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\chapter{A BRIEF ACCOUNT OF MODERN INVARIANT THEORY VS CLASSICAL INVARIANT THEORY}
\label{cha:invariant}

\vspace{-12pt}
\section{Modern Invariant Theory}\label{sec:modern}

In modern invariant theory, we have 
 $\Gamma$,  a subgroup of $GL(\mathbb{C}^n)$. Given a polynomial $f \in \mathbb{C}[x_1, \ldots, x_n]$ then every linear transformation $\pi \in \Gamma$ transforms $f$ into a new polynomial function $f\circ \pi.$ The goal is to  compute the ring of invariants for $\Gamma$: \[ \mathbb{C}[x_1, \ldots, x_n]^{\Gamma}:=\{f \in \mathbb{C}[x_1, \ldots, x_n]: \forall \pi \in \Gamma (f = f\circ \pi)\}.  \]
\begin{example}[Symmetric polynomials]
 We let $S_n$ be the group of permutation matrices in  $GL(\mathbb{C}^n)$. Its invariant ring $\mathbb{C}[x_1, \ldots, x_n]^{S_n}$ is the subring of symmetric polynomials  $\mathbb{C}[\sigma_1, \ldots, \sigma_n]$, by definition of symmetric polynomials.\\
 \end{example} In classical invariant theory we consider the homogeneous polynomial (also known as a form) of degree $n$, 
\[\displaystyle \mathcal{F}(x,y) = \sum_{i+k = n} c_{ik} \binom{n}{i} x^iy^k .\]

For $A \in GL_{2}(\mathbb{C})$, we map the variables \[ \begin{bmatrix} x \\ y \end{bmatrix} \mapsto A \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x' \\ y' \end{bmatrix}. \]

Then, \[  \mathcal{F}(x,y) \mapsto \mathcal{F}(x', y') = \sum_{i+k = n} {c'}_{ik}\binom{n}{i}{x}^i{y}^k .  \]

\begin{definition}  An invariant of the form $\mathcal{F}$ is a polynomial $\mathcal{I}(c_{ik})$ such that \[ \mathcal{I}({c'}_{ik}) = \det(A)^{p} \cdot \mathcal{I}(c_{ik}), \] for some $p \in \mathbb{N}$\end{definition}
\begin{example}[The Quadratic Binomial Case] When $n = 2$,  let $A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} \in GL_{2}(\mathbb{C}).$ 

Then, \[ \begin{bmatrix} x \\ y \end{bmatrix} \mapsto A \begin{bmatrix} x \\ y \end{bmatrix} =  \begin{bmatrix} a_{11}x + a_{12} y \\ a_{21}x + a_{22} y \end{bmatrix} =  \begin{bmatrix} x' \\ y' \end{bmatrix}. \]
$\displaystyle \mathcal{F}(x,y) = c_0x^2+2c_1xy+c_2y^2 \text{ becomes }\mathcal{F}(x', y')= c'_0x^2+2c_1'xy+c_2'y^2.$\\

Then one invariant is $c_0c_2-c_1^2$, because 
\begin{align*}
c_0 'c_2 '-c_1 '^2 &= c_0^2(a_{11}^2a_{12}^2 - a_{11}^2a_{12}^2)\\
&+ 2c_0c_1(a_{11}^2a_{12}a_{22}+a_{12}^2a_{11}a_{21} - a_{11}a_{12}(a_{11}a_{22}+a_{12}a_{21}))\\
& + c_0c_2(a_{11}^2a_{22}^2+a_{12}^2a_{21}^2 - 2a_{11}a_{12}a_{21}a_{22})\\
& +c_1^2(4a_{11}a_{21}a_{12}a_{22} - (a_{11}a_{22} + a_{12}a_{21})^2)\\
& + c_2^2(a_{21}^2a_{22}^2 - a_{21}^2a_{22}^2)\\
& +2c_1c_2(a_{11}a_{21}a_{22}^2 + a_{12}a_{22}a_{21}^2- a_{21}a_{22}(a_{11}a_{22}+ a_{12}a_{21}))\\
& =c_0c_2(a_{11}a_{22} - a_{12}a_{21})^2 - c_1^2(a_{11}a_{22}- a_{12}a_{21})^2\\
& = \delta^2 (c_0c_2-c_1^2).\\
\end{align*}

\end{example}
\begin{definition}  A covariant of the form $\mathcal{F}$ is a polynomial in the  $\mathcal{C}(c_{ik}, x, y)$ such that \[ \mathcal{C}({c'}_{ik}, x,y) = \det(A)^{p} \cdot \mathcal{C}(c_{ik},x,y), \] for some $p \in \mathbb{N}$. \end{definition}
The power of the coefficients $c_{ik}$  is called the degree of $\mathcal{C}$ the power of the variables $x,y$  is called the order of $\mathcal{C}$. Note that an invariant is a covariant of order zero.

Invariant Theory was studied by  a lot of great mathematicians in the late 1800s  and early 1900s like Paul Gordan, Felix Klein, Hermann Weyl, Bartel L.van der Waerden, George Boole, Arthur Cayley, David Hilbert, Emmy Noether to name just a few. There had been a ``British School,'' which Boole was part of, but later G\"ottingen became the ``world capital of invariant theory.''
 Paul Gordan was one of the main contributors to invariant theory. He proved the Finiteness Theorem for binary forms in 1868. In this proof Gordan constructed the invariant ring for binary forms and showed that it is finitely generated. People had tried to generalize this result, but nobody could successfully prove it until 1890 when Hilbert published his first proof of the Finiteness Theorem for general forms. It proved that there exists a finite basis for any system of invariants of arbitrary degree and with an arbitrary number of indeterminantes. Gordan did not like Hilbert's proof since it was an existence proof rather than a constructive proof. This is when Gordan said his famous, ``Das ist Theologie, nicht Mathematik,'' which means``This is Theology, not Mathematics.'' Later, Hilbert published a constructive proof of the Finiteness Theorem where he gave an algorithm on how to construct the invariant ring. Gordan was much happier with this proof, and began to see the advantages of the more theoretic approach. Hilbert's constructive proof essentially ``solved'' the problems in classical invariant theory by producsing an algorithm to find the invariants. 

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