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Polishing under way, sections 1.1 and 1.3 ok, section 1.2 needs a bit more work
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<title>Invariant Theory</title>
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<introduction>
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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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This chapter is co-authored by Francesca Gandini, Sumner Strom, Al Ashir Intisar.
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In this chapter we will present an overview of the theory behind the algorithms implemented in the <url href="https://www.macaulay2.com/doc/Macaulay2/share/doc/Macaulay2/InvariantRing/html/index.html">InvariantRing</url> software package
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in the open-source Computer Algebra System <url href="http://www2.macaulay2.com">Macaulay2 (M2)</url>.
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</p>
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<p>
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You can access an online version of this chapter with live code cell at <url href="https://fragandi.github.io/CURITutorialDevelopment2025/"></url>.
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There you can also learn how to set up a virtual machine on Github with Codespaces so that you write and run M2 code from anywhere.
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A turn-key repository for creating a Codespace
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<!-- (<xref ref="sec-codespaces"/>) -->
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for Macaulay2 is available at
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<url href="https://github.com/fragandi/M2-codespace">fragandi/M2-codespace</url>.
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</p>
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<p>
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We also include some background on orbit sums necessary to implement an algorithm to compute invariants for permutations actions.
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We have worked with a group of collaborators on the first version of the code for this algorithm at the Macaulay2 Workshop at Tulane University in April 2025
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and plan to further test it and release it with Macaulay2 in Fall 2025.
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</p>
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<p>
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We finish the chapter with a selection of examples that illustrate the current capabilities of the InvariantRing package.
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You can run the provided code in your local installation of M2 or go to the online version and execute the code cells on your browser.
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Works well even on mobile devices!
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</p>
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</introduction>
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<section xml:id="sec-invariantrings-theory">
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<title>Invariant Theory</title>
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<subsection xml:id="subsec-introduction">
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<section xml:id="sec-invariantrings">
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<title>A concrete introduction to invariant rings</title>
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<!-- <subsection xml:id="subsec-introduction">
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<title>Introduction to invariant theory</title>
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<p>
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With our research we also had a large focus on making the theory behind invariant rings accessible
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within invariant rings.
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</p>
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</subsection>
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</subsection> -->
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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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<p>
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We can think of a (linear) action of a group on a vector space
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concretely by interpreting each group element as a matrix and describing the action as matrix multiplication on vectors.
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We can then evaluate any polynomial on a vector and its image after the action.
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</p>
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<example>
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<p>
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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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This gives <m>M \bar x = \begin{pmatrix}
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x \\
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-y \\
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\end{bmatrix}</m>. Thus for the polynomial <me> f(\bar x) = f(\begin{bmatrix}
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\end{pmatrix}</m>. Thus for the polynomial <me> f(\bar x) = f(\begin{pmatrix}
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x \\
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y \\
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\end{bmatrix}) = x+y</me> and we have, <me>f(M\bar x) = f(\begin{bmatrix}
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\end{pmatrix}) = x+y</me> and we have, <me>f(M\bar x) = f(\begin{pmatrix}
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x \\
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-y \\
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\end{bmatrix})= x-y</me>.
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\end{pmatrix})= x-y</me>.
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</p>
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<p>
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<definition><p> <m>G \leq GL_m(\mathbb{K}), |G| &lt; \infty</m>, then <m>G</m> is a finite matrix group.
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</example>
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<p>
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More formally, for <m> G </m> a finite group we will consider a linear representation of <m> G </m>
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via its action on a finite dimensional vector space <m>V </m> over a field <m>K </m> of characteristics zero.
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In general, most of the results in this chapter hold in the non-modular case
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i.e., when the characteristics of the field does not divide the order of the group.
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However finite fields are not fully supported by the current version of the InvariantRing package
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and the development of such functionalities is an active area of development.
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</p>
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<p>
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If <m> V </m> is faithful representation of <m> G </m> of dimension <m> m</m>, the image of the representation is isomorphic to <m> G </m>
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and so we consider <m> G </m> as a finite <term>matrix group</term>.
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</p>
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<definition><p> Suppose <m>|G| &lt; \infty</m> and <m>G \leq GL_m(\mathbb{K})</m>, then <m>G</m> is a finite matrix 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: <me>\left\langle \begin{bmatrix}
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<example>
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<p>
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Let us consider a two-dimesional representation of <m>C_2</m>, the cyclic group of order 2. <me>\left\langle \begin{pmatrix}
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1 \amp 0 \\
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0 \amp -1 \\
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\end{bmatrix} \right\rangle = \{ \begin{bmatrix}
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\end{pmatrix} \right\rangle = \left\{ \begin{pmatrix}
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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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\end{pmatrix},\begin{pmatrix}
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1 \amp 0 \\
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0 \amp 1 \\
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\end{bmatrix}\} \cong C_2</me>
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\end{pmatrix} \right \} \cong C_2</me>
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</p>
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</example>
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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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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>GL_m(\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> We will work with a polynomial ring in <m>n</m> variables over the field <m>\mathbb{K}</m>.
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We use the
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notation <m>\bar x = (x_1, x_2,..., x_n)</m> and abuse it by saying <m>\mathbb{K}[x_1,x_2,...,x_n]=\mathbb{K}[\bar x]</m> and
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<m>f(x_1,x_2,...,x_n)=f(\bar x)</m> for <m>f \in R = \mathbb{K}[\bar x]</m>.
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<definition> <p> Let <m>G</m> be a finite matrix group within <m>GL_n(\mathbb{K})</m>. We say that
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<m>f\in \mathbb{K}[\bar x]</m> is invariant under the action of <m>G</m> if and only if
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<me>
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f(A\bar x) = f(\bar x),
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</me>
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for all <m>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 <me>C_2 = \left\langle\begin{bmatrix}
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</p>
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</subsection>
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</section>
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<section xml:id="sec-degree-bounds-algorithms">
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<title>Degree bounds and algorithms</title>
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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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</subsection>
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</section>
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<subsection xml:id="subsec-references">
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<title>References</title>
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<p>
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Mara D. Neusel, Texas Tech University, Lubbock, TX. Publication: The Student Mathematical Library. Publication Year 2007: Volume 36
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. ISBNs: 978-0-8218-4132-7 (print); 978-1-4704-2147-2 (online)
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. DOI: http://dx.doi.org/10.1090/stml/036
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<section xml:id="sec-invariantrings-packages">
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<title>InvariantRings package </title>
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<p>
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We conclude with references to the algorithms implemented in the InvariantRing package and examples of its implementation.
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Version 2.0 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2023-09-14, in the article The InvariantRing package for Macaulay2
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(DOI: 10.2140/jsag.2024.14.5). That version can be obtained from the journal or from the Macaulay2 source code repository.
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</p>
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<subsection xml:id="subsec-references">
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<title>References for the implemented algorithms</title>
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<ul>
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<li>
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<p> An elimination theory algorithm that computes the Hilbert ideal for any linearly reductive group: Derksen, H. and Kemper, G. (2015). Computational Invariant Theory. Heidelberg: Springer. Algorithm 4.1.9, pp 159-164
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</p>
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</li>
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<li>
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<p>A simple and efficient algorithm for invariants of tori based on: Derksen, H. and Kemper, G. (2015). Computational Invariant Theory. Heidelberg: Springer. Algorithm 4.3.1 pp 174-177
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</p>
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</li>
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<li>
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<p>An adaptation of the tori algorithm for invariants of finite abelian groups based on: Gandini, F. Ideals of Subspace Arrangements. Thesis (Ph.D.)-University of Michigan. 2019. ISBN: 978-1392-76291-2. pp 29-34.
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</p>
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<p> An elimination theory algorithm that computes the Hilbert ideal for any linearly reductive group: Derksen, H. and Kemper, G. (2015). Computational Invariant Theory. Heidelberg: Springer. Algorithm 4.1.9, pp 159-164
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</p><p>A simple and efficient algorithm for invariants of tori based on: Derksen, H. and Kemper, G. (2015). Computational Invariant Theory. Heidelberg: Springer. Algorithm 4.3.1 pp 174-177
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</p><p>An adaptation of the tori algorithm for invariants of finite abelian groups based on: Gandini, F. Ideals of Subspace Arrangements. Thesis (Ph.D.)-University of Michigan. 2019. ISBN: 978-1392-76291-2. pp 29-34.
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</p><p>King's algorithm and the linear algebra method for invariants of finite groups: Derksen, H. and Kemper, G. (2015). Computational Invariant Theory. Heidelberg: Springer. Algorithm 3.8.2, pp 107-109; pp 72-74
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</p><p>The algorithms for primary and secondary invariants, and Molien series of finite groups implemented in version 1.1.0 of this package by: Hawes, T. Computing the invariant ring of a finite group. JSAG, Vol. 5 (2013). pp 15-19. DOI: 10.2140/jsag.2013.5.15
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</li>
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<li>
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<p>King's algorithm and the linear algebra method for invariants of finite groups: Derksen, H. and Kemper, G. (2015). Computational Invariant Theory. Heidelberg: Springer. Algorithm 3.8.2, pp 107-109; pp 72-74
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</p>
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</li>
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<li>
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<p>The algorithms for primary and secondary invariants, and Molien series of finite groups implemented in version 1.1.0 of this package by: Hawes, T. Computing the invariant ring of a finite group. JSAG, Vol. 5 (2013). pp 15-19. DOI: 10.2140/jsag.2013.5.15
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</p>
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</li>
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<li>
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<p>
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The orbit sum approach is under development following Mara D. Neusel, Texas Tech University, Lubbock, TX. Publication: The Student Mathematical Library. Publication Year 2007: Volume 36,
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DOI: <url href="http://dx.doi.org/10.1090/stml/036"/>
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</p>
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</li>
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</ul>
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</section>
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<section xml:id="sec-invariantrings-packages">
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<title>InvariantRings package </title>
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<section xml:id="sec-invariantring-demoss">
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<subsection xml:id="sec-invariantring-demos">
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<subsubsection>
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