem}
\begin{exam}
\label{exam:guiding_lie_alg_example}
An key example to illustrate all of these notational conventions is to start with the Lie algebra $\mathfrak k = \mathfrak u(d)$.
In this case, we have
\begin{itemize}
\item $\mathfrak k = \mathfrak u(d)$ (skew-Hermitian $d\times d$ matrices),
\item $\mathfrak p = i\mathfrak k = i\mathfrak u(d)$ (Hermitian $d\times d$ matrices),
\item $\mathfrak g = \mathfrak k \oplus \mathfrak p = \mathfrak {gl}(d, \mathbb C)$ ($d\times d$ complex matrices),
\item $\mathfrak t = \mathfrak u(1)^{\times d} \cong i\mathbb R^{d}$ ($d\times d$ diagonal matrices with imaginary entries),
\item $\mathfrak a = i\mathfrak t = i\mathfrak u(1)^{\times d} \cong \mathbb R^{d}$ ($d\times d$ diagonal matrices with real entries), and
\item $\mathfrak h = \mathfrak t \oplus \mathfrak a \cong \mathbb C^{d}$ ($d\times d$ diagonal matrices with complex entries).
\end{itemize}
\end{exam}

\begin{defn}
A Lie algebra, $\mathfrak g$, is called \defnsty{reductive} if
\begin{equation}
\mathfrak g = \mathfrak k_{\mathbb C}
\end{equation}
where $\mathfrak k$ is the Lie algebra of a compact matrix Lie group $K$.
If $\mathfrak g$ is reductive and has trivial center, i.e., $\mathfrak z(\mathfrak g) = 0$, then it is \defnsty{semisimple}.
\end{defn}
\begin{exam}
Note that every complex reductive Lie algebra $\mathfrak g$ can be decomposed as a direct sum
\begin{equation}
\label{eq:reductive_semisimple_decomp}
\mathfrak g = \mathfrak z(\mathfrak g) \oplus \mathfrak s,
\end{equation}
where $\mathfrak z(\mathfrak g)$ is its center and $\mathfrak s$ is semisimple~\cite[Prop. 7.6]{hall2015lie}.
For example, the Lie algebra of $d\times d$ complex matrices, $\mathfrak{gl}(d, \mathbb C)$, is reductive as it is the complexification of $\mathfrak{gl}(d, \mathbb R)$ or $\mathfrak u(d)$, but it is not semisimple because its center,
\begin{equation}
\mathfrak{z}(\mathfrak{gl}(d, \mathbb C)) = \{ z \ident_{d} \mid z \in \mathbb C \} \cong \mathbb C,
\end{equation}
consists of the set of scalar $d\times d$ matrices.
In this case, \cref{eq:reductive_semisimple_decomp} becomes
\begin{equation}
\mathfrak{gl}(d, \mathbb R) \cong \mathbb C \oplus \mathfrak{sl}(d, \mathbb C),
\end{equation}
where semisimple factor, $\mathfrak{sl}(d, \mathbb C)$, consists of all traceless $d \times d$ complex matrices.
\end{exam}

Although finite-dimensional representations of semisimple Lie algebras can be shown to be completely reducible~\cite[Thm. 10.9]{hall2015lie}, the same is not true of finite-dimensional representations of reductive Lie algebras~\cite[Ex. 10.9.1]{hall2015lie}\footnote{A standard example is the Lie algebra representation $\arep : \mathbb C \to \mathfrak{gl}(2, \mathbb C)$ sending $z \in \mathbb C$ to $\arep(z) = \begin{pmatrix} 0 & z \\ 0 & 0 \end{pmatrix}$ which contains non-trivial invariant subspaces but is not expressible as the direct sum of smaller representations of $\mathbb C$.}.
Given that finite-dimensional representations of semisimple Lie algebras are completely reducible~\cite[Thm. 10.9]{hall2015lie}, it becomes worth classifying their finite-dimensional irreducible representations.
This classification is provided by the theorem of highest weights (\cref{thm:hwt_alg}) and depends on a handful of supporting concepts and terminology to be covered over the next few pages.

The crucial reason for taking $\mathfrak k$ in the definition of a reductive Lie algebra to be the Lie algebra of a compact Lie group $K$ is to import Weyl's unitary trick mentioned in the proof of \cref{thm:compact_completely_reducible} for the purposes of establishing the following result~\cite[Prop. 7.4]{hall2015lie}.
\begin{lem}
\label{lem:inner_prod_alg}
Let $\mathfrak g = \mathfrak k_{\mathbb C}$ be a reductive Lie algebra.
Then there exists an inner product $\braket{\cdot, \cdot} : \mathfrak g \times \mathfrak g \to \mathbb C$ on $\mathfrak g$ which is (i) real-valued when restricted to $\mathfrak k \subseteq \mathfrak g$, and (ii) satisfies for all $X,Y,Z \in \mathfrak g$,
\begin{equation}
\braket{\ad(X)(Y), Z} = \braket{Y, \ad(X^{*})(Z)},
\end{equation}
where $\ad : \mathfrak g \to \mathfrak{gl}(\mathfrak g)$ is the adjoint representation of $\mathfrak g$ on itself, and $X^{*}$ is defined as in \cref{eq:star_involution}.
\end{lem}

Recall, at this stage, that the roots of a Lie algebra are the weights (simultaneous eigenvalues) of its adjoint representation (recall \cref{defn:weights_alg} and \cref{exam:roots_alg}).
In the context of reductive Lie algebras, roots must interact with the complex structure in a particular way.
\begin{lem}
\label{lem:root_complex_structure}
Let $\alpha \in \mathfrak h^{*}$ be a root for a reductive Lie algebra $\mathfrak g = \mathfrak k \oplus i \mathfrak k$ relative to a maximal abelian subalgebra $\mathfrak h = \mathfrak t \oplus i \mathfrak t$.
If $H = H_1 + i H_2 \in \mathfrak h$ for $H_1, H_2 \in \mathfrak t$, then
\begin{equation}
\alpha(H)^{*} = - \alpha(H_1) + i \alpha(H_2).
\end{equation}
\end{lem}
\begin{proof}
Let $X \in \mathfrak g$ be a non-zero eigenvector of the adjoint representation of $H \in \mathfrak t \subseteq \mathfrak k$ with eigenvalue $\alpha \in \mathbb C$, meaning
\begin{equation}
\ad(H)(X) = [H, X] = \alpha X.
\end{equation}
Since the inner product in \cref{lem:inner_prod_alg} is anti-linear in the first argument and linear in the second argument, and $H \in \mathfrak t \subseteq \mathfrak k$ satisfies $H^{*} = - H$, one concludes
\begin{align}
\begin{split}
\alpha^{*}\braket{X, X}
&= \braket{\alpha X, X}, \\
&= \braket{\ad(H)(X), X}, \\
&= \braket{X, \ad(H^{*})(X)}, \\
&= -\braket{X, \ad(H)(X)}, \\
&= -\alpha \braket{X, X}.
\end{split}
\end{align}
Therefore, for every $H \in \mathfrak t$, its adjoint representation, $\ad(H)$, has purely imaginary eigenvalues, $\alpha^{*} = - \alpha$.
Since roots are just simultaneous eigenvectors of the adjoint representation of $\mathfrak h$, one concludes, for all $H \in \mathfrak t$, that $\alpha(H) \in i \mathbb R$ is imaginary. Therefore,
\begin{align}
\alpha(H)^{*} &= \alpha(H_1 + i H_2)^{*} = (\alpha(H_1) + i \alpha(H_2))^{*}, \\
&= \alpha(H_1)^{*} - i \alpha(H_2)^{*} = - \alpha(H_1) + i \alpha(H_2).
\end{align}
\end{proof}
\begin{rem}
\label{rem:real_root}
In light of \cref{lem:root_complex_structure}, a root of a reductive Lie algebra, while having the type of a complex-linear function $\alpha : \mathfrak h \to \mathbb C$ (i.e. $\alpha \in \mathfrak h^{*}$), is essentially determined by its restriction to $\mathfrak t$ or $i\mathfrak t$. Specifically, every root can be identified with a real-linear function on $i\mathfrak t$, i.e. $\mathfrak h : i \mathfrak t \to \mathbb R$.
\end{rem}

Understanding the algebraic structure of the roots and associated root spaces yields the most significant result in the representation theory of semisimple Lie algebras.
\begin{thm}
\label{thm:root_space_decomp}
Let $\mathfrak g = \mathfrak k_{\mathbb C}$ be a complex semisimple Lie algebra with fixed maximal abelian subalgebra $\mathfrak h$.
Let $R \subseteq \mathfrak h^{*}$ denote the set of all non-zero roots of $\mathfrak g$.
Then $\mathfrak g$ can be decomposed as the direct sum
\begin{equation}
\mathfrak g = \mathfrak h \oplus \bigoplus_{\alpha \in R} \mathfrak g_{\alpha},
\end{equation}
where $\mathfrak g_{\alpha} \subseteq \mathfrak g$ is the root space with associated to $\alpha \in \mathfrak h^{*}$.\\
\begin{itemize}
\item for all $\alpha, \beta \in \mathfrak h^{*}$, $[g_{\alpha}, g_{\beta}] \subseteq g_{\alpha + \beta}$, meaning
\begin{equation}
X \in g_{\alpha}, Y \in g_{\beta} \implies [X,Y] \in g_{\alpha, \beta},
\end{equation}
\item if $\alpha \in R$ is a root with root vector $X \in g_{\alpha}$, then $-\alpha \in \mathfrak h^{*}$ is also a root with root vector $X^{*} \in g_{-\alpha}$, and moreover $\alpha$ and $-\alpha$ are the \textit{only} non-zero roots proportional to $\alpha$,
\item the roots span all of $\mathfrak h^{*}$,
\begin{equation}
\mathrm{span}(R) = \mathfrak h^{*},
\end{equation}
\item for each root $\alpha \in R$, the root space, $\mathfrak g_{\alpha}$, is one-dimensional,
\begin{equation}
\dim(\mathfrak g_{\alpha}) = 1.
\end{equation}
\end{itemize}
\end{thm}
\begin{proof}
The statement of the above theorem is a combination of a number of results~\cite[Prop. 7.16-7.18, Thm. 7.19, Thm 7.23]{hall2015lie}.
\end{proof}

\begin{thm}
\label{thm:sl2_inside}
Let $\mathfrak g = \mathfrak k_{\mathbb C}$ be a reductive Lie algebra.
Then there exists a subalgebra $\mathfrak s^{\alpha} \subseteq \mathfrak g$ spanned by $H_{\alpha} \in \mathfrak h$, $X_{\alpha} \in \mathfrak g_{\alpha}$, and $Y_{\alpha} = X_{\alpha}^{*} \in \mathfrak g_{-\alpha}$ which satisfies
\begin{equation}
[H_{\alpha}, X_{\alpha}] = 2 X_{\alpha},
\qquad
[H_{\alpha}, Y_{\alpha}] = 2 Y_{\alpha},
\qquad
[X_{\alpha}, Y_{\alpha}] = H_{\alpha},
\end{equation}
and thus $\mathfrak s^{\alpha}$ is isomorphic to $\mathrm{sl}(2, \mathbb C)$ from~\cref{exam:HXY_basis}.
Moreover, $H_{\alpha} \in \mathfrak h$ is uniquely determined by $\alpha$ and called the \defnsty{coroot} of $\alpha$.
\end{thm}
\begin{proof}
Let $\braket{\cdot, \cdot}$ be the inner product on $\mathfrak g$ provided by \cref{lem:inner_prod_alg}
Then let $X \in g_{\alpha}$ be a non-zero root vector for the root $\alpha \in \mathfrak h^{*}$, meaning
\begin{equation}
\forall H \in \mathfrak h : \ad(H)(X) = [H,X] = \alpha(H) X.
\end{equation}
By \cref{thm:root_space_decomp}, $X^{*} \in g_{-\alpha}$ is a root vector for the root $-\alpha \in \mathfrak h^{*}$.
Therefore, by \cref{lem:inner_prod_alg}, for all $H \in \mathfrak h$,
\begin{equation}
\braket{[X,X^*], H} = \braket{X^*, [X^*, H]} = - \braket{X^{*}, [H, X^{*}]} = \alpha(H) \braket{X^{*}, X^{*}}.
\end{equation}
As $X^{*}$ is non-zero, so is $\braket{X^{*}, X^{*}}$, which means $\alpha$ is proportional to $[X,X^{*}]$ in the sense that
\begin{equation}
\forall H \in \mathfrak h : \alpha(H) = \frac{\braket{[X,X^*], H}}{\braket{X^{*}, X^{*}}}.
\end{equation}
Again using \cref{thm:root_space_decomp}, $[X, X^{*}] \in g_{0} = \mathfrak h$ and therefore
\begin{equation}
\alpha([X,X^*]) = \frac{\norm{[X,X^*]}^{2}}{\norm{X^{*}}^{2}} > 0
\end{equation}
is well-defined, real and positive.
The first claim of the proof follows from defining
\begin{equation}
H_{\alpha} = \frac{2}{\alpha([X,X^*])}[X,X^*],
\qquad
X_{\alpha} = \sqrt{\frac{2}{\alpha([X,X^*])}}X,
\qquad
Y_{\alpha} = \sqrt{\frac{2}{\alpha([X,X^*])}}X^*,
\end{equation}
which satisfy the claimed commutation relations.
To prove that $H_{\alpha}$ is independent of the choice of non-zero root vector $X \in g_{\alpha}$ initially chosen, it suffices to note that, by \cref{thm:root_space_decomp}, $\dim(g_{\alpha}) = 1$, and thus the only other choice would be a scalar multiple of $X$ which leaves the definition of $H_{\alpha}$ unaffected as $\alpha$ is proportional to $H_{\alpha}$,
\begin{equation}
\alpha(H) \propto \braket{H_{\alpha}, H},
\end{equation}
with proportionality constant $\norm{[X,X^*]}^{2}/(2\norm{X^{*}}^{4})$ which is independent of scalar multiples of $X$.
\end{proof}
\begin{cor}
\label{cor:integral_rep}
Let $\mathfrak g = \mathfrak k_{\mathbb C}$ be a finite-dimensional reductive Lie algebra, let $\alpha, \beta \in \mathfrak h^{*}$ be roots of $\mathfrak g$ and let $H_{\beta} \in \mathfrak h$ be the coroot of $\beta$ as defined in \cref{thm:sl2_inside}.
Then $\alpha(H_{\beta})$ is an integer,
\begin{equation}
\alpha(H_{\beta}) \in \mathbb Z.
\end{equation}
\end{cor}
\begin{proof}
Given that the coroot $H_{\beta} \in \mathfrak h$ as defined by \cref{thm:sl2_inside}, belongs to subalgebra of $\mathfrak g$ isomorphic to $\mathfrak{sl}(2, \mathbb C)$, we can apply \cref{exam:reps_sl2C} to see that the eigenvalues of $\arep(H_{\beta})$ for any finite-dimensional representation must be integers.
Finally, given that $\mathfrak g$ is assumed finite-dimensional, and that $\alpha(H_{\beta})$ is an eigenvalue of $\ad(H_{\beta})$, it must be that $\alpha(H_{\beta})$ is an integer.
\end{proof}

\begin{rem}
The results of \cref{thm:root_space_decomp},\cref{cor:integral_rep} and \cite[Thm. 7.26]{hall2015lie} imply that the roots of any finite-dimensional semisimple complex Lie algebra constitute an abstract \textit{root system} (see \cref{sec:roots_systems}).
Using the inner product provided by \cref{lem:inner_prod_alg}, it is possible to identify $\mathfrak h$ with its dual vector space $\mathfrak h^{*}$ and therefore every root $\alpha \in \mathfrak h^{*}$ with an unique element $H_{\alpha}' \in \mathfrak h$ such that
\begin{equation}
\alpha(H) = \braket{H'_{\alpha}, H},
\end{equation}
holds for every $H \in \mathfrak h'$.
By \cref{thm:sl2_inside} we see that the element $H_{\alpha}' \in H$ would be \textit{proportional} to the coroot $H_{\alpha} \in H$.
The convention adopted in this thesis is to consider roots as functionals, i.e. $\alpha \in \mathfrak h^{*}$, in which case the enveloping vector space for the abstract root systems considered in \cref{sec:roots_systems} should be taken to be $\s E = \mathfrak h^{*}$ with inner product, $\braket{\cdot, \cdot}'$ on $\mathfrak h'$, \textit{dual} to the inner product provided by \cref{lem:inner_prod_alg}, i.e., for all $\alpha, \beta \in \mathfrak h^{*}$ let
\begin{equation}
\braket{\alpha, \beta}' \coloneqq \braket{H'_{\alpha}, H'_{\beta}}.
\end{equation}
Of course, in the setting of finite-dimensional inner product spaces, which convention one chooses is purely a matter of preference.
\end{rem}
\subsection{Roots systems}
\label{sec:roots_systems}

\begin{defn}
\label{defn:root_system}
A \defnsty{root system} $R$ is a finite set of non-zero vectors, called \defnsty{roots}, of a finite dimensional real inner product space $(\s E, \braket{\cdot, \cdot})$ such that
\begin{enumerate}[i)]
\item the roots span $\s E$, i.e. $\mathrm{span}(R) = \s E$,
\item if $\alpha \in R$, then $s \alpha \in R$ if and only if $s \in \{-1,+1\}$,
\item if $\alpha, \beta \in R$, then $c_{\alpha}(\beta)$ defined by
\begin{equation}
\label{eq:cartan_integer}
c_{\alpha}(\beta) \coloneqq 2\frac{\braket{\alpha, \beta}}{\braket{\alpha, \alpha}},
\end{equation}
in an integer (called a \defnsty{Cartan integer}), and
\item if $\alpha, \beta \in R$, then $r_{\alpha}(\beta) \in R$ where
\begin{equation}
\label{eq:root_reflect}
r_{\alpha}(\beta) \coloneqq \beta - c_{\alpha}(\beta) \alpha.
\end{equation}
\end{enumerate}
The \defnsty{rank of a root system} $R$ is the dimension of $\s E$.
\end{defn}

\begin{rem}
\label{rem:weyl_group}
For any root $\alpha \in R$ of a root system $R$, \cref{eq:root_reflect} defines a \textit{reflection} $r_{\alpha} : R \to R$ sending each root $\beta \in R$ to the root $r_{\alpha}(\beta) \in R$ obtained by reflecting $\beta$ through the hyperplane $\s E_{\alpha} \subset \s E$ (of dimension $\dim(\s E)-1$) orthogonal to $\alpha$ defined by
\begin{equation}
\label{eq:root_hyperplane}
\s E_{\alpha} \coloneqq \{ v \in \s E \mid \braket{\alpha, v} = 0 \}.
\end{equation}
The collection of all such reflections $W = \{r_{\alpha} \mid \alpha \in R\}$ generates a finite subgroup of the orthogonal group on $\s E$ known as the \defnsty{Weyl group} of the root system $R$.
\end{rem}

\begin{defn}
\label{defn:weyl_chambers}
Let $R \subset \s E$ be a root system of the real inner product space $(\s E, \braket{\cdot, \cdot})$ and let $S : \s E \to \{-1, 0, +1\}^{R}$ be the function assigning to each element $\mu \in \s E$ the function, $S_{\mu} : R \to \{-1,0,+1\}$, mapping each root $\alpha \in R$ to the sign of $\braket{\mu, \alpha}$:
\begin{equation}
S_{\mu}(\alpha) = \mathrm{sign}(\braket{\mu, \alpha}) =
\begin{cases}
-1 & \text{if }\braket{\mu, \alpha} < 0, \\
\hspace{0.75em}0 & \text{if }\braket{\mu, \alpha} = 0, \\
+1 & \text{if }\braket{\mu, \alpha} > 0.
\end{cases}
\end{equation}
Since there are only finitely many $\{-1, 0, +1\}$-valued functions on $R$, the map $\mu \mapsto S_{\mu}$ partitions $\s E$ into finitely many disjoint regions.
Those regions where $S_{\mu}$ is strictly non-zero for all roots,
\begin{equation}
\forall \alpha \in R : S_{\mu}(\alpha) = \mathrm{sign}(\braket{\mu, \alpha}) \neq 0,
\end{equation}
are called the \defnsty{Weyl chambers}.
In other words, the Weyl chambers are the open connected components of the set-wise difference $\s E \setminus \bigcup_{\alpha \in R} \s E_{\alpha}$ where $\s E_{\alpha}$ is the hyperplane orthogonal to the root $\alpha$ as defined by \cref{eq:root_hyperplane}.

The Weyl group, $W$, acts transitively on the Weyl chambers~\cite[Prop. 8.23]{hall2015lie}.
\end{defn}

The following remark demonstrates that for every root system $R$ one can always pick a subset of roots $\Delta \subset R$ called a \textit{base} for $R$ that has a number of desirable properties~\cite[Thm. 8.16]{hall2015lie}.
Choosing a base for a root system, as we shall see, is equivalent to choosing a particular Weyl chamber to be the \textit{positive} one.
\begin{rem}
\label{rem:base_root_sys}
Let $R \subset \s E$ be a root system in $\s E$ and fix a hyperplane $\s W \subset \s E$ of dimension $\dim(\s E) - 1$, called the \defnsty{separating hyperplane}, such that no root $\alpha \in R$ is contained within $\s W$.
That such a hyperplane actually exists follows from choosing a Weyl chamber, denoted by $C$, to be considered as the \defnsty{positive Weyl chamber}, and then choosing a vector $\omega \in C$ belonging to the positive Weyl chamber to serve as the normal vector of $\s W$, i.e. $\s W$ is defined by $\omega \in C$ by
\begin{equation}
\s W = \{ v \in \s E \mid \braket{\omega, v} = 0 \}.
\end{equation}
As $\omega$ belongs to the interior of a Weyl chamber, it is not contained in any of the hyperplanes $\s E_{\alpha}$ defined by \cref{eq:root_hyperplane}~\cite[Prop. 8.14]{hall2015lie}.
The separating hyperplane $\s W$ derives its name because every root $\alpha \in R$ belongs exactly one of the two connected components in $\s E \setminus \s W$.
In fact, given a hyperplane $\s W$ with normal $\omega$, a root $\alpha \in R$ is said to be a \defnsty{positive root} if $\braket{\alpha,\omega} > 0$ or a \defnsty{negative root} if $\braket{\alpha, \omega} < 0$.
The set of positive roots in $R$ is denoted $R_{+}$ and the set of negative roots is denoted $R_{-}$ such that $R$ is the disjoint union $R = R_{+} \sqcup R_{-}$.
While this bipartition of the root system into positive and negative roots evidently depends on the initial choice of positive Weyl chamber, it is independent of the choice of separating hyperplane.

With respect to this decomposition of the root system, the subset $\Delta \subset R_+ \subset R$ of positive roots that cannot be written as a sum of two or more positive roots is called the \defnsty{base} of the root system.
Using this construction, one can additionally show that i) the elements of the base $\Delta$ are linearly independent and ii) every positive (respectively negative) root is expressible as a non-negative (respectively non-positive) integer linear combination of elements in $\Delta$~\cite[Thms. 8.16]{hall2015lie}.
These final two conditions are sometimes taken as an axiomatic definition for a \textit{base of a root system}, but it can be shown that every subset $\Delta \subset R$ satisfying these conditions arises under the construction presented here for some (non-unique) choice of separating hyperplane $\s W$ and normal $\omega$~\cite[Thms. 8.16 \& 8.17]{hall2015lie}.
\end{rem}

The next two definitions are concerned with elements $\mu \in \s E$ of the enveloping vector space that are not necessarily roots.
\begin{defn}
Let $R \subset \s E$ be a root system with inner product $\braket{\cdot, \cdot}$.
For each root, $\alpha \in R$, its \defnsty{coroot} is defined by
\begin{equation}
H_{\alpha} \coloneqq 2 \frac{\alpha}{\braket{\alpha, \alpha}}.
\end{equation}
An element $\mu \in \s E$ is called an \defnsty{algebraically integral element} if for all roots $\alpha \in R$,
\begin{equation}
\braket{\mu, H_{\alpha}} = 2\frac{\braket{\mu, \alpha}}{\braket{\alpha, \alpha}} \in \mathbb Z.
\end{equation}
\end{defn}

\begin{defn}
Let $R \subset \s E$ be a root system with inner product $\braket{\cdot, \cdot}$ and let $\Delta \subset R$ be a base for $R$ (\cref{rem:base_root_sys}).
An element $\mu \in \s E$ is called \defnsty{dominant} if
\begin{equation}
\forall \alpha \in \Delta : \braket{\mu, \alpha} \geq 0,
\end{equation}
and \defnsty{strictly dominant} if,
\begin{equation}
\forall \alpha \in \Delta : \braket{\mu, \alpha} > 0.
\end{equation}
\end{defn}
\begin{rem}
Relative to a given choice of base, $\Delta$, for a root system, $R \subset \s E$, it is not too difficult to see that an element $\mu \in \s E$ is strictly dominant if and only if it belongs to the positive Weyl chamber (\cref{rem:base_root_sys}) and dominant if and only if it belongs to the closure of the positive Weyl chamber.
\end{rem}

\begin{defn}
Let $R \subset \s E$ be a root system with bases $\Delta = \{\alpha_1, \ldots, \alpha_r\} \subset R$.
An element $\mu \in \s E$ is \defnsty{higher} than an element $\nu \in \s E$, expressed as
\begin{equation}
\mu \geq \nu,
\end{equation}
if there exists non-negative integers $\{c_1, \ldots, c_r\}$ such that
\begin{equation}
\mu - \nu = c_1 \alpha_1 + \cdots + c_r \alpha_r.
\end{equation}
The relation, $\geq$, defines a partial ordering on $\s E$.
\end{defn}

\begin{exam}
Let $n \in \mathbb N$ and let $\{e_1, e_2, \ldots, e_{n}, e_{n+1}\}$ be the standard orthonormal basis for $\mathbb R^{n+1}$ and let $\s E \simeq \mathbb R^{n}$ be the subspace of vectors $v$ with components $v_j = \braket{v, e_j}$ summing to zero.
Then the $A_n$ root system is the root system consisting of all vectors of the form $\alpha_{ij} = e_i - e_j$ for some $i \neq j \in [n+1]$.
The Cartan integer is given by
\begin{equation}
c_{\alpha_{ij}}(\alpha_{kl}) = 2 \frac{\braket{\alpha_{ij}, \alpha_{kl}}}{\braket{\alpha_{ij}, \alpha_{ij}}} = \braket{\alpha_{ij}, \alpha_{kl}} = \delta_{ik} + \delta_{jl} - \delta_{il} - \delta_{jk}.
\end{equation}
The base $\Delta$ for the $A_n$ root system is typically taken to be the of roots $\alpha_{ij}$ with $j = i + 1$:
\begin{equation}
\Delta = \{\alpha_{12}, \alpha_{23}, \ldots, \alpha_{n-1,n}, \alpha_{n,n+1} \}.
\end{equation}
With respect to this standard base for $A_n$, the positive roots are those roots $\alpha_{ij}$ with $i > j$ and the negative roots are those with $i < j$.
The Weyl group $W$ for the $A_n$ is given by the faithful representation of the symmetric group $S_{n}$ which acts on the roots, $\alpha_{ij}$, for $\pi \in S_n$ by $\alpha_{ij} \mapsto \alpha_{\pi(i)\pi(j)}$.
This representation of $S_n$ is also known as the \textit{standard representation} of $S_{n}$.
The root system $A_n$ is also the root system associated to the Lie algebra $\mathfrak{sl}(n+1, \mathbb C)$.
\end{exam}

\subsection{Highest weights}
\label{sec:highest_weights}

The finite-dimensional irreducible representations of complex semisimple Lie algebras are determined, up to isomorphism, by their highest weights.
The following theorem is one part of the theorem of highest weights for complex semisimple Lie algebras.
\begin{thm}
\label{thm:hwt_alg}
Let $\mathfrak g$ be a complex semisimple Lie algebra, let $\mathfrak h \subseteq \mathfrak g$ be a fixed maximal abelian subalgebra and let $R$ be the root system of $\mathfrak g$ (relative to $\mathfrak h$), and let $\Delta \subset R$ be a base for $R$ with positive roots $R_+$ and negative roots $R_-$.
If $\arep : \mathfrak g \to \mathfrak{gl}(\s V)$ is an irreducible representation of $\mathfrak g$ on a finite-dimensional complex vector space $\s V$, then exists a weight $\lambda \in \mathfrak h^{*}$ for $\arep$ with weight vector $v_{\lambda} \in \s V_{\lambda} \subset \s V$, meaning
\begin{equation}
\forall H \in \mathfrak h : \arep (H) v_{\lambda} = \lambda(H) v_{\lambda},
\end{equation}
such that $\lambda$ is a \defnsty{highest weight}, meaning for all positive roots $\alpha \in R_+$ and corresponding root vectors $X \in g_{\alpha}$,
\begin{equation}
\label{eq:pos_roots_annhilate}
\arep(X) v_{\lambda} = 0.
\end{equation}
Moreover, the highest weight $\lambda \in \mathfrak h^{*}$ is a dominant, algebraically integral element relative to the root system $R$ and base $\Delta$.
Furthermore, if $\arep' : \mathfrak g \to \mathfrak{gl}(\s V')$ is another finite-dimensional irreducible representation of $\mathfrak g$ with highest weight $\lambda$, then $\arep'$ and $\arep$ are isomorphic representations of $\mathfrak g$.
\end{thm}
\begin{proof}
Let $\lambda \in \mathfrak h^*$ be any weight for $\arep$ with weight vector $v \in \s V$.
If $w = \arep(X) v \neq 0$ was non-zero for some root vector $X \in \mathfrak g_{\alpha}$ with root $\alpha \in R$, it would constitute a weight vector with weight $\alpha + \lambda \in \mathfrak h^{*}$ because for all $H \in \mathfrak h$,
\begin{equation}
\arep(H) \arep(X) v = \arep([H,X]) v + \arep(X) \arep(H) v = \alpha(H) \arep(X) v - \lambda(H) \arep(X) v,
\end{equation}
and thus
\begin{equation}
\arep(H) w = (\alpha+\lambda)(H) w.
\end{equation}
By the finite-dimensionality of $\s V$, can only be finitely many weights $\lambda \in \mathfrak h^{*}$ for the representation $\arep$ and therefore there must exist a weight that is highest in the sense that for all \textit{positive} roots $\alpha \in R_+$, \cref{eq:pos_roots_annhilate} holds.

That $\lambda \in \mathfrak h^*$ is a dominant, algebraically integral element with respect to the root system $R$ with base $\Delta$ follows from first recalling \cref{cor:integral_rep} (which guarantees $\lambda$ is an algebraically integral element) and then second noting that if $\lambda$ is not dominant, e.g. $\lambda(H_{\alpha}) < 0$ for some root $\alpha \in \Delta$ in the base $\Delta$, then $\arep(X) v \neq 0$ is necessarily a weight vector with weight $\lambda + \alpha$, contradicting \cref{eq:pos_roots_annhilate}.

That irreducible representations with the same highest weight are isomorphic follows from noting that if $v_{\lambda} \in \s V$ and $v_{\lambda}' \in \s V'$ are highest weight vectors with the same weight $\lambda \in \mathfrak h'$, then $v_{\lambda} \oplus v_{\lambda}'$ is a highest weight vector with weight $\lambda$ for the direct sum representation $\phi \oplus \phi'$ of $\mathfrak g$ on $\s V \oplus \s V'$.
By restricting $\phi \oplus \phi'$ to the invariant subspace $\s U \subseteq \s V \oplus \s V'$ containing $v_{\lambda} \oplus v_{\lambda}'$ and applying a version of Schur's lemma for Lie algebras, we see that both $\s V$ and $\s V'$ are isomorphic to the restriction of $\phi \oplus \phi'$ onto $\s U$ and thus are isomorphic to each other~\cite[Prop 6.15]{hall2015lie}.
\end{proof}

\begin{rem}
\label{rem:weights_complex_structure}
Although weights, $\lambda : \mathfrak h \to \mathbb C$, assign complex numbers, $\lambda(H)$, to elements $H \in \mathfrak h$, of the complexified Lie algebra $\mathfrak h = \mathfrak t \oplus i \mathfrak t$, they are uniquely determined by their assignment of purely imaginary numbers to $\mathfrak t$, or equivalently, by their assignment of real numbers to $i\mathfrak t$, i.e.
\begin{align}
H \in \mathfrak t &\implies \lambda(H) \in i \mathbb R, \\
H \in i\mathfrak t &\implies \lambda(H) \in \mathbb R.
\end{align}
This observation is analogous to the result of \cref{lem:root_complex_structure} for roots.
In this manner, weights can be identified by their restriction to either $\mathfrak t$ or $i\mathfrak t$.
In the latter case, where $\lambda$ is identified with its restriction to $i\mathfrak t$, denoted by $\lambda \in i \mathfrak t \to \mathbb R$, the weight $\lambda$ is called a \textit{real weight} and written as $\lambda \in (i \mathfrak t)^{*}$.
\end{rem}

It can also be shown that a converse to \cref{thm:hwt_alg} also holds and states that for every dominant algebraically integral element $\lambda \in \mathfrak h^{*}$ is the highest weight of some finite-dimensional irreducible representation~\cite[Thm. 9.5]{hall2015lie}.
\Cref{thm:hwt_alg} and its converse, when taken together, states that the isomorphism classes of finite-dimensional irreducible representations of complex semisimple Lie algebras are in bijection with the set of dominant algebraically integral elements $\lambda \in \mathfrak h^{*}$.

Not only does the theorem of highest weights provide a complete classification of the finite-dimensional irreducible representations for semisimple Lie algebras, it can be extended to provide a complete classification of the finite-dimensional irreducible representations of compact connected Lie groups and their complexifications.
The reason for this correspondence lies with the observation that if $K$ is a compact connected Lie group with Lie algebra $\mathfrak k$, the exponential map $\exp : \mathfrak k \to K$, which assigns to each element $X \in \mathfrak k$ of the Lie algebra an element $\exp(X) \in K$ of the Lie group, is \textit{surjective}~\cite[Cor. 11.10]{hall2015lie}, i.e.
\begin{equation}
\label{eq:surjective}
K = \exp(\mathfrak k) \coloneqq \{\exp(X) \in K \mid X \in \mathfrak k\}.
\end{equation}
If $K$ is not compact\footnote{For a counterexample in the case of non-compact $K$, consider that $\begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix} \in \SL(2, \mathbb C)$ is not the exponential of any matrix $X \in \mathfrak{sl}(2, \mathbb C)$.} or connected, then this is no longer true.

To begin, let $T$ to be an abelian Lie group and $\grep : T \to \GL(\s V)$ a representation of $T$ on a finite-dimensional complex vector space $\s V$.
As $T$ is abelian, $[\grep(t),\grep(t')] = 0$ for all $t,t' \in T$, and therefore Schur's lemma dictates that every irreducible representation of $T$ is one-dimensional whose images are subgroups of the complex multiplicative group $\wozero{\mathbb C}$, e.g. $\grep : T \to \wozero{\mathbb C}$.
If the group $T$ is both abelian and \textit{compact}, then an irreducible representation $\grep$ must map $T$ into the circle group $\U(1) \subset \wozero{\mathbb C}$.
If $\mathfrak t$ is the Lie algebra of $T$, then every irreducible representation $\grep : T \to \U(1)$ of $T$ satisfies for all $A \in \mathfrak t$,
\begin{equation}
\label{eq:toral_irrep}
\grep( \exp(A) ) = e^{\lambda(A)}
\end{equation}
where $\lambda : \mathfrak t \to i \mathbb R$ is a purely imaginary linear function on $\mathfrak t$.
Alternatively, if $H \in i \mathfrak t$, then the irreducible representation $\grep : T \to \U(1)$ must be of the form
\begin{equation}
\grep( \exp(i H) ) = e^{i \lambda(H)}
\end{equation}
where $\lambda : i \mathfrak t \to \mathbb R$ is a real linear function on $i \mathfrak t$.

Of course, if $H \in i\mathfrak t$ is such that $\exp(iH)$ is the identity element in $T$, then $\lambda : i \mathfrak t \to \mathbb R$ must map $H$ to an integer multiple of $2\pi$ in order to ensure that $\grep$ is indeed a representation.

Alternatively, if one considers those elements $H' \in i \mathfrak t$ such that $\exp(2\pi i H')$ is the identity in $T$, then for the same reason $\lambda : i \mathfrak t \to \mathbb R$ must map $H'$ to an integer $\lambda(H') \in \mathbb Z$.
\begin{defn}
Let $T$ be a compact connected abelian Lie group, called a \defnsty{torus}, with Lie algebra $\mathfrak t$ and identity element $e \in T$.
Let $\Gamma \subset i \mathfrak t$ be defined by
\begin{equation}
\Gamma \coloneqq \{ H \in i \mathfrak t : \exp(2 \pi i H) = e\}.
\end{equation}
A linear function $\lambda : i \mathfrak t \to \mathbb R$ is said to be \defnsty{analytically integral} if for all $H \in \Gamma \subset i \mathfrak t$,
\begin{equation}
\lambda(H) \in \mathbb Z.
\end{equation}
\end{defn}
In this manner, every irreducible representation of a torus $T$ can be identified with an analytically integral element $\lambda : i \mathfrak t \to \mathbb R$.
\begin{lem}
Let $\grep : T \to \U(1)$ be an irreducible representation of a torus $T$.
Then there exists an analytically integral linear function $\lambda : i \mathfrak t \to \mathbb R$ satisfying for all $H \in i \mathfrak t$,
\begin{equation}
\grep( \exp(i H) ) = e^{i \lambda(H)}.
\end{equation}
Moreover, $\lambda : i \mathfrak t \to \mathbb R$ is a weight for the Lie algebra representation, $\phi : \mathfrak t \to i\mathbb R$, induced by $\grep : T \to \U(1)$ and $i \lambda(H) = \phi(i H)$.
\end{lem}
\begin{rem}
Every torus $T$ is isomorphic to the standard torus $\U(1)^{r}$ for some $r \in \mathbb N$ called the \textit{rank} of $T$.
The Lie algebra $\mathfrak t$ of a torus $T$ of rank $r$ is isomorphic to $\mathfrak t \cong i\mathbb R^{r}$, and thus $i \mathfrak t \cong \mathbb R^{r}$.
As every element of $\U(1)^{r}$ can be expressed as
\begin{equation}
(e^{i \theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_r})
\end{equation}
for some $(\theta_1, \ldots, \theta_r) \in [0, 2\pi)^{r} \subset \mathbb R^{r} = i \mathfrak t$, every irreducible representation of $\U(1)^{r}$ is of the form
\begin{equation}
\grep(e^{i \theta_1}, e^{i\theta_2}, \ldots, e^{i\theta_r}) = e^{ i (\lambda_1 \theta_1 + m_2 \theta_2 + \cdots + m_r \theta_r)}
\end{equation}
for some $(m_1, \ldots, m_r) \in \mathbb Z^{r}$.
In this way, the analytically integral weight, $\lambda : \mathbb R^{r} \to \mathbb R$, is the function
\begin{equation}
\lambda(\theta_1, \theta_2, \cdots, \theta_r) = m_1 \theta_1 + m_2 \theta_2 + \cdots + m_r \theta_r.
\end{equation}
\end{rem}

\begin{rem}
A torus $T$ is called a \defnsty{maximal torus} of a compact connected Lie group $K$ if it is a subgroup $T \subseteq K$ and if it is not a proper subgroup of any other torus in $K$.
If $T$ is a maximal torus of $K$, then the Lie algebra of $T$, $\mathfrak t$, is a maximal abelian subalgebra of the Lie algebra of $K$, $\mathfrak k$.
\end{rem}

Using \cref{eq:surjective}, it can be shown that the highest weight representation theorem for semisimple Lie algebras, $\mathfrak g$, (\cref{thm:hwt_alg}) extends to a highest weight representation theorem for compact connected Lie groups, $K$~\cite[Thm. 12.6]{hall2015lie}.
\begin{thm}
\label{thm:hwt_grp}
Let $K$ be a connected, compact matrix Lie group and $T$ a fixed maximal torus in $K$. Then
\begin{enumerate}
\item every irreducible representation of $K$ has a highest weight,
\item two irreducible representations of $K$ with the same highest weight are isomorphic,
\item the highest weight of each irreducible representation is dominant and analytically integral, and
\item for every dominant, analytically integral element, $\lambda$, there exists and irreducible representation with highest weight $\lambda$.
\end{enumerate}
\end{thm}
\begin{proof}
See \cite[Sec. 12.5]{hall2015lie}.
\end{proof}
\begin{rem}
The discrepancy between the condition of being an \textit{algebraically integral} weight (appearing in \cref{thm:hwt_alg}) and the stronger condition of being an \textit{analytically integral} weight (appearing in \cref{thm:hwt_grp}) is a consequence of the fact that there can exist representations of semisimple Lie algebras, $\mathfrak g = \mathfrak k_{\mathbb C}$, which are not induced by a representation of compact connected Lie group with Lie algebra $\mathfrak k$.
For instance (\cite[Ex. 12.11]{hall2015lie}), consider $K = \mathrm{SO}(3)$ and $\mathfrak k = \mathfrak{so}(3)$, with basis
\begin{equation}
F_1 = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & -1 \\
0 & 1 & 0
\end{pmatrix},
\qquad
F_2 = \begin{pmatrix}
0 & 0 & 1 \\
0 & 0 & 0 \\
-1 & 0 & 0
\end{pmatrix},
\qquad
F_3 = \begin{pmatrix}
0 & -1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}
\end{equation}
and maximal abelian subalgebra $\mathfrak t$ spanned by the asymmetric matrix $F_3$.
Then the only positive root is fixed by $\alpha(F_3) = i$ (recall that $\alpha$ is purely imaginary on $\mathfrak t$ by \cref{lem:root_complex_structure}) with root vector $X = F_1 - i F_2 \in \mathfrak{so}(3)_{\mathbb C}$ because
\begin{equation}
[F_3, X] = [F_3, F_1] - i [F_3, F_2] = F_2 + i F_1 = i X = \alpha(F_3) X.
\end{equation}
Since the associated coroot is
\begin{equation}
H_{\alpha} = \frac{2[X,X*]}{\alpha([X,X^*])} = 2 i F_3 \in i \mathfrak t,
\end{equation}
a weight $\lambda : i \mathfrak t \to \mathbb R$ is algebraically integral ($\lambda(H_{\alpha}) = 2 \lambda(i F_3) \in \mathbb Z$) if and only if $\lambda(i F_3) \in \mathbb R$ is an integer or half integer:
\begin{equation}
2 \lambda(i F_3) \in \mathbb Z.
\end{equation}
On the other hand, as $\exp(2 \pi i \theta F_3) = \ident_3$ is the identity in $\mathrm{SO}(3)$ if and only if $\theta \in \mathbb Z$, a weight $\lambda \in i \mathfrak t \to \mathbb R$ is analytically integral if and only if $\lambda(i F_3)$ is an integer:
\begin{equation}
\lambda(i F_3) \in \mathbb Z.
\end{equation}
\end{rem}

To conclude this section, we introduce a definition of a \textit{complex reductive group} which will be used throughout the thesis.
Although the exact definition of a reductive group can be quite varied depending on the mathematical setting~\cite{lee2001structure,milne2014algebraic,knapp2001representation}, our definition is well-suited for out purposes.
Just as we considered the complexification of a Lie algebra in \cref{sec:complexification}, there is a related notion of complexification for a compact Lie groups which relies on a universal construction.
The \textit{complexification} of a compact Lie group $K$, denoted $G = K_{\mathbb C}$, is a complex Lie group equipped with an inclusion map $K \xhookrightarrow{} G$ such that every smooth Lie group homomorphism from $K$ to a complex Lie group $Q$ lifts uniquely to a holomorphic homomorphism $G \to Q$.
The key idea here is simply that representations of $K$, $\grep : K \to \GL(\s V)$, on complex vector spaces $\s V$, gives rise to a representation $\grep : G \to \GL(\s V)$ of the larger group $G$.
\begin{defn}
\label{defn:complex_reductive_group}
A \defnsty{complex reductive group} is a group $G$ that is the complexification, $G = K_{\mathbb C}$, of a compact connected Lie group $K$.
If $\mathfrak k$ is the Lie algebra of $K$, then the Lie algebra of $G$ is the complexification $\mathfrak g = \mathfrak k_{\mathbb C} = \mathfrak k \oplus i \mathfrak k$.
\end{defn}

Since $K$ is a compact and connected Lie group we have $K = \exp(\mathfrak k)$. Similarly we can let $\mathfrak p = i \mathfrak k$ and define a subset $P \coloneqq \exp(i\mathfrak)$ of $G$.
Then the notion of a local Cartan involution, $\theta : \mathfrak g \to \mathfrak g$, on a complex reductive Lie algebra, $\mathfrak g$ can be generalized to an involutive group isomorphism, $\Theta : G \to G$, on complex reductive group $G$ called the \defnsty{global Cartan involution}.
This isomorphism $\Theta$ fixes $K \subset G$ and take inverses on $P \subset G$.
As a quick summary,
\begin{align}
\begin{split}
A \in \mathfrak k &\implies \theta(A) = + A, \\
H \in \mathfrak p = i \mathfrak k &\implies \theta(H) = - H, \\
\end{split}
\begin{split}
k \in K = \exp(\mathfrak k) &\implies \Theta(k) = k, \\
p \in P = \exp(i \mathfrak k) &\implies \Theta(p) = p^{-1}. \\
\end{split}
\end{align}
In general, for any element $g \in G$ of a complex reductive group, we define $g^{*} \in G$ by
\begin{equation}
g^{*} \coloneqq \Theta(g)^{-1}.
\end{equation}
This notation is consistent with the previously defined $*$-involution operation on $X \in \mathfrak g$ by $X^{*} = -\theta(X)$ in the sense that $\exp(X^{*}) = \exp(X)^{*}$.
Furthermore, the universal property of the complexification of $K$ means that any finite-dimensional unitary representation $\grep : K \to \U(\s H)$ of $K$ extends uniquely to a representation (also denoted by $\grep$) $\grep : G \to \GL(\s H)$ such that $\grep(g^{*}) = \grep(g)^{*}$ where the latter $*$-operation is simply the complex conjugate relative to the inner product on $\s H$.

\begin{exam}
Let $d \in \mathbb N$ be a fixed finite dimension.
For example if $K = \SU(d)$ is the group of unitary matrices with determinant one, then the complexification of $K = \SU(d)$ is $G = \SL(d, \mathbb C)$, the group of complex matrices with determinant one.
The Lie algebra of $K = \SU(d)$ is $\mathfrak k = \mathfrak{su}(d)$ the algebra of skew-Hermitian traceless matrices, while $i \mathfrak k \simeq i\mathfrak{su}(d)$ is the algebra of Hermitian traceless matrices and $P = \exp(i\mathfrak{su}(d))$ the set of positive definite matrices and $\mathfrak g = \mathfrak{sl}(d, \mathbb C)$ the Lie algebra of traceless matrices.
In this example, the $*$-involution on either $X \in \mathfrak{sl}(d, \mathbb C)$ or $g \in \SL(d, \mathbb C)$ is complex conjugation of matrices.
\end{exam}

\begin{rem}
\label{rem:polar_decomp}
In general, the group multiplication in $G$ gives rise to a diffeomorphism (but not group isomorphism) $G \simeq K \times P$ called the \defnsty{Cartan decomposition}.
Specifically, as $P \coloneqq \exp(\mathfrak p) = \exp(i\mathfrak k)$, there exists an invertible map $K \times \mathfrak p \to G$ given by
\begin{equation}
(k, X) \mapsto k \cdot \exp(X).
\end{equation}
In particular, we have the following relationships between $k, X$ and $g$:
\begin{align}
\begin{split}
g &= k \cdot \exp(X), \\
g^{-1} &= \exp(-X) \cdot k^{-1}, \\
\Theta(g) &= k \cdot \exp(-X), \\
g^{*} &= \exp(X)\cdot k^{-1}, \\
\end{split}
\end{align}
Therefore, $X$ and $k$ are uniquely determined by $g$ since
\begin{equation}
g^{*} \cdot g = \exp(2X) \implies \exp(X) = \sqrt{g^{*} g},
\end{equation}
and thus
\begin{equation}
k = \Theta(g) \sqrt{g^{*} g}.
\end{equation}
When $K = \U(1)$ and thus $G = \wozero{\mathbb C}$, this diffeomorphism states that every non-zero complex number $z \in \wozero{\mathbb C}$ can be written in polar form, $z = e^{i\theta}\abs{z}$ where $e^{i\theta} \in \U(1)$ and $\abs{z} > 1$ is a positive real number.
More generally, if $K = \U(d)$ and thus $G = \GL(d, \mathbb C)$, this diffeomorphism states that every $d \times d$ invertible complex matrix $M \in \GL(d, \mathbb C)$ can be written uniquely in the form $M = U P$ where $U$ is a unitary matrix and $P$ is a positive definite matrix.
\end{rem}

A more detailed decomposition of elements in a complex reductive group is offered by the \textit{Iwasawa decomposition} of $G$.
Consider a generic complex reductive group, $G = K_{\mathbb C}$, which is the complexification of a connected compact Lie group $K$.
A \defnsty{Borel subgroup} $B \subseteq G$ is a maximal solvable subgroup (which are all conjugate to each other by some elements of $K$).
Once a particular Borel subgroup $B \subseteq G$ is chosen, its intersection with $K$,
\begin{equation}
T = B \cap K,
\end{equation}
constitutes a maximal torus in $K$.
Moreover, the commutator subgroup of $B$,
\begin{equation}
N = [B, B],
\end{equation}
is a maximal unipotent subgroup of $B$ and moreover $T$ normalizes $N$ inside $B$.
The Lie algebras of $B, N$ and $T$ are denoted by $\mathfrak b, \mathfrak n$ and $\mathfrak t$.
Moreover, let $\mathfrak a \coloneqq i\mathfrak t$ and define $A$ as the image under the exponential map of $\mathfrak a$ in $G$.
%In this way, $\mathfrak a \subseteq \mathfrak p$ is a maximal abelian subalgebra of $\mathfrak p = i \mathfrak k$ in the same way that $\mathfrak t$ is a maximal abelian subalgebra of $\mathfrak k$.

\begin{prop}
\label{prop:iwasawa}
Let $G = K_{\mathbb C}$ be a complex reductive group with fixed Borel subgroup $B \subseteq K$.
Then every group element $g \in G$ can be \textit{uniquely} expressed as
\begin{equation}
\label{eq:iwasawa}
g = k_g \cdot a_g \cdot n_g,
\end{equation}
for unique $k_g \in K$, $a_g \in A$ and $n_g \in N$.
This decomposition of $g \in G$ is known as the \defnsty{Iwasawa decomposition}~\cite{iwasawa1949some}.
\end{prop}

\begin{exam}
For the special case of $K = \U(d)$ and $G = \GL(d, \mathbb C)$, one can choose the Borel subgroup $B$ to be the subgroup of upper triangular matrices and then $T$ the subgroup of diagonal matrices.
The maximal unipotent subgroup $N$ of $B$ is then the upper triangular matrices with ones along the main diagonal and the Iwasawa decomposition is essentially the QR decomposition of an invertible matrix where the diagonal part of the upper triangular matrix is factored out.
When $d=3$ the Iwasawa decomposition can be expressed as
\begin{equation}
\underbrace{\begin{pmatrix}
g_{11} & g_{12} & g_{13} \\
g_{21} & g_{22} & g_{23} \\
g_{31} & g_{32} & g_{33} \\
\end{pmatrix}}_{g}
\underbrace{\begin{pmatrix}
k_{11} & k_{12} & k_{13} \\
k_{21} & k_{22} & k_{23} \\
k_{31} & k_{32} & k_{33} \\
\end{pmatrix}}_{k}
\underbrace{\begin{pmatrix}
a_1 & 0 & 0 \\
0 & a_2 & 0 \\
0 & 0 & a_3 \\
\end{pmatrix}}_{a}
\underbrace{\begin{pmatrix}
1 & n_{12} & n_{13} \\
0 & 1 & n_{23} \\
0 & 0 & 1 \\
\end{pmatrix}}_{n},
\end{equation}
where $g \in \GL(3, \mathbb C)$ is any invertible $3\times 3$ complex matrix, $k \in \U(3)$ is a unitary $3 \times 3$ matrix, the diagonal of $a$ is strictly positive $(a_1, a_2, a_3) \in \mathbb R_{> 0}^{3}$, and $(n_{12}, n_{13}, n_{23}) \in \mathbb C^{3}$.
\end{exam}

\clearpage
\section{Glossary of Notation}
\label{sec:notation}
Throughout this thesis, all logarithms are taken to be base $e$.

\begin{table}[htb]
\caption{Notation relating to Hilbert spaces.}
\begin{tabularx}{\textwidth}{p{0.12\textwidth}X}
\hline
\hline
$\s H$ & a complex inner product space \\
$\langle\cdot,\cdot\rangle$ & the (sesquilinear) inner product associated to $\s H$ \\
$\norm{\cdot}$ & the norm induced by the $\langle\cdot,\cdot\rangle$ \\
$\wozero{\s H}$ & the non-zero elements of $\s H$ \\
$\s H^{*}$ & the dual vector space of linear functions on $\s H$ \\
$\End(\s H)$ & all (linear) operators acting on $\s H$ \\
$\s B(\s H)$ & bounded operators on $\s H$ \\
$\s B_{\geq 0}(\s H)$ & positive semi-definite operators on $\s H$ \\
$\GL(\s H)$ & invertible linear operators on $\s H$  \\
$\U(\s H)$ & unitary operators on $\s H$  \\
%$\mathfrak{u}(\s H)$ & skew-Hermitian operators on $\s H$  \\
%$i\mathfrak{u}(\s H)$ & Hermitian operators on $\s H$  \\
$\mathbb P \s H$ & the projective space associated to $\s H$ \\
$\psi$ & a generic ray in the projective space $\mathbb P \s H$ \\
$P_{\psi}$ & the rank-one projection operator onto the ray $\psi$ \\
$\rho$ & a generic density operator \\
$\varphi$ & the functional form of a state on $\s H$ ($\varphi : \bound(\s H) \to \mathbb C$) \\
$\s S(\s H)$ & the set of all density operators on $\s H$ \\
\hline
\hline
\end{tabularx}
\end{table}

\begin{table}[htb]
\caption{Notation relating to probability measures.}
\begin{tabularx}{\textwidth}{p{0.12\textwidth}X}
\hline
\hline
$X$ & a topological space (typically a Polish space) \\
$\borel{X}$ & the Borel $\sigma$-algebra on $X$ \\
$\Delta$ & a generic element in the Borel sigma algebra $\borel{X}$ \\
$(X,\borel{X})$ & a standard Borel space for $X$ \\
$\mu$ & a probability measure $\mu : \borel{X} \to [0,1]$ on $X$ \\
$\probs(X)$ & the set of all probability measures on $X$ \\
$f_{*}\mu$ & the push-forward of a measure $\mu$ through a measurable function $f$ \\
$I$ & a rate function for $X$ ($I : X \to [0,\infty]$) \\
\hline
\hline
\end{tabularx}
\end{table}

\begin{table}[htb]
\caption{Notation relating to Lie groups, Lie algebras and representations.}
\begin{tabularx}{\textwidth}{p{0.12\textwidth}X}
\hline
\hline
$K$ & a compact, sometimes connected, Lie group  \\
$k$ & a generic group element $k \in K$ \\
$\mathfrak k$ & the Lie algebra of the Lie group $K$ \\
$G$ & the complexification of $K$ ($G = K_{\mathbb C}$) \\
$g$ & a generic group element $g \in G$ \\
$\mathfrak g$ & the Lie algebra of the complexification of $K$ ($\mathfrak g = \mathfrak k \oplus i \mathfrak k$) \\
$[\cdot,\cdot]$ & the Lie bracket of a Lie algebra $\mathfrak g$ \\
$\Theta$ & the global Cartan involution on $G$ ($\Theta : G \to G$) \\
$\theta$ & the local Cartan involution on $\mathfrak g$ ($\theta : \mathfrak g \to \mathfrak g$) \\
$X^{*}$ & the star involution on $\mathfrak g$ ($X^{*} = -\theta(X)$) \\
$g^{*}$ & the star involution on $G$ ($g^{*} = \Theta(g)^{-1}$) \\
\hline
$\grep$ & a group representation of $G$ on $\s H$ ($\grep : G \to \GL(\s H)$) \\
$\arep$ & a Lie algebra representation of $\mathfrak g$ on $\s H$ ($\arep : \mathfrak k \to \mathfrak{gl}(\s H)$) \\
$\hwsub{\lambda}{\grep}$ & subspace of highest weight vectors with weight $\lambda$ for $\grep$ \\
$\hwsub{\lambda,k}{\grep}$ & the projection operator $\hwsub{\lambda}{\grep}$ conjugated by $k$ \\
$\isosub{\lambda}{\grep}$ & the isotypic subspace of highest weight $\lambda$ for $\grep$ \\
$\capacity_{\grep}$ & the capacity map $\capacity_{\grep} : \s H \to [0,\infty]$ for $\grep$  \\
$\momap_{\grep}$ & the moment map $\momap_{\grep} : \proj \s H \to i \mathfrak k^{*}$ for $\grep$  \\
\hline
\hline
\end{tabularx}
\end{table}

\begin{table}[htb]
\caption{Commonly used symbols for the complexification of Lie algebras.}
\begin{tabularx}{\textwidth}{p{0.12\textwidth}p{0.12\textwidth}p{0.2\textwidth}X}
\hline
\hline
Group & Algebra & Relationship(s) & Comment \\
\hline
$K$ & $\mathfrak k$ &  & a compact Lie group \\
$G$ & $\mathfrak g$ & $\mathfrak g \coloneqq \mathfrak k \oplus i \mathfrak k$ & the complexification of $K$ \\
$T$ & $\mathfrak t$ & $\mathfrak t \subseteq \mathfrak k \subseteq \mathfrak g$ & the maximal torus of $K$ \\
$H$ & $\mathfrak h$ & $\mathfrak h \coloneqq \mathfrak t \oplus i \mathfrak t$ & $\mathfrak h$ a Cartan subalgebra of $\mathfrak g$ \\
$P$ & $\mathfrak p$ & $\mathfrak p \coloneqq i\mathfrak k$ & \\
$B$ & $\mathfrak b$ & $T = B \cap K$ & a maximal solvable (Borel) subgroup of $G$ \\
$N$ & $\mathfrak n$ & $N = [B, B] \subseteq B$ & the maximal unipotent subgroup \\
$A$ & $\mathfrak a$ & $\mathfrak a \coloneqq i \mathfrak t$ & \\
\hline
\hline
\end{tabularx}
\end{table}

\chapter{Non-commutative optimization}
\label{chap:invariant_theory}
\section{Introduction}
\label{sec:bounding_asymptotic_likelihoods}

In \cref{chap:estimation_theory}, we will be interested in subject of quantum tomography which aims to estimate the properties of an unknown quantum system empirically by performing measurements on a large number of identical copies that system.
Consequently, it will be useful to develop mathematical techniques to study probabilities of the form,
\begin{equation}
p_n(x \mid \rho) = \Tr(\rho^{\otimes n} E^{x}_{n}),
\end{equation}
where $n$ is a large positive integer, $\rho$ is a density operator acting on a Hilbert space $\s H$ describing an unknown quantum state, and $E^{x}_{n}$ a positive semidefinite operator acting on $\s H^{\otimes n}$ describing the event of obtaining the estimate $x$ upon performing a collective measurement on $n$ identical copies of $\rho$. The purpose of this chapter, therefore, is to develop these mathematical methods using tools from invariant theory to study the large $n$ limit of probabilities of the above form.

Before doing so, here we aim to provide a sketch for how the representation theory of non-compact groups will be used in this context.
Consider a non-compact group $G$ together with a non-unitary representation, $\grep : G \to \GL(\s H)$, of $G$ on the Hilbert space $\s H$.
Now let a group element, $g \in G$, act on the density operator, $\rho$, by conjugation, sending $\rho$ to the operator $g \cdot \rho$, defined by
\begin{equation}
g \cdot \rho \coloneqq \grep(g) \rho \grep(g)^{*},
\end{equation}
where $\grep(g)^{*}$ is the complex conjugate of the operator $\grep(g)$.
Similarly, let $g \in G$ act on the measurement effect $E_{n}^{x}$ via
\begin{equation}
g \cdot E_n^{x} \coloneqq \grep^{\otimes n}(g)^{*} E_n^{x} \grep^{\otimes n}(g).
\end{equation}
If, in addition, there happens to exists a function, $\chi_x : G \to (0, \infty)$, independent of $n$, such that
\begin{equation}
g \cdot E_n^{x} \leq (\chi_x(g^{-1}))^{n} \ident^{\otimes n},
\end{equation}
then a quick calculation reveals an upper bound on $p_n(x \mid \rho)$ which holds for all $g \in G$:
\begin{align}
p_n(x \mid \rho)
= \Tr(\rho^{\otimes n} E^{x}_{n})
= \Tr( (g\cdot \rho)^{\otimes n} (g^{-1} \cdot E_n^{x}) )
\leq [\chi_x(g)\Tr(g\cdot \rho)]^{n}.
\end{align}
Therefore, the likelihood of producing the estimate $x$ decays at an exponential rate with respect to increasing $n$,
\begin{equation}
p_n(x \mid \rho) \leq \exp( - n I_\rho(x)),
\end{equation}
where the rate $I_{\rho}(x) \in [0, \infty]$ is obtained by optimizing over all $g \in G$:
\begin{equation}
I_{\rho}(x) = - \ln \inf_{g \in G} \chi_x(g)\Tr(g\cdot \rho).
\end{equation}
From the perspective of property tomography, therefore, it becomes desirable to find a sequence of measurements such that a quantum state, $\rho$, has vanishing rate, $I_{\rho}(x) = 0$, if and only if, the state $\rho$ has property $x$.
Finding examples of such measurements schemes will be the subject of \cref{chap:estimation_theory}.

In this chapter, we will investigate a general class of optimization problems over a non-compact, non-commutative group $G$, similar to the one defined above.
Our focus here will be on the general theory of non-commutative optimization from the perspective of geometric invariant theory; for instance the Kempf-Ness theorem provides a correspondence between extremal values and the vanishing of a generalized gradient, called the moment map~\cref{thm:kempf_ness_theorem}.
Algorithmic implementations and complexity theoretic aspects of non-commutative optimization theory can be found elsewhere~\cite{burgisser2019towards}.
The final result of this chapter, covered in \cref{sec:strong_duality}, is the \textit{strong duality} result of Ref.~\cite{franks2020minimal}, which serves as the foundation for many of the results in subsequent chapters.

\section{Invariants \& norm minimization}

\subsection{Group orbits \& stability}
\label{sec:orbits_stability}

The purpose of this section is to introduce the concept of group orbits associated to group representations and to study their topological closures.
In particular, various notions of stability, borrowed from the subject of geometric invariant theory are directly related to whether or not the orbit of a vector under the action of a group is closed.

\begin{defn}
\label{defn:group_orbits}
Let $\s H$ be a complex finite-dimensional Hilbert space and let $\grep : G \to \GL(\s H)$ be a representation of a reductive group $G$ such that the inner product $\braket{\cdot, \cdot}$ on $\s H$ is invariant under the action of the maximal compact subgroup $K \subseteq G$.
Let $v \in \s H$ be a vector in $\s H$.
The \defnsty{orbit of $v$}, $G \cdot v$, is the set
\begin{equation}
G \cdot v = \grep(G)v \coloneqq \{ \grep(g)v \in \s H \mid g \in G\}.
\end{equation}
The \defnsty{closure} of the orbit $G \cdot v$, denoted by $\overline{G \cdot v}$ or $\overline{\grep(G)v}$, is the topological closure of $G \cdot v$ with respect to the norm-induced topology on $\s H$.
\end{defn}
\begin{exam}
As representations are linear group actions, the orbit of the origin $0 \in \s H$ is the singleton set $\grep(G)0 \coloneqq \{ 0 \}$.
This orbit will be called the \defnsty{trivial orbit}, while all other orbits are considered \defnsty{non-trivial orbits}.
\end{exam}

A particularly useful tool for verifying that two points belong to distinct orbits are invariant polynomials.
We begin by recalling that the \defnsty{ring of complex polynomials}, denoted $\mathbb C[\s V]$, over a finite-dimensional complex vector space $\s V$, can be defined either in a coordinate-full or coordinate-free manner.
The coordinate-free definition for $\mathbb C[\s V]$ proceeds by considering the commutative ring of functions from $\s V$ to $\mathbb C$ generated by elements of the dual space $\s V^{\ast}$ of $\mathbb C$-linear functions on $\s V$.
From this perspective, $\mathbb C[\s V]$ is also sometimes called the symmetric algebra on $\s V$.
Alternatively, a coordinate-full definition proceeds by identifying some basis $\{e_1, \ldots, e_d\}$ for $\s V$ with the indeterminate variables $\{x_1, \ldots, x_d\}$ such that the polynomials in $\mathbb C[\s V]$ can be identified with the set of complex polynomials in $d$ variables, denoted $\mathbb C[x_1, \ldots, x_d]$.
In either case, any representation of a group $G$ on $\s V$ induces an action of $G$ on $\mathbb C[\s V]$ and subsequently gives rise to the notion of an invariant polynomial.

\begin{defn}
\label{defn:invariant_polynomials}
Let $\s V$ be a finite-dimensional complex vector space and $\mathbb C[\s V]$ the complex polynomial ring on $\s V$.
Given a representation $\grep : G \to \GL(\s V)$ of a group $G$ on $\s V$, there is a natural action of the group $G$ on $\mathbb C[\s V]$, denoted here by $\grep_*$, and defined for $p \in \mathbb C[\s V]$ and $g \in G$ by
\begin{equation}
(\grep_{*}(g)(p))(v) \coloneqq p(\grep(g^{-1}) v).
\end{equation}
A polynomial $p \in \mathbb C[\s V]$ is said to be a \defnsty{invariant} (or sometimes $G$-invariant) polynomial if for all $g \in G$, $\grep_{*}(g)(p) = p$.
The subset of all invariant polynomials forms a subring of $\mathbb C[\s V]$, called the \defnsty{ring of invariant polynomials} and is denoted by $\mathbb C[\s V]^\grep$.
\end{defn}

\begin{rem}
Recall that in this setting, every group $G$ acts \textit{linearly} on the vector space $\s V$ through the representation $\grep : G \to \GL(\s H)$.
Consequently, the degree of a homogeneous polynomial $p \in \mathbb C[\s V]$ is always preserved by the action of $G$ meaning,
\begin{equation}
\forall g \in G : \deg(\grep_{*}(g) p ) = \deg(p).
\end{equation}
Therefore, the ring of $G$-invariant polynomials, $\mathbb C[\s V]^\grep$, naturally forms a \textit{graded ring} in the sense that
\begin{equation}
\mathbb C[\s V]^\grep = \bigoplus_{n=0}^{\infty}\mathbb C[\s V]^\grep_n
\end{equation}
where each summand, $\mathbb C[\s V]^\grep_n$, is the vector space of homogeneous degree-$n$ $G$-invariant polynomials, and furthermore for every $n,m \in \mathbb N$, the product of a degree-$n$ and degree-$m$ $G$-invariant polynomial is a degree-$(n+m)$ $G$-invariant polynomial:
\begin{equation}
\mathbb C[\s V]^\grep_n \otimes \mathbb C[\s V]^\grep_m \subseteq \mathbb C[\s V]^\grep_{n+m}.
\end{equation}
\end{rem}

\begin{rem}
\label{rem:proto_link_algebra_geo}
The definition of a $G$-invariant polynomial $p \in \mathbb C[\s V]^\grep$ immediately implies that $p$ takes constant values when evaluated on orbits because $p(\grep(g)v) = p(v)$ for all $g \in G$.
Moreover, since polynomials are continuous functions, they are also constant when evaluated on the closures of orbits, i.e., for all $w \in \overline{\grep(g)v}$, we have $p(w) = p(v)$ also.
The contrapositive version of this observation is quite useful; if $v,w \in \s V$ are vectors such that $p(v) \neq p(w)$ for some $G$-invariant polynomial $p$, one concludes that $v$ and $w$ must belong to distinct orbit closures, i.e., the intersection of their orbit closures is empty: $\overline{\grep(g)v} \cap \overline{\grep(g)w} = \emptyset$.
\end{rem}

\begin{exam}
Perhaps the simplest non-trivial example of an invariant polynomial arises under the action of the finite multiplicative group $\mathbb Z_{2} = \{-1, +1\}$ on the one-dimensional complex vector space $\s V \cong \mathbb C$ where $-1 \in \mathbb Z_{2}$ acts on $x \in \mathbb C$ by reflection $x \mapsto - x$.
In this example, the polynomial ring $\mathbb C[\s V] \simeq \mathbb C[x]$ is univariate and consists of polynomials of the form $c_0 + c_1 x + c_2 x^2 + \cdots$.
Under the action of $\mathbb Z_{2}$ by $x \mapsto - x$, the invariant polynomials $\mathbb C[x]^{\mathbb Z^{2}}$ are precisely those containing only even-degree terms, i.e. $c_0 + c_2 x^2 + \cdots$.
\end{exam}

\begin{exam}
\label{exam:prototype_orbits}
For another example involving a continuous group, consider the two-dimensional vector space $\s V = \mathbb C^2$ and them multiplicative group of non-zero complex numbers $G = \wozero{\mathbb C}$ acting on $(v_1, v_2) \in \mathbb C^{2}$ by sending $z \in \wozero{\mathbb C}$ to the linear transformation $\grep(z) \in \End(\mathbb C^{2})$ defined by
\begin{equation}
(v_1, v_2) \mapsto (z v_1, z^{-1} v_2).
\end{equation}
The orbits under the above action fall into one of three qualitatively distinct categories (depicted in \cref{fig:orbit_scaling_example}) which can be determined by considering a representative element $(v_1, v_2)$ of the orbit:
\begin{enumerate}[i)]
\item If $(v_1, v_2) = (0,0)$, then the orbit is simply the trivial orbit $\grep(\wozero{\mathbb C}) (0, 0) = \{ (0,0) \}$.
\item If $v_1 = a \neq 0$ but $v_2 = 0$, then the orbit consists of the $v_2 = 0$ axis in $\mathbb C^{2}$ where the origin has been removed $\grep(\wozero{\mathbb C}) (a, 0) = \{ (z,0) \mid z \in \wozero{\mathbb C} \} \simeq \wozero{\mathbb C}$ (or analogously if $v_1 = 0$ and $v_2 = b \neq 0$).
\item Or if $(v_1, v_2) = (a,b)$ are both non-zero, then the orbit consists of all points such that $v_1 v_2 = ab$, i.e. $\grep(\wozero{\mathbb C}) (a, b) = \{ (v_1, v_2) \mid v_1 v_2 = ab \}$.
\end{enumerate}
Moreover, the ring of invariant polynomials, $\mathbb C[v_1, v_2]^{\wozero{\mathbb C}}$, is generated from a single invariant polynomial of homogeneous degree two, namely $v_1 v_2$.
Indeed the third type of orbit above is completely characterized as the solution set to the polynomial equation $v_1 v_2 = c$ where $c \in \mathbb C$ is a non-zero complex constant.
By comparison, it can be shown that the orbits of type i) or ii) cannot be expressed as the solution set of invariant polynomials in $\mathbb C[v_1, v_2]^{\wozero{\mathbb C}}$.
\end{exam}
\begin{figure}
\centering
\includegraphics{figures_orbit_scaling_example.pdf}
\caption{Three distinct types of group orbits under the group action sending the pair of complex numbers $(v_1, v_2)$ to $z \cdot (v_1, v_2) = (z v_1, z^{-1} v_2)$, visualized using the moduli of the complex coordinates, $(|v_1|, |v_2|)$.}
\label{fig:orbit_scaling_example}
\end{figure}

In the previous example (\cref{exam:prototype_orbits}), it was shown that the orbits of a group acting on a vector space cannot always be characterized as the solutions to a family of invariant polynomials.
In general, any non-zero group orbit whose topological closure contains the origin of the vector space cannot be separated from the zero orbit using invariant polynomials invariant (or more generally invariant continuous functions).
Therefore, distinguishing between different group orbits is highly sensitive to whether or not the topological closure of the group orbit contains the origin.
This observation gives rise to the notion of stability that will be used subsequently.
\begin{defn}
\label{defn:stability}
Here $\s V$ will be a complex finite-dimensional Hilbert space, and $\grep : G \to \GL(\s V)$ a representation of a reductive group $G$ acting on $\s V$. Let $v \in \s V$ be a fixed vector.
Then $v$ is said to be
\begin{enumerate}[i)]
\item \defnsty{unstable} if $0 \in \overline{\grep(G)v}$,
\item \defnsty{semistable} if $0 \not \in \overline{\grep(G)v}$,
\item \defnsty{polystable} if $v$ is semistable and $\grep(G)v$ is closed, i.e. $\grep(G)v = \overline{\grep(G)v}$,
\item and \defnsty{stable} if $v$ is polystable and additionally the stabilizer of $v$, $G_{v}$, has finite order.
\end{enumerate}
\end{defn}
\begin{rem}
Linguistically, one might expect the notion of a \textit{stable} vector to be any vector that is \textit{not} unstable.
However, this is not the case; the logical opposite of an unstable vector is the notion of a \textit{semistable} vector.
\end{rem}
\begin{exam}
Returning to \cref{exam:prototype_orbits} and consulting \cref{fig:orbit_scaling_example}, one can see that with respect to the operation $(v_1, v_2) \mapsto (z v_1, z^{-1} v_2)$ for $z \in \wozero{\mathbb C}$, vectors $(v_1,v_2)$ lying on either the $v_1$ or $v_2$ axes are unstable because their orbit closures contain the origin.
By comparison, if $(v_1,v_2)$ does not lie on either $v_1$ or $v_2$ axes, then $(v_1, v_2)$ is in fact semistable because their orbit closure excludes the origin.
Moreover, in this example, the semistable points are additionally polystable (because their orbits are topologically closed) and stable (because their stabilizer is the trivial group and has order one which is finite).
\end{exam}
These examples bring us to a fundamental result of geometric invariant theory which can be seen as a strengthening of the link between the intersections of orbit closures and invariant polynomials previously mentioned in \cref{rem:proto_link_algebra_geo}.
%\textcolor{red}{TODO:find a primary reference for this orbit-separation result (something more direct than \cite{acuaviva2022minimal}). Something newer than Hilbert's 1893 result.}
\begin{prop}
\label{prop:mumford}
Let $G$ be a complex reductive group, let $\grep : G \to \GL(\s V)$ be a rational representation of $G$ on a complex finite-dimensional vector space $\s V$, and fix $v,w \in \s V$.
Then $\overline{\grep(G)v}\cap\overline{\grep(G)w} \neq \emptyset$ if and only if $p(v) = p(w)$ for all invariant polynomials $p \in \mathbb C[\s V]^{\grep}$.
\end{prop}
If one of the two vectors in \cref{prop:mumford} is taken to be the zero vector, one obtains the following corollary which relates the semistability of a vector (\cref{defn:stability}) to the values it takes on homogeneous invariant polynomials.
\begin{cor}
Let $\grep : G \to \GL(\s V)$ be as in \cref{prop:mumford} and fix $v \in \s V$.
Then $0 \in \overline{\grep(G)v}$ if and only if $p(v) = p(0)$ for all invariant polynomials $p \in \mathbb C[\s V]$.
Equivalently, $v$ is unstable if and only if $p(v) = 0$ for all homogeneous invariant polynomials $p \in \bigoplus_{n=1}^{\infty} \mathbb C[\s V]^{\grep}_n$.
\end{cor}

\begin{defn}
\label{defn:null_cone}
The \defnsty{null cone}, denoted by $\s N$, is the set of all vectors $v \in \s V$ which cannot be separated from the origin by invariant polynomials, i.e.
\begin{equation}
\s N \coloneqq \{v \in \s H \mid 0 \in \overline{\grep(G) v} \}.
\end{equation}
\end{defn}

\subsection{Invariant and fixed subspaces}
\label{sec:invariant_and_fixed_subspaces}

Recall, from \cref{defn:invariant_subspace}, that a subspace $\s W \subseteq \s V$ is invariant under the action of a group representation $\grep : G \to \GL(\s V)$ if for all group elements $g \in G$, the linear operation, $\grep(g)$, maps every vector $w \in \s W$ inside $\s W$ to another vector $\grep(g) w \in \s W$ inside $\s W$.
There is a stronger notion of invariance for subspaces that requires each \textit{vector} $w \in \s W$ itself to be invariant.
This stronger notion of invariance will be important in \cref{sec:occasionality}.
Note that in some references, e.g. \cite{franks2020minimal}, this subspace of invariant vectors is referred to as \textit{the} invariant subspace.
In this thesis, we instead prefer to use the terminology of the \textit{fixed subspace}.
\begin{defn}
Let $\grep : G \to \GL(\s H)$ be a representation of a group $G$ on a complex finite-dimensional Hilbert space $\s H$.
The \defnsty{fixed subspace} is the subspace $\s H^\grep \subseteq \s H$ consisting of all vectors on which $\grep$ acts trivially:
\begin{equation}
\s H^\grep \coloneqq \{ v \in \s H \mid \forall g \in G, \grep(g)v = v \}.
\end{equation}
\end{defn}
The fixed subspace $\s H^\grep$ can also be identified with the multiplicity space for the trivial subrepresentation of $\grep$, meaning
\begin{equation}
\s H^\grep \simeq \Hom_{G}(1_{G},\grep)
\end{equation}
where the symbol, $1_{G}$, here denotes the trivial representation of $G$ sending $g \in G$ to the multiplicative unit $1_{G}(g) = 1 \in \mathbb C$.
\begin{defn}
\label{defn:fixed_subspace_n}
Let $\grep : G \to \GL(\s H)$ be a representation of a group $G$ on a complex finite-dimensional Hilbert space $\s H$.
The \defnsty{fixed subspace of degree $n$} is fixed subspace $(\s H^{\otimes n})^\grep \subseteq \s H^{\otimes n}$ of the $n$th tensor power representation $\grep^{\otimes n}$:
\begin{equation}
(\s H^{\otimes n})^\grep \coloneqq \{ w \in \s H^{\otimes n} \mid \forall g \in G, \grep^{\otimes n}(g) w = w \}.
\end{equation}
The projection operator onto the fixed subspace of degree $n$, $(\s H^{\otimes n})^\grep \subseteq \s H^{\otimes n}$, will always be denoted by $\fsub{\grep^{\otimes n}}$.
\end{defn}

\begin{rem}
\label{rem:fixed_graded}
For any positive integers $n, m \in \mathbb N$, there exists a map sending $w_{n} \in (\s H^{\otimes n})^\grep$ and $w_{m} \in (\s H^{\otimes m})^\grep$ to the vector
\begin{equation}
w_{n} \otimes w_{m} \in (\s H^{\otimes (n+m)})^\grep,
\end{equation}
and through this map, $(\s H^{\otimes n})^\grep \otimes (\s H^{\otimes m})^\grep$ can be viewed as a subspace of $(\s H^{\otimes (n+m)})^\grep$:
\begin{equation}
(\s H^{\otimes n})^\grep \otimes (\s H^{\otimes m})^\grep \subseteq (\s H^{\otimes (n+m)})^\grep.
\end{equation}
If $\fsub{\grep^{\otimes n}} \in \End(\s H^{\otimes n})$ is the projection operator onto the subspace $(\s H^{\otimes n})^\grep \subset \s H^{\otimes n}$, then we obtain the operator inequality
\begin{equation}
\fsub{\grep^{\otimes n}} \otimes \fsub{\grep^{\otimes m}} \leq \fsub{\grep^{\otimes (n+m)}}.
\end{equation}
\end{rem}

\begin{lem}
\label{lem:projection_fixed_subspace}
Let $G$ be a complex reductive group which is the complexification of a compact, connected Lie group $K$, i.e. $G = K_{\mathbb C}$.
Let $\s H$ be a complex finite-dimensional Hilbert space and $\grep : G \to \GL(\s H)$ a representation of $G$ on $\s H$.
Let $\fsub{\grep} \in \End(\s H)$ denote the projection operator onto the subspace $\s H^\grep \subseteq \s H$ fixed by $G$.
Then $\fsub{\grep}$ can be expressed as an integral over the compact group $K$,
\begin{equation}
\label{eq:fixed_projector}
\fsub{\grep} = \int_{k \in K} \grep(k) \diff \mu (k),
\end{equation}
where $\mu : \borel{K} \to [0,1]$ denotes the normalized $K$-invariant Haar measure.
\end{lem}
\begin{proof}
From the $K$-invariance of the Haar measure $\diff u$ and Schur's lemma, one can conclude that the right-hand side of \cref{eq:fixed_projector} is equal to the projector operator onto the subspace fixed by the restriction of $\grep$ onto $K$, denoted by $\grep|_{K}$.
To prove \cref{lem:projection_fixed_subspace}, it suffices to show that the subspaces fixed by $G$ (via $\grep$) and $K$ (via $\grep|_{K}$) indeed coincide.
This follows by noticing that when $v$ is fixed by $K$, for all $X$ in the Lie algebra $\mathfrak k$ of $K$, $\arep(X)v = 0$.
Moreover, since $X \in \mathfrak k$ is arbitrary and $G$ is the complexification of $K$, $\arep(X+iY)v = \arep(X)v + i \arep(Y)v = 0$ for all $X,Y \in \mathfrak k$ and thus $\arep(Z)v = 0$ holds for all $Z \in \mathfrak g = \mathfrak k \oplus i \mathfrak k$ and thus $v$ is also $\grep$-invariant.
\end{proof}
\begin{cor}
Let $\grep : G \to \GL(\s H)$ be as in \cref{lem:projection_fixed_subspace}. Then the projection operator onto the fixed subspace of degree $n$, $(\s H^{\otimes n})^\grep$ may be expressed as
\begin{equation}
\label{eq:stable_projector_degree_n}
\fsub{\grep^{\otimes n}} = \int_{k \in K} \grep^{\otimes n}(k)\diff \mu(k).
\end{equation}
\end{cor}
\begin{proof}
The claim follows by applying \cref{lem:projection_fixed_subspace} to the $n$th tensor power representation $\grep^{\otimes n} : G \to \GL(\s H^{\otimes n})$.
\end{proof}

\begin{rem}
\label{rem:homo_poly}
Note that the notion of an invariant polynomial of degree $n$ (encountered in \cref{defn:invariant_polynomials}) is intimately related to the notion of an element of the fixed subspace of degree $n$.
Recall that a homogeneous degree $n$ polynomial $p \in \mathbb C[\s H]_{n}$ is characterized by the fact that $p(x v) = x^n p(v)$ for all $x \in \mathbb C$.
Furthermore every homogeneous degree $n$ polynomial, $p \in \mathbb C[\s H]_{n}$, can be identified by its coefficients with respect to a basis of monomials (of degree $n$), and thus it can be uniquely associated with an element of the $n$th symmetric subspace $\mathrm{Sym}^{n}(\s H)$ in the sense that for each $p \in \mathbb C[\s H]_{n}$ there exists a $w \in \mathrm{Sym}^{n}(\s H)$ such that $p(v) = \langle w, v^{\otimes n}\rangle$ holds for all $v \in \s H$.
\end{rem}
\begin{rem}
For any representation $\grep : G \to \GL(\s H)$, the representation $\phi^{\otimes n} : G \to \GL(\s H)$ naturally commutes with the permutation of the tensor factors in $\s H^{\otimes n}$. In view of \cref{rem:homo_poly}, the space of homogeneous degree $n$, $G$-invariant polynomials, $\mathbb C[\s H]_n^\grep$, can therefore be identified with the space of symmetric, $G$-invariant vectors in $\s H^{\otimes n}$. In particular, we have the following relationship:
\begin{equation}
\mathbb C[\s H]_{n}^\grep \simeq (\s H^{\otimes n})^{\grep} \cap \mathrm{Sym}^{n}(\s H).
\end{equation}
\end{rem}

\subsection{Capacities \& moment maps}
\label{sec:capacities_moment_maps}

While \cref{sec:orbits_stability} was primarily concerned with whether the orbit of a vector $v \in \s H$ under the action of a group representation $\grep : G \to \GL(\s H)$ contains the origin (or gets arbitrarily close to the origin), the purpose of this section is to introduce concepts which characterize these notions of stability more quantitatively.

Specifically, since the Hilbert space $\s H$ is equipped with a positive-definite norm $\norm{\cdot} : \s H \to \mathbb R_{\geq 0}$, the magnitude of $\norm{\grep(g)v}$ as $g$ varies through the group $G$ serves as a measure of the distance between $\grep(g)v$ and the origin $0 \in \s H$.
By minimizing over the whole group $G$ we obtain a measure of distance between the orbit $\grep(G) v$ and the origin called the \textit{capacity of $v$}.
\begin{defn}
\label{defn:capacity}
Let $\grep : G \to \GL(\s V)$ and let $v \in \s V$. The \defnsty{capacity of the vector $v$} is defined as
\begin{equation}
\capacity(v) \coloneqq \inf_{g \in G} \norm{\grep(g) v} = \min_{w \in \overline{\grep(G)v}} \norm{w}.
\end{equation}
\end{defn}
\begin{exam}
Again returning to \cref{exam:prototype_orbits} and consulting \cref{fig:orbit_scaling_example}, one can see that the capacity of a vector $(v_1, v_2) \in \mathbb C^{2}$ under the group acting $(v_1, v_2) \mapsto (z v_1, z^{-1} v_2)$ for $z \in \wozero{\mathbb C}$, defined as
\begin{equation}
\capacity((v_1,v_2)) = \inf_{z\in\wozero{\mathbb C}} \norm{(zv_1, z^{-1}v_2)} = \inf_{z\in\wozero{\mathbb C}} \sqrt{\abs{zv_1}^{2}+ \abs{z^{-1}v_2}^2},
\end{equation}
depends on which of the three types of orbits the vector $(v_1, v_2)$ belongs.
First, the capacity of the zero vector is evidently zero, $\capacity((0,0)) = 0$, as the norm of the zero vector is always zero.
Second, the capacity of any vector $(v_1, v_2)$ lying on, say, the $v_1=0$ axis has zero capacity because in the limit as $\abs{z}$ tends to infinity, the value of $\abs{z^{-1} v_2}$ tends to zero and thus for all $v_2 \in \mathbb C$, $\capacity((0,v_2)) = 0$.
Finally, the capacity of a vector $(v_1, v_2)$ which satisfies both $v_1 \neq 0$ and $v_2 \neq 0$ has a positive capacity equal to
\begin{equation}
\capacity((v_1,v_2)) = \inf_{x\in\mathbb R_{>0}} \sqrt{x\abs{v_1}^{2}+ x^{-1}\abs{v_2}^{2}} = \sqrt{2\abs{v_1}\abs{v_2}},
\end{equation}
and this value is attained when $x = \abs{z} = \abs{v_2}/\abs{v_1}$ (see \cref{fig:orbit_scaling_example}).
\end{exam}

\begin{rem}
In general, the value of the capacity of a vector $v$ is evidently related to the stability properties of $v$.
Indeed one can readily verify that $\capacity(v) = 0$ if and only if $v$ is unstable, or equivalently, $\capacity(v) > 0$ if and only if $v$ is semistable.
The specific magnitude of the capacity when $\capacity(v) > 0$ is therefore unimportant for the purposes of assessing stability.
While the definition given above is consistent with references~\cite{burgisser2019towards, franks2020minimal}, some authors, e.g. \cite{amendola2021invariant}, prefer to define capacity as $\inf_{g \in G} \norm{\grep(g) v}^{2}$.
The specific value of the capacity, beyond whether or not it vanishes, will become important in \cref{sec:estimating_moment_maps}.
\end{rem}
\begin{rem}
\label{rem:proj_cap}
Notice that by linearity of the representation $\grep : G \to \GL(\s V)$, the capacity of any vector $v \in \s V$ satisfies the following scaling property:
\begin{equation}
\forall z \in \mathbb C : \abs{z}\capacity(v) = \capacity(zv).
\end{equation}
Since this property holds for all groups $G$, it becomes reasonable to define the capacity for the ray $\psi \in \proj \s H$, termed the \defnsty{projective capacity} of $\psi$, by
\begin{equation}
\projcapacity(\psi) \coloneqq \inf_{g \in G} \Tr(P_{\psi} \grep(g^{*} g))^{\frac{1}{2}}.
\end{equation}
The capacity of a non-zero vector, $v \in \wozero{\s V}$, and the capacity of the ray $[v] \in \proj \s H$ containing $v$ are thus related by
\begin{align}
\begin{split}
\projcapacity([v])
&= \inf_{g \in G} \Tr(P_{[v]} \grep(g^{*} g))^{\frac{1}{2}}, \\
&= \norm{v}^{-1} \inf_{g \in G} \sqrt{\braket{v, \grep(g^{*} g)v}}, \\
&= \norm{v}^{-1} \inf_{g \in G} \sqrt{\braket{\grep(g)v, \grep(g)v}}, \\
&= \norm{v}^{-1} \inf_{g \in G} \norm{\grep(g) v}, \\
&= \norm{v}^{-1} \capacity(v).
\end{split}
\end{align}
Although the notion of projective capacity is  well-defined and useful, the capacity of a vector, $\capacity(v)$, has the conceptually useful property that the infimum of $\norm{\grep(g) v}$ over $g \in G$ always holds for some vector $w \in \overline{\grep(G)v}$ in the orbit closure of $v$ so that $\capacity(v) = \norm{w}$, whereas the closure of the corresponding projective orbit is not well defined when $\projcapacity(\psi) = 0$.
\end{rem}
\begin{rem}
For any vector $v \in \s H$, the capacity $\capacity(v)$ is lower-bounded by zero (as the norm is non-negative) and upper bounded by the norm $\norm{v}$ (because the identity element $e \in G$ yields $\norm{\grep(e) v} = \norm{v}$):
\begin{equation}
0 \leq \capacity(v) \leq \norm{v}.
\end{equation}
If $v$ happens to be such that $\capacity(v) = \norm{v}$, then by definition, $v$ attains the minimal norm in its orbit, $\grep(G)v$, and thus the norm of $\grep(g)v$ is always equal to or greater than the norm of $v$ itself, i.e. $\norm{\grep(g) v} \geq \norm{v}$ for all $g \in G$.
\end{rem}
\begin{rem}
\label{rem:norm_changing_subset}
Consider the case where the group $G$ is the complexification, $G = K_{\mathbb C}$, of a connected compact Lie group $K$ (see \cref{sec:rep_theory}), and additionally where the norm $\norm{\cdot} : \s H \to \mathbb R_{\geq 0}$ on the representation space $\s H$ is $K$-invariant in the sense that $\norm{\grep(k) v} = \norm{v}$ for all $k \in K$.
The $K$-invariance of the norm means that the function sending a group element $g \in G$ to the norm $\norm{\grep(g) v}$ is constant over left cosets $Kg \in K \backslash G$ where
\begin{equation}
Kg \coloneqq \{ kg \in G \mid k \in K \},
\end{equation}
simply because for all elements $h \in Kg$ of the left coset $Kg$ satisfies
\begin{equation}
\norm{\grep(h) v} = \norm{\grep(k) \grep(g) v} = \norm{\grep(g) v}.
\end{equation}
Therefore, in computing the capacity of a vector $v$, one need not optimize $\norm{\grep(g)v}$ as $g$ varies over the whole group, but instead, one only needs to optimize $\norm{\grep(g)v}$ as $g$ varies over distinct representatives from the set of all left cosets, denoted by $K \backslash G$.
Furthermore, recall from \cref{sec:highest_weights} that the Cartan decomposition $G \simeq K \times P$ of $G$ means the set of left cosets $K \backslash G$ is diffeomorphic to the subset $P \subseteq G$ in the sense that the left coset $Kg$ is identified with the value of $g^{*} g \in P$.
Therefore, capacity of a vector $v$ can also be understood as the optimum of the function sending $p \in P$ to $\sqrt{\braket{v, \grep(p) v}}$.
\end{rem}
\begin{rem}
The problem of computing the capacity of a vector $v \in \s H$ can be conceptualized as an optimization problem over the group $G$ where the objective function whose output is being optimized is the function $g \mapsto \norm{\grep(g) v}$ or equivalently, the function $g \mapsto \log \norm{\grep(g) v}$, also known as the \textit{Kempf-Ness function}.
\end{rem}
\begin{defn}
\label{defn:kempf_ness_function}
Let $\grep : G \to \GL(\s H)$ be a representation of a complex reductive group $G$ on a complex finite-dimensional Hilbert space $\s H$.
Let $v \in \wozero{\s H}$ be non-zero. The \defnsty{Kempf-Ness function} $F_v : G \to \mathbb R$ is defined for $g \in G$ by
\begin{equation}
F_v(g) = \log \norm{\grep(g) v}.
\end{equation}
\end{defn}
\begin{rem}
\label{rem:geodesic_convex_kempf_ness}
In order to characterize the extremal points of the Kempf-Ness function $F_v : G \to \mathbb R$, it will be useful to consider the derivative of the function $f_{v} : \mathbb R \to \mathbb R$ for $Z \in \mathfrak g$ defined for $t \in \mathbb R$ as follows, where $Z \in \mathfrak g$ is an element of the Lie algebra of $G$, and
\begin{equation}
f(t) \coloneqq F_{v}(\exp(tZ)) = \log \norm{\grep(\exp(tZ)) v} = \log \norm{\exp(t \arep (Z))v}.
\end{equation}
When $G$ is the complexification of a compact Lie group $K$, the Lie algebra of $G$ is $\mathfrak g = \mathfrak k \oplus i \mathfrak k$ where $\mathfrak k$ is the Lie algebra of $K$.
If, in addition, the norm $\norm{\cdot}$ is $K$-invariant, then $f_{v}$ is a constant function in the direction $X \in \mathfrak k$ corresponding to the Lie algebra of compact Lie group $K$ (this statement is the equivalent of \cref{rem:norm_changing_subset} from the Lie algebraic perspective).
Therefore, the Kempf-Ness function only changes along curves $t \mapsto \exp(t X)$ where $X$ belongs to the subspace $i \mathfrak k \subset g$ (i.e., when $X$ satisfies $X^{*} = X$).
In this setting, one obtains
\begin{equation}
f(t) \coloneqq F_{v}(\exp(tX)) = \log \norm{\exp(t \arep (X))v} = \log \norm{v_{t}}.
\end{equation}
where $v_t = \exp(t \arep (X))v$.
Letting $u_t$ denote the unit vector $v_t/\norm{v_t}$, the first two derivatives of $f(t)$ are \cite[Eq. (3.13)]{burgisser2019towards}
\begin{align}
f'(t) &= \langle u_t, \arep(X) u_t\rangle, \quad \text{and} \\
f''(t) &= 2 (\langle \arep(X)u_t, \arep(X) u_t\rangle - \langle u_t,\arep(X)u_t\rangle^{2}) = \norm{ u_t \wedge\arep(X)  u_t }^{2},
\end{align}
where $u_t \wedge \arep(X) u_t$ is the antisymmetric vector defined by
\begin{equation}
u_t \otimes\arep(X) u_t -\arep(X) u_t \otimes u_t.
\end{equation}
There are two important conclusions to be drawn from the above calculation.
First, the non-negativity of the second derivative of $f$ demonstrates that $f$ is a \textit{convex} function.
In the language of \cite{burgisser2019towards}, this proves that the Kempf-Ness function, $F_v$, is geodesically convex.
Second, the first derivative of $f$ evaluated at $t = 0$, $f'(0)$, when viewed as being functionally dependent on the value of $X \in i \mathfrak k$, gives rise to the notion of the moment map, defined below.
\end{rem}

\begin{defn}
\label{defn:moment_map}
Let $\grep : G \to \GL(\s H)$ be the representation of reductive group $G$ with maximal compact subgroup $K$ and Lie algebra $\mathfrak g = \mathfrak k \oplus i \mathfrak k$ and $\arep : \mathfrak g \to \mathfrak{gl}(\s H)$ the induced Lie algebra representation of $\mathfrak g$.
The \defnsty{moment map} associated to the representation $\grep$ is the function
\begin{equation}
\momap : \proj \s H \to i\mathfrak k^{*},
\end{equation}
which assigns to each ray $\psi \in \proj \s H$ the linear function $\momap(\psi) : i \mathfrak k \to \mathbb R$ defined by
\begin{equation}
\forall X \in i \mathfrak k, \quad \momap(\psi)(X) = \Tr(P_{\psi}\arep (X)) = \frac{\braket{v, \arep(X) v}}{\braket{v, v}},
\end{equation}
where $v \in \wozero{\psi}$ is any non-zero representative vector in $\psi$.
\end{defn}

In this way, the moment map $\momap : \proj \s H \to i\mathfrak k^{*}$ can be interpreted as a kind of non-commutative gradient of the Kempf-Ness function.
Evidently, the minimum value of the Kempf-Ness function, which is related to the capacity of the vector $v$, and the gradient of the Kempf-Ness function, which is captured by the moment map evaluated on the ray containing $v$, are intimately related.
The following result, known as the Kempf-Ness theorem, solidifies this connection and follows directly from observations made in \cref{rem:geodesic_convex_kempf_ness}.
\begin{thm}[Kempf-Ness]
\label{thm:kempf_ness_theorem}
Let $G = K_{\mathbb C}$ be the complexification of a compact connected Lie group $K$, let $\grep : G \to \GL(\s H)$ be a rational representation of $G$ on a complex finite-dimensional Hilbert space $\s H$ with $K$-invariant inner product, and let $\momap : \proj \s H \to i\mathfrak k^{*}$ be the moment map associated to the representation $\grep : G \to \GL(\s H)$.
Then, for all non-zero vectors $v \in \wozero{\s H}$,
\begin{equation}
\forall g \in G : \norm{\grep(g) v} \geq \norm{v} \quad \Longleftrightarrow \quad \momap([v]) = 0.
\end{equation}
Equivalently, $\capacity(v) = \norm{v}$ if and only if $\momap([v]) = 0$.

Additionally, if $\momap([v]) = 0$ and $w \in \grep(G)v$ has the same norm as $v$ ($\norm{w} = \norm{v}$), then $v$ and $w$ necessarily belong to the same $K$-orbit, $w \in \grep(K) v$.
\end{thm}

\begin{prop}
\label{prop:torus_convex_hull_weights}
Let $r \in \mathbb N$ and let $K = \U(1)^{\times r}$ be the $r$-dimensional torus.
The complexification of $K$ is therefore $G = \wozero{\mathbb C}^{r}$.
Let $\grep : G \to \GL(\s V)$ be a representation of $G$ where $\s V$ admits of the decomposition $\s V = \bigoplus_{\lambda \in \Lambda} \s V_{\lambda}$ where $\Lambda = \mathbb Z^{r}$ be a finite subset $r$-tuples of integers and where $\s V_{\lambda}$ is the multiplicity space for the irreducible representation of $\wozero{\mathbb C}^{r}$ with weight $\lambda$.
Let $v \in \s V$ be a vector with weight-space decomposition $v = \sum_{\lambda} v_{\lambda}$ (where $v_{\lambda} \in \s V_{\lambda}$) and define the support of $v$ to be
\begin{equation}
\Lambda_v \coloneqq \{ \lambda \in \mathbb Z^{r} \mid v_{\lambda} \neq 0 \}.
\end{equation}
Then the capacity of the vector $v$ vanishes if and only if the convex hull of $\Lambda_v \subset \mathbb Z^{r}$, viewed as a subset of $\mathbb R^{r}$, \textit{excludes} zero.
\end{prop}
\begin{proof}
In this setting, the Lie algebra of $K = \U(1)^{\times r}$ is $\mathfrak k = (i \mathbb R)^{r}$ and thus $i \mathfrak k = \mathbb R^{r}$.
Now a vector $v \in \s V$ has weight $\lambda \in \mathbb Z^{r}$ if for all $(z_1, \ldots, z_r) \in \wozero{\mathbb C}^{r}$,
\begin{equation}
\label{eq:rank_r_torus_rep}
\grep(z_1, \ldots, z_r)v = z_1^{\lambda_1}\cdots z_r^{\lambda} v.
\end{equation}
The subspace consisting of all vectors in $\s V$ with weight $\lambda$ is the (possibly empty) weight space $\s V_{\lambda}$.
Using the decomposition of $v$ into distinct weight spaces, the capacity squared of $v$ can be expressed as
\begin{equation}
\label{eq:capacity_squared_torus}
\capacity^{2}(v)
= \inf_{z \in \wozero{\mathbb C}^{r}} \norm{\grep(z) v}^{2}
= \inf_{z \in \wozero{\mathbb C}^{r}} \sum_{\lambda \in \Lambda_v} \norm{z_1^{\lambda}\cdots z_r^{\lambda} v_{\lambda}}^{2}
= \inf_{x \in \mathbb R^{r}} \sum_{\lambda \in \Lambda_v} e^{\lambda \cdot x} \norm{v_{\lambda}}^{2}.
\end{equation}
where $x = (x_1, \ldots, x_r) \in \mathbb R^{r}$ has $i$th component $x_i = 2 \log \abs{z_i}$ and where
\begin{equation}
\lambda \cdot x = \braket{\lambda, x}_{\mathbb R^{r}} = \sum_{i=1}^{r} w_{i} x_{i} = \sum_{i=1}^{r} 2w_{i} \log \abs{z_i}.
\end{equation}
Notice that the summand, $e^{\lambda \cdot x}\norm{v_{\lambda}}^{2}$, in \cref{eq:capacity_squared_torus} is strictly positive, $\norm{v_{\lambda}}^{2} > 0$, and therefore $\capacity(v) = 0$ if and only if the value of $\lambda \cdot x$ can be made arbitrarily negative by some $x \in \mathbb R^{r}$ simultaneously for all $\lambda \in \Lambda_v$.
In other words, $\capacity(v) = 0$ if for all $L \in \mathbb R_{\geq 0}$, there exists an $x \in \mathbb R^{r}$ such that $\lambda \cdot x < - L$ holds for $\lambda \in \Lambda_v$.
This condition, in turn, can be seen to be equivalent to the condition that the convex hull of $\Lambda_v$ in $\mathbb R^{r}$ does \textit{not} contain the origin.
This is because if there exists a $c \in [0,1]^{\abs{\Omega(v)}}$ such that $\sum_{\lambda \in \Lambda_v} c_{\lambda} \lambda = 0 \in \mathbb Z^{r}$ and $\sum_{\lambda \in \Lambda_v} c_{\lambda} = 1$ (that is, the origin is in the convex hull of $\Lambda_v$), then one concludes for any $x \in \mathbb R^{r}$, that $\lambda \cdot x \geq 0$ holds for some $\lambda \in \Lambda_v$ because
\begin{equation}
0 = \sum_{\lambda \in \Lambda_v} (c_{\lambda} \lambda) \cdot x = \sum_{\lambda \in \Lambda_v} c_{\lambda} (\lambda \cdot x)
\end{equation}
On the other hand, if there exists a hyperplane $h \in \mathbb R^{r}$ separating $\Lambda_v$ from the origin, i.e. $\lambda \cdot h > 0$ holds for all $\lambda \in \Lambda_v$, then taking $x = -s h$ for $s > 0$ arbitrarily large yields for all $\lambda \in \Lambda_v$,
\begin{equation}
\lim_{s \to \infty} \lambda \cdot x = \lim_{s \to \infty} -s \lambda \cdot h = - \infty.
\end{equation}
In summary, we have $\capacity(v) = 0$ if and only if the convex hull of $\Lambda_v \subset \mathbb Z^{r}$ (viewed as a subset of $\mathbb R^{r}$) excludes zero.
\end{proof}
\begin{rem}
\label{rem:convex_combo_as_moment_map}
Regarding the moment map of the representation of the $r$-dimensional torus, $\U(1)^{\times r}$, defined by \cref{eq:rank_r_torus_rep}, one has for all vectors $v \in \s V$ and associated rays $[v] \in \proj \s V$ and $x \in i \mathfrak k = \mathbb R^{r}$ the following relationship:
\begin{equation}
\momap([v])(x) = \sum_{\lambda \in \Lambda_v}\frac{\norm{v_{\lambda}}^{2}}{\norm{v}^{2}}\lambda \cdot x.
\end{equation}
In other words, the moment map evaluated on the ray $[v]$, $\momap([v])$, can be identified with an element in the convex hull of the weights $\Lambda_v$ supporting $v$.
Moreover, the particular convex weighting which identifies $\momap([v])$ is given by the coefficients $\norm{v_{\lambda}}^{2}/\norm{v}^{2}$.
\end{rem}
\begin{rem}
Another foundational result in geometric invariant theory is that the image of the moment map $\momap: \proj \s H \to i\mathfrak k^{*}$ evaluated on the orbit closure $\overline{\grep(G)v}$ always intersects the closed fundamental Weyl chamber, $i\mathfrak t^{*}_+$, (viewed as a subset of $i\mathfrak k^{*}$ by using an inner product on $i\mathfrak k$ which is invariant under the adjoint action $\Ad$ of $K$ on $i\mathfrak k$) forms a convex polytope known as the \textit{moment polytope} of $v$, denoted by $\Delta_v \coloneqq \momap(\overline{\grep(G)v}) \cap i\mathfrak t^{*}_+$.
In \cref{sec:moment_polytope}, we will return to the topic of moment polytopes.
\end{rem}

The purpose of the remainder of this section is to highlight the various compositional aspects and symmetries of moment maps and capacities associated to a representation $\grep : G \to \GL(\s H)$ of a complex reductive group $G = K_{\mathbb C}$ with maximal compact subgroup $K$.
To begin, notice that as the capacity of a vector, $v \in \s H$, is defined as an optimization over all of $G$, the action of $G$ on $v$ does not modify the capacity of $v$, i.e., $\capacity(\grep(g) v) = \capacity(v)$ for all $g \in G$.
Unlike the invariance of the capacity map, the moment map, $\momap([v])$, of the ray containing $v$ does vary with the action of $G$.
\begin{lem}
\label{lem:momap_equivariance}
The moment map $\momap : \proj \s H \to i\mathfrak k^{*}$ associated to the representation $\grep : G \to \GL(\s H)$ (where $G = K_{\mathbb C}$ and $K$ is a compact Lie group) is $K$-equivariant, meaning for all $\psi \in \proj\s H$, and $k \in K$ the moment map satisfies
\begin{equation}
\Ad^{*}(k)(\momap(\psi)) = \momap(\grep(k) \cdot \psi)
\end{equation}
where $\Ad^{*} : G \to \GL(\mathfrak g^{*})$ is the dual of the adjoint representation.
\end{lem}
\begin{proof}
Recall from \cref{sec:rep_theory} that the adjoint representation $\Ad : G \to \GL(\mathfrak g)$ satisfies $\Ad(g)(X) = g X g^{-1}$ and is furthermore represented by $\arep(\Ad(g)(X)) = \arep(g X g^{-1}) = \grep(g) \arep(X) \grep(g^{-1})$.
Since $k \in K$ is represented as a unitary, $\grep(k^{-1}) = \grep(k)^{*}$, we have for generic $X \in \mathfrak i \mathfrak k$, the following chain of equivalences
\begin{align}
\begin{split}
\Ad^{*}(k)(\momap(\psi))(X)
&= \momap(\psi)(\Ad(k^{-1})(X)),\\
&= \Tr(P_{\psi} \arep(\Ad(k^{-1})(X)) ), \\
&= \Tr(P_{\psi} \arep(k^{-1}X k) ), \\
&= \Tr(P_{\psi} \grep(k^{-1})\arep(X) \grep (k) ), \\
&= \Tr(P_{\grep(k) \cdot \psi}\arep(X)), \\
&= \momap(\grep(k) \cdot \psi)(X).
\end{split}
\end{align}
\end{proof}

\subsection{Composition of capacities \& moment maps}

\begin{rem}
The following collection of results are concerned with the moment map or capacity map of tensor-products of multiple distinct representations, as defined in \cref{sec:composing_representations}.
In order to distinguish between the moment maps or capacity map associated to distinct representations, a subscript will be added wherever appropriate to avoid confusion.
Specifically, the moment map and capacity map associated to the representation $\grep : G \to \GL(\s H)$ will be respectively written as
\begin{align}
\begin{split}
\momap_{\grep} &: \proj \s H \to i\mathfrak k^{*}, \\
\capacity_{\grep} &: \s H \to \mathbb R_{\geq 0}
\end{split}
\end{align}
In summary, it will be shown that moment maps are additive along tensor products while capacity maps are supermultiplicative.
\end{rem}

\begin{lem}
\label{lem:cap_sup_mult}
Let $\grep_1 : G \to \GL(\s V_1)$ and $\grep_2 : G \to \GL(\s V_2)$ be representations and let $v_1 \otimes v_2 \in \s V_1 \otimes \s V_2$. Then
\begin{equation}
\capacity_{\grep_1 \otimes \grep_2}(v_1 \otimes v_2) \geq \capacity_{\grep_1}(v_1)\capacity_{\grep_2}(v_2) = \capacity_{\grep_1 \boxtimes \grep_2}(v_1 \otimes v_2).
\end{equation}
\end{lem}
\begin{proof}
The proof follows from the injectivity of the copying map $\Delta : G \to G \times G$ sending $g$ to $\Delta(g) = (g,g)$ which connects the internal and external tensor products of $\grep_1$ and $\grep_2$ first encountered in \cref{rem:int_ext_copy}.
\begin{align}
&\capacity_{\grep_1 \otimes \grep_2}(v_1 \otimes v_2)
=\inf_{g\in G}\norm{(\grep_1(g) \otimes \grep_2(g))(v_1 \otimes v_2)}\\
&\quad\geq\inf_{\substack{g_1\in G\\ g_2 \in G}}\norm{\grep_1(g_1)v_1}\norm{\grep_2(g_2)v_2}
\end{align}
\end{proof}
\begin{lem}
\label{lem:moment_map_ext}
Let $\grep_1 : G_1 \to \GL(\s V_1)$ and $\grep_2 : G_2 \to \GL(\s V_2)$ be representations and let $v_1 \otimes v_2 \in \s V_1 \otimes \s V_2$. Then the moment map for the external tensor product representation $\grep_1 \boxtimes \grep_2$ is of the form $\momap_{\grep_1 \boxtimes \grep_2} : \proj(\s V_1 \otimes \s V_2) \to (i\mathfrak k_1 \oplus i \mathfrak k_2)^{*}$ where
\begin{equation}
\momap_{\grep_1 \boxtimes \grep_2}([v_1 \otimes v_2]) = \momap_{\grep_1}([v_1])\oplus \momap_{\grep_2}([v_2]).
\end{equation}
\end{lem}
\begin{proof}
If $\mathfrak g_1$ and $\mathfrak g_2$ are the Lie algebras of $G_1$ and $G_2$, then the Lie algebra of $G_1 \times G_2$ is $\mathfrak g_1 \oplus \mathfrak g_2$. Moreover, if $\arep_1$ and $\arep_2$ are the respective induced Lie algebra representations, then $\grep_1 \boxtimes \grep_2$ induces the Lie algebra representation $\arep_1 \boxtimes \arep_2 : \mathfrak g_1 \oplus \mathfrak g_2 \to \mathfrak{gl}(\s V_1 \otimes \s V_2)$ which for $X_1 \oplus X_2 \in \mathfrak g_1 \oplus \mathfrak g_2$ is defined by
\begin{equation}
(\arep_1 \boxtimes \arep_2)(X_1 \oplus X_2) = \arep_1(X_1) \otimes I_{\s V_2} + I_{\s V_1} \otimes \arep_2(X_2).
\end{equation}
Applying this result to the definition of the moment map in \cref{defn:moment_map} when $v = v_1 \otimes v_2$ yields the claim.
\end{proof}
\begin{lem}
\label{lem:moment_map_int}
Let $\grep_1 : G \to \GL(\s V_1)$ and $\grep_2 : G \to \GL(\s V_2)$ be representations and let $v_1 \otimes v_2 \in \s V_1 \otimes \s V_2$. Then the moment map for the internal tensor product representation $\grep_1 \otimes \grep_2 : G \to \GL(\s V_1 \otimes \s V_2)$ is of the form $\momap_{\grep_1 \otimes \grep_2} : \proj (\s V_1 \otimes \s V_2) \to i\mathfrak k^{*}$ where
\begin{equation}
\momap_{\grep_1 \otimes \grep_2}([v_1 \otimes v_2]) = \momap_{\grep_1}([v_1])+\momap_{\grep_2}([v_2]).
\end{equation}
\end{lem}
\begin{proof}
If $\mathfrak g$ is the Lie algebra of $G$, then the representation of $\mathfrak g$ induced by the internal tensor product representation $\grep_1 \otimes \grep_2$ of $G$ on $\s V_1 \otimes \s V_2$ is simply $\arep_1 \otimes \arep_2 : \mathfrak g \to \mathfrak{gl}(\s V_1 \otimes \s V_2)$ defined for $X \in \mathfrak g$ by
\begin{equation}
(\arep_1 \otimes \arep_2)(X) = \arep_1(X) \otimes I_{V_2} + I_{V_1} \otimes \arep_2(X).
\end{equation}
Applying this to the definition of the moment map in \cref{defn:moment_map} when $v = v_1 \otimes v_2$ yields the claim.
\end{proof}

\section{Occasionality \& semistability}

\subsection{Cumulants of quantum observables}

This section considers the moment generating function associated to the random variable formed by a quantum state and quantum observable pair.
The main result, \cref{cor:cumulant_ineqs}, is the derivation of an error-bound on the second order expansion of this moment generating function.

\begin{lem}
\label{lem:first_three_cumulants}
Let $X$ be a self-adjoint operator, $X^{*} = X$, acting on a complex finite-dimensional Hilbert space $\s H$.
Let $\End(\s H)$ be a $C^*$-algebra of linear maps on $\s H$ and let $\varphi : \End(\s H) \to \mathbb C$ be a state.
Let $R$ be the discrete random variable with distribution $\mathrm{Prob}(R = x) = \varphi(P_x)$ where $P_x \in \End(\s H)$ projects onto the eigenspace of $X$ with eigenvalue $x \in \mathbb R$ and let $M : \mathbb R \to \mathbb R$ be its moment generating function:
\begin{equation}
M(t) = \mathbb E(\exp(tR)) = {\sum}_{x} e^{t x} \varphi(P_x) = \varphi(\exp(t X))
\end{equation}
Whenever $t \in \mathbb R$ is such that $\varphi(\exp(t X)) \neq 0$ let $\varphi_t$ denote the state sending any $Y \in \End(\s H)$ to
\begin{equation}
\varphi_t(Y) \coloneqq \frac{\varphi(e^{t \frac{X}{2}} Y e^{t \frac{X}{2}})}{\varphi(e^{tX})}.
\end{equation}
Then the first three derivatives of the cumulant generating function $K(t) = \log M(t) = \log \varphi (\exp(tX))$ are:
\begin{align}
K^{(1)} &= \varphi_t(X), \\
K^{(2)} &= \varphi_t(X^2) - \varphi_t(X)^2 = \varphi_t((X - \varphi_t(X))^{2}), \\
K^{(3)} &= \varphi_t(X^3) - 3\varphi_t(X^2)\varphi_t(X) + 2 \varphi_t(X)^3 = \varphi_t((X - \varphi_t(X))^{3}). \\
\end{align}
\end{lem}
\begin{proof}
Throughout this proof, we make use of the assumption that $M(t) = \varphi(e^{tX}) \neq 0$ for all $t$ in a neighborhood of zero so that $K(t) = \log M(t)$ is well-defined.
\begin{align}
K^{(1)}(t) &= \frac{\varphi(Xe^{tX})}{\varphi(e^{tX})} \\
K^{(2)}(t) &= \frac{\varphi(X^2e^{tX})\varphi(e^{tX}) - \varphi(Xe^{tX})^2}{\varphi(e^{tX})^2} \\
\begin{split}
&= \frac{(\varphi(X^3e^{tX})\varphi(e^{tX}) - \varphi(X^2e^{tX})\varphi(Xe^{tX}))\varphi(e^{tX})^2}{\varphi(e^{tX})^4} \\
&\qquad - \frac{2(\varphi(X^2e^{tX})\varphi(e^{tX}) - \varphi(Xe^{tX})^2)\varphi(e^{tX})\varphi(Xe^{tX})}{\varphi(e^{tX})^4}
\end{split} \\
&= \frac{\varphi(X^3e^{tX})\varphi(e^{tX})^2 - 3\varphi(X^2e^{tX})\varphi(Xe^{tX})\varphi(e^{tX}) + 2\varphi(Xe^{tX})^{3}}{\varphi(e^{tX})^3}
\end{align}
The claim follows from noting that $X^{p}e^{tX} = e^{t\frac{X}{2}}X^{p}e^{t\frac{X}{2}}$ holds for any exponent $p \in \mathbb N$ because $[e^{tX}, X] = 0$ and therefore
\begin{equation}
\frac{\varphi(X^pe^{tX})}{\varphi(e^{tX})} = \varphi_{t}(X^{p}).
\end{equation}
\end{proof}
\begin{cor}
\label{cor:cumulant_ineqs}
Let everything be as defined by \cref{lem:first_three_cumulants}.
The following inequalities hold for any $t \in \mathbb R$ where $\varphi(e^{tX}) \neq 0$ so that $K(t)$ is well-defined.
\begin{equation}
0 \leq \abs{K^{(3)}(t)} \leq 2 \norm{X}_{\mathrm{op}} K^{(2)}(t), \qquad 0 \leq K^{(2)}(t) \leq 4 \norm{X}_{\mathrm{op}}^{2}.
\end{equation}
\end{cor}
\begin{proof}
Let $S_t = X - \varphi_t(X)$ be the self-adjoint operator, $S_t^* = S_t$, obtained by shifting $X$ about its mean $\varphi_t(X)$.
By subadditivity of the operator norm, we have
\begin{equation}
\norm{S_t}_{\mathrm{op}} \leq \norm{X}_{\mathrm{op}} + \varphi_t(X) \norm{1}_{\mathrm{op}} \leq 2 \norm{X}_{\mathrm{op}}.
\end{equation}
Furthermore, the first few moments of $S_t$ with respect to $\varphi_t$ are
\begin{equation}
\varphi_t(S_t) = 0, \qquad \varphi_t(S_t^2) = K^{(2)}, \qquad \varphi_t(S_t^3) = K^{(3)}.
\end{equation}
Then by \cref{eq:state_op_bound} and positivity of $\varphi_t$,
\begin{equation}
0 \leq \varphi_t(S_t^2) \leq \norm{S_t^2}_{\mathrm{op}} \leq 4 \norm{X}_{\mathrm{op}}^{2}
\end{equation}
Furthermore by \cref{cor:cumulant_ineqs},
\begin{equation}
\abs{\varphi_t(S_t^3)} \leq \varphi_t(S_t^2)^{\frac{3}{2}} \leq 2 \norm{X}_{\mathrm{op}} K^{(2)}.
\end{equation}
\end{proof}

\begin{rem}
Before stating and proving the next result, let $\epsilon > 0$ be small and let $X \in \mathrm{End}(\s H)$ be self-adjoint.
For any state, $\varphi : \mathrm{End}(\s H) \to \mathbb C$, one can make the following approximation:
\begin{equation}
\varphi(\exp(\epsilon X)) \approx 1 + \epsilon \varphi(X) + \frac{\epsilon^2}{2} \varphi(X^2).
\end{equation}
If additionally $\varphi(X) = 0$, one obtains the approximation $\varphi(\exp(\epsilon X)) \approx 1 + \frac{\epsilon^2}{2} \varphi(X^2)$ and therefore up to terms of order $\epsilon^{3}$, we have
\begin{equation}
\log\varphi(\exp(\epsilon X)) \approx \frac{1}{2}\epsilon^2\varphi(X^2).
\end{equation}
The following result quantifies the error introduced by this approximation.
\end{rem}

\begin{lem}
\label{lem:CLT}
Let $X^{*} = X$ be a self-adjoint operator and let $\varphi$ be a state.
Then the moment generating function satisfies
\begin{equation}
\varphi(\exp(t X)) = \exp\left[\varphi(X) t + (\varphi(X^2) - \varphi(X)^2)\frac{t^2}{2} + r(t)\right]
\end{equation}
where $r : \mathbb R \to \mathbb C$ is a remainder term that satisfies for all $t \in \mathbb R$,
\begin{equation}
\abs{r(t)} \leq \frac{4}{3} \norm{X}_{\mathrm{op}}^{3} t^3.
\end{equation}
Moreover when $\varphi(X) = 0$,
\begin{equation}
\lim_{n \to \infty} \varphi(\exp\left(\frac{X}{\sqrt{n}}\right))^{n} = \exp\left[\frac{\varphi(X^2)}{2}\right]
\end{equation}
\end{lem}
\begin{proof}
The proof relies on taking a Taylor series of the cumulant generating function $K(t) = \log \varphi(e^{tX})$ for $t > 0$ about $t = 0$ to second order
\begin{equation}
K(t) = \kappa_1 t + \frac{\kappa_2}{2!} t^2 + r(t),
\end{equation}
where (i) the coefficient $\kappa_n$ is the cumulant of degree $n$, i.e., the $n$th derivative of $K(t)$ evaluated at $t = 0$, and (ii) the remainder term $r(t)$ is (for a fixed $t > 0$) of the form
\begin{equation}
r(t) = \frac{K^{(3)}(c)}{3!} t^3.
\end{equation}
for some $c \in [0,t]$.
Furthermore using \cref{cor:cumulant_ineqs}, the remainder term may be bounded uniformly with respect to $t$ by
\begin{equation}
\label{eq:remainder_bound}
\abs{r(t)} \leq \frac{1}{6}\abs{K^{(3)}(c)} t^3 \leq \frac{4}{3} \norm{X}_{\mathrm{op}}^{3} t^3.
\end{equation}
Now using \cref{lem:first_three_cumulants} together with the fact that $\varphi_t$ evaluated at $t = 0$ is merely $\varphi$ yields
\begin{equation}
\kappa_1 = \varphi(X), \qquad \kappa_2 = \varphi((X - \varphi(X))^2).
\end{equation}
Therefore if $\varphi(X) = 0$, $\kappa_1 = 0$ and $\kappa_2 = \varphi(X^2)$ which means $K(t)$ is to leading order quadratic in $t$:
\begin{equation}
K(t) = \varphi(X^2)\frac{t^2}{2} + r(t).
\end{equation}
By appropriately scaling the cumulant generating function
\begin{equation}
K(t) \mapsto \lambda^2 K(\lambda^{-1} t)
\end{equation}
for some large $\lambda > 0$, one can suppress the contribution of the remainder term while leaving the second order term unaffected in the sense that
\begin{equation}
\lambda^2 K(\lambda^{-1} t) = \varphi(X^2)\frac{t^2}{2} + \lambda^{2}r(\lambda^{-1}t),
\end{equation}
where by \cref{eq:remainder_bound},
\begin{equation}
\label{eq:remainder_scaled_bound}
\abs{\lambda^{2}r(\lambda^{-1}t)} \leq \lambda^{-1} \frac{4}{3}\norm{H}_{\mathrm{op}}^{3} t^3.
\end{equation}
Setting $t = 1$ and taking the limit as $\lambda \to \infty$ produces
\begin{equation}
\lim_{\lambda \to \infty} \lambda^2 K(\lambda^{-1}) = \frac{\varphi(X^2)}{2}
\end{equation}
as claimed.
\end{proof}

\subsection{Typical, occasional \& exceptional behaviours}
\label{sec:typical_occasional_exceptional}

In the forthcoming sections, namely \cref{sec:occasionality} and \cref{sec:strong_duality}, we will consider the asymptotics of sequences of probabilities $\{p_n \in [0,1] \mid n \in \mathbb N\}$.
In particular, we will be interested in the sequence of probabilities that arises from applying a quantum state $\varphi : \End(\s H) \to \mathbb C$ to the sequence of fixed subspaces (\cref{sec:invariant_and_fixed_subspaces}) of increasing degree $n$, $\{(\s H^{\otimes n})^\grep \mid n \in \mathbb N\}$ associated to the representation $\grep : G \to \GL(\s H)$ of a group $G$.
Specifically, the sequence of probabilities will have the form
\begin{equation}
\label{eq:prob_seq_target}
p_n = \varphi^{\otimes n}(\fsub{\grep^{\otimes n}})
\end{equation}
where $\fsub{\grep^{\otimes n}} \in \End(\s H^{\otimes n})$ is the projection operator on the fixed subspace $(\s H^{\otimes n})^\grep$.
It will be shown that the qualitative behaviour of this sequence of probabilities in the limit of large $n$ encodes information about the relationship between the state $\varphi$ and the group $G$.
The purpose of this section is to define three types of asymptotic behaviours which we refer to as \textit{typical}, \textit{occasional} and \textit{exceptional}.

\begin{defn}
A sequence $\{p_n \in [0,1] \mid n \in \mathbb N\}$ of probability assignments is said to describe
\begin{enumerate}[i)]
\item a \defnsty{typical} behaviour if
\begin{equation}
\lim_{n\to\infty} p_n = 1,
\end{equation}
\item an \defnsty{occasional} behaviour if there exists constants $\alpha > 0, \beta > 0$ such that
\begin{equation}
\label{eq:occasional}
\limsup_{n\to\infty} n^{\alpha} p_n \geq \beta,
\end{equation}
\item an \defnsty{exceptional} behaviour if there exists a constant $\gamma \in [0, 1)$ such that
\begin{equation}
\label{eq:exceptional}
\limsup_{n\to\infty} p_n^{\frac{1}{n}} \leq \gamma.
\end{equation}
\end{enumerate}
\end{defn}
\begin{rem}
A \textit{typical} behaviour is essentially any sequence of events with probabilities $\{p_n \in [0,1] \mid n \in \mathbb N\}$ that one can be arbitrarily certain will eventually occur because for any small $\epsilon > 0$, we have $p_n > 1 - \epsilon$ for sufficiently large $N$.
\end{rem}
\begin{rem}
While the notion of an \textit{occasional} behaviour is more subtle, it faithfully captures the intuitive idea that a fair coin occasionally yields and equal number of heads and tails.
Indeed, the probability of flipping a fair coin $n$ times and obtaining an equal number of heads and tails is equal to
\begin{equation}
p_n = \begin{cases}
\frac{1}{2^{n}}\binom{n}{\frac{n}{2}} & n \text{ even} \\
0 & n \text{ odd}.
\end{cases}
\end{equation}
Although obtaining an equal number of heads and tails is \textit{atypical} in the sense that $p_n \to 0$ as $n \to \infty$, one expects to obtain an equal number of heads and tails \textit{occasionally} because one can show (for $n$ even) that $p_n \geq (2n)^{-1/2}$ and therefore \cref{eq:occasional} holds for $\alpha = \frac{1}{2}$ and $\beta = \frac{1}{\sqrt{2}}$.
\end{rem}
\begin{rem}
The terminology of an \textit{exceptional} behaviour is justified because as $n \to \infty$, the probability $p_n$ decays to zero exponentially fast.
To see this, let $\gamma \in [0, 1)$ be as in \cref{eq:exceptional} and let $\epsilon > 0$ be such that $\gamma + \epsilon < 1$. Then for sufficiently large $N \in \mathbb N$, the limit in \cref{eq:exceptional} implies $\sup_{n \geq N} p_n^{\frac{1}{n}} \leq \gamma + \epsilon$.
This implies that $p_n \leq (\gamma+\epsilon)^{n}$ which means $p_n$ eventually decays to zero at a rate that is at least exponential in $n$ with exponent $\log(\gamma+\epsilon) < 0$.
In other words, there exists an $r \in (0,1)$ such that $p_n \leq r^{n}$ holds for sufficiently large $n \geq N$.
\end{rem}

When the sequence of probabilities under investigation arises from the application of a tensor power state $\varphi^{\otimes n}$ to the projection operator $\fsub{\grep^{\otimes n}}$ of a fixed subspace $(\s H^{\otimes n})^\grep \subset \s H^{\otimes n}$ as in \cref{eq:prob_seq_target}, one can use the operator inequality $\fsub{\grep^{\otimes n}} \otimes \fsub{\grep^{\otimes m}} \leq \fsub{\grep^{\otimes (n+m)}}$ (see \cref{rem:fixed_graded}) to show that the sequence $\{ p_{n} \mid n\in \mathbb N\}$ is \textit{super-multiplicative} in the sense that
\begin{equation}
p_{n+m} = \varphi^{\otimes (n+m)} ( \fsub{\grep^{\otimes (n+m)}}) \geq \varphi^{\otimes (n+m)} ( \fsub{\grep^{\otimes n}} \otimes \fsub{\grep^{\otimes m}}) = \varphi^{\otimes n}(\fsub{\grep^{\otimes n}})\varphi^{\otimes n}(\fsub{\grep^{\otimes n}}) = p_n p_m.
\end{equation}
As it turns out, this property is useful for establishing limits such as those appearing in \cref{eq:exceptional}.
First we recall a powerful lemma for subadditive sequences of real numbers known as Fekete's subadditivity lemma~\cite[Lem. 1.2.1]{steele1997probability}.
\begin{lem}%[Fekete's Subadditive Lemma]
For every sequence $\{a_n\}_{n\in \mathbb N}$ satisfying
\begin{equation}
a_{n+m} \leq a_n + a_m,
\end{equation}
the following limit exists and satisfies
\begin{equation}
\lim_{n\to\infty} \frac{a_n}{n} = \inf_{n \in \mathbb N} \frac{a_n}{n}.
\end{equation}
\end{lem}
\begin{proof}
If the sequence ever reaches $a_m = -\infty$ for finite $m$, then by subadditivity, both sides of the above equation are $-\infty$.
Henceforth assume $a_n > - \infty$ for all $n$.
Now for any $n = mk + r$, subadditivity implies $a_{n} \leq ma_{k} + a_{r}$ and thus
\begin{equation}
\frac{a_{n}}{n} \leq \frac{ma_{k}}{n} + \frac{a_{r}}{n} = \left(1-\frac{r}{n}\right) \frac{a_{k}}{k} + \frac{a_{r}}{n}.
\end{equation}
Moreover for fixed $k$, one may assume for all $n \geq k$ that $n = km + r$ holds for some $r$ satisfying $0 \leq r < k$.
For any $\epsilon > 0$ pick a $k$ such that $\frac{a_{k}}{k}$ is within $\epsilon$ of $L = \inf_{n \in \mathbb N} \frac{a_n}{n}$:
\begin{equation}
\frac{a_{k}}{k} < L + \epsilon.
\end{equation}
Since $k$ is now fixed and $r < k$,
\begin{equation}
\lim_{n \to \infty} \frac{a_{n}}{n} < \lim_{n \to \infty} \left[\left(1-\frac{r}{n}\right)(L + \epsilon) + \frac{a_{r}}{n}\right] = L+ \epsilon.
\end{equation}
Since this holds for any $\epsilon > 0$, the lemma holds by taking $\epsilon \rightarrow 0$.
\end{proof}
\begin{rem}
Note that every super-multiplicative sequence $\{b_n\}_{n\in\mathbb N}$, meaning $b_{n+m} \geq b_n b_m$, gives rise to a subadditive sequence with $a_n = - \log b_n$ (assuming $b_n > 0$ for all $n$).
This implies an analogue of Fekete's lemma for super-multiplicative sequences.
\end{rem}
\begin{cor}
For every sequence $\{b_n\}_{n\in \mathbb N}$ satisfying
\begin{equation}
b_{n+m} \geq b_nb_m,
\end{equation}
the following limit exists and satisfies
\begin{equation}
\limsup_{n\to\infty} b_n^{\frac{1}{n}} = \sup_{n \in \mathbb N} b_n^{\frac{1}{n}}.
\end{equation}
\end{cor}
\begin{proof}
By supermultiplicativity, if $b_k > 0$ for some $k$, then $b_{mk} \geq b_k^{m} > 0$ for all $m$. Therefore $a_m \mapsto - \log b_{mk}$ is a well-defined subadditive sequence which implies $\lim_{m\to\infty} \frac{a_m}{m} = \inf_{m} \frac{a_m}{m}$. Therefore for all $k$ such that $b_k > 0$,
\begin{equation}
\lim_{m\to\infty} b_{mk}^{\frac{1}{mk}} = \sup_{m \in \mathbb N} b_{mk}^{\frac{1}{mk}} \geq b_{k}^{\frac{1}{k}}.
\end{equation}
This implies that
\begin{equation}
\limsup_{n\to\infty} b_n^{\frac{1}{n}} \geq \sup_{n \in \mathbb N} b_n^{\frac{1}{n}},
\end{equation}
and the reverse inequality holds trivially.
\end{proof}
\begin{rem}
Note that one cannot necessarily replace $\limsup$ with $\lim$ in the above corollary because it remains possible to find a subsequence $\{n_j\}_{j \in \mathbb N}$ contained outside of all sub-semigroups $\{mk | k \in \mathbb N\}$ of $\mathbb N$ such that $b_{n_j} = 0$ holds for all $j$. See the footnote in \cite{franks2020minimal} for further explanation of this subtlety.
\end{rem}

\subsection{The occasionality theorem}
\label{sec:occasionality}

The purpose of this section is to develop and prove an important relationship between the constant subspaces and semistable vectors of a representation $\grep : G \to \GL(\s H)$ and its tensor powers $\grep^{\otimes n} : G \to \GL(\s H)$ (\cref{thm:occasionality}).
The proof presented here is taken directly from \citeauthor{franks2020minimal}~\cite{franks2020minimal}.
Our only deviation from \cite{franks2020minimal} is to emphasize the role of cumulants (\cref{sec:moments_cumulants}), as noticed by \cite[Rem. 3.16]{burgisser2019towards}.
As this result serves as the foundation for many other results in this thesis, the proof is represented here purely for completeness.

Before doing so, it will be necessary to distinguish between subgroups of $G$ which stabilizer a given non-zero vector $v \in \wozero{\s H}$ from the subgroups of $G$ which stabilize a given ray $\psi \in \proj \s H$.
To clarify this subtle distinction, we consider the following example.
\begin{exam}
Let $\s H_+$ and $\s H_-$ be two orthogonal and complementary subspaces of a complex finite-dimensional Hilbert space
\begin{equation}
\s H = \s H_+ \oplus \s H_-,
\end{equation}
and let $P_\pm$ be the orthogonal projection operator onto $\s H_{\pm}$.
Moreover, let $X = P_+ - P_- \in \End(\s H)$ be the operator which reflects vectors through the subspace $\s H_+ \subset \s H$, i.e., $X = \ident_{\s H} - 2 P_-$.
Then consider any vector $v \in \wozero{\s H}$ with non-zero component in both $\s H_+$ and $\s H_-$.
Let $\grep : \wozero{\mathbb C} \to \GL(\s H)$ the representation of $\wozero{\mathbb C}$ defined by $\grep(e^{z}) = \exp(z X)$.
The action of $\wozero{\mathbb C}$ on $v = v_+ + v_-$ thus
\begin{equation}
\grep(e^{z}) v = e^{+z}v_+ + e^{-z}v_-,
\end{equation}
and therefore the stabilizer of $v$ is the trivial subgroup $\{1\} \subset \wozero{\mathbb C}$ while the stabilizer of the ray containing $v$ is the subgroup $\mathbb Z_2 \simeq \{-1, +1\} \subset \wozero{\mathbb C}$.
\end{exam}

The following result, found in \cite[Lem. 2.2]{mumford1984stratification}, demonstrates the connection between geometric and algebraic notions of stability.
\begin{lem}
\label{lem:finite_projective_quotient}
Let $\psi \in \proj \s H$ be a ray and let $v \in \wozero{\psi}$ be non-zero so that $\mathbb C v = \psi$.
Let $\grep : G \to \GL(\s H)$ be a representation of a reductive group $G$ and let $G_{\psi}$ denote the subgroup of $G$ which leaves the subspace $\psi$ invariant and $G_v$ the subgroup of $G_{\psi}$ which leaves the vector $v$ invariant.
Then either $G_{\psi}/G_{v} \simeq \wozero{\mathbb C}$ or $G_{\psi}/G_{v}$ is finite (and thus $\dim G_{\psi} = \dim G_{v}$).
Moreover if $G_{\psi}/G_{v} \simeq \wozero{\mathbb C}$ then $v$ is necessarily unstable.
\end{lem}
\begin{proof}
The quotient $G_{\psi} / G_{v}$ of $G_{\psi}$ by $G_{v}$ considered by the lemma arises when considering the sequence
\begin{equation}
\label{eq:stabilizing_exact_sequence}
\begin{tikzcd}
G_{v} \arrow[r] &
G_{\psi} \arrow[r] &
\wozero{\mathbb C}
\end{tikzcd}
\end{equation}
where the map $G_{v} \rightarrow G_{\psi}$ is simply a subgroup inclusion (whose image is the subgroup $G_{v} \subseteq G_{\psi}$) while $G_{\psi} \rightarrow \wozero{\mathbb C}$ is the map sending $g \in G_{\psi}$ to the unique non-zero complex number $c_g$ satisfying $\grep(g) v = c_g v$.
Evidently, the kernel of the latter map $G_{\psi} \rightarrow \wozero{\mathbb C}$, i.e., those $g \in G_{\psi}$ such that $\grep(g) v = v$ is equal to $G_{v}$, and thus the above sequence is an example of an \textit{exact} sequence.
In any case, the quotient $G_{\psi} / G_{v}$ is precisely the image of the action of $G_{\psi}$ on $v$ and is necessarily an algebraic subgroup of $\wozero{\mathbb C}$ (as $G$ itself is reductive).
Being an algebraic subgroup of $\wozero{\mathbb C}$, $G_{\psi} / G_{v}$ must either be isomorphism to i) the entire group $\wozero{\mathbb C}$ or ii) a finite subgroup of $\wozero{\mathbb C}$.
If $G_{\psi}/ G_{v} \simeq \wozero{\mathbb C}$, then there exists an element $X$ in the Lie algebra of $G_{\psi}$ such that $\grep(\exp(tX)) v = e^{tx}v$ for some non-zero $x \in \wozero{\mathbb R}$.
Therefore, in the limit of either large or small $t \in \mathbb R$,
$\grep(\exp(tX)) v = e^{tx} v \rightarrow 0$,
and thus the vector $v$ must be unstable.
\end{proof}
A very useful corollary of \cref{lem:finite_projective_quotient}, which applies to semistable vectors, is the following result.
\begin{cor}
\label{cor:semistable_finite_unitary}