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<p>
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This chapter is co-authored by Francesca Gandini, Sumner Strom, Al Ashir Intisar
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</p>
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</introduction>
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<!-- <section xml:id="sec-template">
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<title>Template </title>
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<p>
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This is the introduction to the template.
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</p>
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<subsection xml:id="subsec-template">
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<title>Template Subsection</title>
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<p>
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This is the template subsection.
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</p>
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</subsection>
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</section> -->
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<section xml:id="sec-notes">
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<title>Notes</title>
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<subsection xml:id="subsec-notes">
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<title>Notes Subsection</title>
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<p>
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To build in pretext use the template subsection and pretext environment features.
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Make sure that you are building and compiling from main using preview code chat.
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Additionally, when running if the compilation fails run the command "pretext build web" in terminal.
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Make sure that all of the xml have unique ids so that you dont run into compilation errors.
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Follow the other chapters to find out how things are used.
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Make sure that the sections are properly linked within the main.ptx. Additionally, use the subsections and sections for organization.
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If something doesnt display there is a paragraph line missing.
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</p>
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</subsection>
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</section>
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</introduction>
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<section xml:id="sec-m2-codespace">
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<title>Creating a M2 Codespace</title>
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</section>
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</section>
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<section xml:id="sec-invariantrings-theory">
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<title>InvariantRings Theory</title>
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<subsection xml:id="subsec-finite-matrix-groups">
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<title>Finite Matrix Groups</title>
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<p>Example:
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Consider <me>M = \begin{pmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{pmatrix} </me> and the vector <m>\bar x = \begin{pmatrix} x\\ y\\ \end{pmatrix}</m>
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This gives <m>M \bar x = \begin{bmatrix}
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x \\
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-y \\
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\end{bmatrix}</m>. Thus for the polynomial <m> f(\bar x) = f(\begin{bmatrix}
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x \\
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y \\
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\end{bmatrix}) = x+y</m> we have <m>f(M\bar x) = f(\begin{bmatrix}
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x \\
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-y \\
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\end{bmatrix})= x-y</m>.
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</p>
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<p>
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<definition><p> <m>G \leq GLm(\mathbb{K}), |G| &lt; \infty</m>, then <m>G</m> is a finite matrix group.
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(In other words if <m>G</m> is a group of actions under which <m>\mathbb{K}</m> remains invariant
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under then it is smaller or equal to the total amount of group actions <m>GLm(\mathbb{K})</m> that
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would keep the polynomial invariant. AND the <m>|G|</m> is finite then <m>G</m> is a finite matrix
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group?)
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</p></definition>
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</p>
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<p>
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NOTE: An action of a finite group <m>G \curvearrowright \mathbb{K}^n</m> given a realization of <m>G</m> as a finite matrix group. </p><p>
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Example: <m>\langle \begin{bmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{bmatrix} \rangle = \{ \begin{bmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{bmatrix},\begin{bmatrix}
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1 \amp 0 \\
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0 \amp 1 \\
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\end{bmatrix}\} \cong C_2</m>
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</p>
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</subsection>
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<subsection xml:id="subsec-invariant-rings">
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<title>Invariant Rings</title>
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<p>
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\textbf{Notation <m>\bar x = (x_1, x_2,..., x_n)</m>, with <m>R = \mathbb{K}[x_1,x_2,...,x_n]</m>}
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<definition> <p> <m>G</m> is a finite matrix group within <m>GLm(\mathbb{K})</m> when?
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<m>f\in \mathbb{K}[x_1,x_2,...,x_n]</m> is invariant under the action of <m>G</m> if and only if
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<m>f(A\bar x) = f(\bar x)</m>, <m>\forall A \in G</m>.
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</p></definition>
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</p><p>
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Ex. <m>f(\bar x)=x</m> and <m>f(\bar x) = x +y^2</m> in <m>\mathbb{K}[x_1,x_2,...,x_n]</m> is invariant under <m>C_2 = \langle\begin{bmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{bmatrix} \rangle</m>\\
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However <m>f(\bar x)=x+y</m> is not. What are others?
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</p>
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<p>
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<definition><p> <m>R^G : = \{f \in R \, | f(A\bar x) = f(\bar x), \forall A \in G\} \subseteq R</m>
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is the invariant ring for the action of <m>G</m>
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</p></definition>
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</p>
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<p>
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Show this is a subring.</p>
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<p> How does on find generators for <m>R^G</m>?</p>
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<p> Is <m>R^G</m> even finitely generated?</p>
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<p> Work through Hilbert's proof.
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</p>}
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</subsection>
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<subsection xml:id="subsec-reynolds-operator">
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<title>Reynolds Operator</title>
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<p>
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Idea: "Averaging" over the action of <m>G</m> we get an invariant
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</p>
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<p>
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<definition> <p> <m>R_G: R \xrightarrow{} R^G</m> <me>R_G(f) = \frac{1}{|G|} \sum_{A\in G} f(A \bar x) </me>
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Exercise: <m>R_G</m> has many nice properties? \textbf{WHAT MORE TO SAY HERE}</p></definition>
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</p>
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<p>Example: <m>C_2 = \langle\begin{bmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{bmatrix}\rangle</m> <me>R_G(x+y) = \frac{1}{2} ((x+y) + (x-y)) = x\in R^G</me>
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</p>
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</subsection>
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<subsection xml:id="subsec-noether-degree-bound">
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<title>Nöether Degree Bound(NDB)</title>
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<p>
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<theorem><p> (Noether): <me>R^G = \mathbb{K} [ R_G(\bar x^{\bar \beta}) | \; |\bar \beta| \leq |G|]</me>
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<m>\implies</m> NDB : The ring of invariants is generated in degrees <m>\leq |G|</m>
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</p></theorem>
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</p>
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<p>
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Note: This is a computational tool! We can apply <m>R_G</m> to all the finitely many monomials in degrees <m>\leq |G|</m> to get generators for <m>R^G</m>.
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Exercise: Try this for <m>C_4</m> ... show!
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</p>
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</subsection>
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<subsection xml:id="subsec-hilbert-ideal">
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<title>Hildbert Ideal</title>
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<p>
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Note: In general for <m>\{ f_1,..., f_s\} \subseteq \R</m>,\\ <m>\{f_1,...f_s\}</m> and <m>\R</m> can be quite different objects
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Exercise?
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</p>
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<p>
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<theorem><p> Let <m>J_G = R(R^G)_t</m>, ideal generated by all positive degree invariants.\\
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If <m>J_G = (f_1,...,f_s)</m> and <m>f_i\in R^G, \,\, \forall i</m>
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(apply <m>R^G</m> if it is not), then <m>R^G = \mathbb{K}[f_1,...f_s]</m>
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</p></theorem>
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</p>
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</subsection>
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<subsection xml:id="subsec-presentations">
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<title>Presentations</title>
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<p>
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<definition><p>Definition: Let <m>S = \mathbb{K}[f_1,...f_s] \subset R</m>.
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A presentation of <m>S</m> is a map, <me>T=: \mathbb{K}[u_1,...u_s] \xrightarrow{\phi}S</me>
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such that <m>\frac{T}{ker(\phi)} \cong S</m> With the syzygies of <m>f_i</m>'s
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giving the presentation ideal.
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</p></definition>
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</p>
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<p>
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<proposition><p>(Elimination Theory): In <m>S \bigotimes \mathbb{K}[u_1,...,u_s = \mathbb{K}[x_1,...,x_n,u_1,...u_s]</m> consider the ideal,
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<me>I = (u_i - f_x(\bar x) | \, \langle f_i\rangle = S</me>
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Then,
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<me>ker \phi = I \cap \mathbb{K}[u_1,...,u_s]</me>
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</p></proposition>
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</p>
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<p>
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<algorithm><p> compute a Groebner Basis <m>G</m> for <m>I</m> with elimination order for the <m>x</m>'s.\\
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Then, <m>G \cap \mathbb{K}[y_1,...y_s]</m> is the Groebner Basis for <m>ker \phi</m>
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</p></algorithm>
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</p>
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</subsection>
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<subsection xml:id="subsec-graph-of-linear-actions">
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<title>Graph of Linear Actions</title>
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<p>
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<definition> <p>let <m>G \leq GL_n(\mathbb{K}), \,\, G\curvearrowright \mathbb{K}^n =:V, \,\, |G|\infty</m>.
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For <m>A\in G</m> consider,
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<me>V_A = \{(\bar v, A\bar v)|\,\,v\in V\} \subseteq V\bigotimes V</me>
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Then <m>A_G = \cup_{A\in G}V_A</m> is the subspace arrangement associated to the action of G.
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</p></definition>
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</p>
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<p>
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Note: <m>V_A</m> is a linear subspace, <m>\mathbb{I}(V_A):=</m> set of polynomials vanishing on <m>\mathbb{V}_A</m> is a linear ideal.
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Example: <me>V_{\begin{bmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{bmatrix}} = \{(x_1,x_2,x_1,-x_2)\} = V(y_1,-x_1, y_2+x_2)</me>
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</p>
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</subsection>
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<subsection xml:id="subsec-subspace-arrangement-approach">
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<title>Subspace Arrangement Approach</title>
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<p>
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<theorem><p>
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(Dekseu): Let <m>I_G = \mathbb{I}(A_G) = \cap_{A\in G}\mathbb{I}(V_A) \subseteq \mathbb{K}[x_1,...x_n,y_1,...y_n].</m>
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\textbf{(ARE n and n or m and n THESE DIFFERENT?)}\\
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Then <me>(I_G +(y_1,...,y_n)) \cap \R = J_G</me> This uses elimination theory and the Hilbert ideal.
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</p> </theorem>
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</p>
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<p>
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Note: The same approach works in the exterior algebra!
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</p>
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<p>
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<theorem><p>Let <m>I_G^{'} = \cap_{A\in G} \mathbb{I}(V_A) \subseteq \Lambda(\bar x, \bar y)</m>.\\
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Then <me>(I_G^{'} +(y_1,...y_n)) \cap \Lambda(x_1,...,x_n) = J_G^{'} : = \Lambda(\bar x)(\Lambda(\bar x)^G)_+</me>
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</p></theorem>
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</p>
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<p>
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Note: This approach is slow for polynomials, but might be fast for skew polynomials.
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</p>
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</subsection>
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<subsection xml:id="subsec-AGWM">
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<title>Abelian GPS and Weight Matrices</title>
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<p>
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Let <m>G \cong \mathbb{Z}_d, \bigoplus....\bigoplus \mathbb{Z}_{dr}, \,\,\,\,\, d_i|d_{i+1}</m> for <m>1 \leq i \leq r-1</m>\\
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<me>\langle g_1\rangle \bigoplus...\bigoplus\langle g_r \rangle, \,\,\,\,\, |g_i| =d_i</me>
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A diagonal action of <m>G</m> on <m>R</m> is given by
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<me>g_i \cdot x_j = \mu_i^{\omega ij}x_j</me>
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for <m> \mu_i : d_i^{th}</m> primitive root of unity and <m>i \in [x]</m>,<m>j \in [n]</m>.
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And encoded in the weight matrix <m>W = (\omega_{ij})_{ij} = ?????????????</m>
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</p>
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<p>
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<theorem><p> <m>\bar x^{\bar \beta} \in R^G \iff W_{\bar \beta}\cong (0,...,0)</m> for
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zeros being the weight of <m>g_1</m> acting on <m>\bar x^{\bar \beta}</m> and being modulo <m>d_i</m>.
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</p></theorem>
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</p>
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<p>
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Note: We can examine all monomials <m>|\bar \beta| \leq |G|</m> and sort them by their weight <m>W\bar \beta</m>.
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The ones with weight <m>\bar 0</m> will be invariant!
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</p>
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<p>
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Question: Does this work for monomials in the exterior algebra?
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</p>
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</subsection>
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</section>
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</chapter>

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