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Algebra</a> <a href="/?q=in%3A234838" title="Articles in this Issue">320, No. 10, 3773-3810 (2008)</a>. </div> <div class="abstract">Let \(F_{r}\) be a free group of rank \(r\), and let \(\mathfrak{R}_{r} =\)Hom\(\left( F_{r},\text{SL}\left( 3,\mathbb{C}\right) \right) \cong \)SL\(\left( 3,\mathbb{C}\right) ^{\times r}.\) Then SL\(\left( 3,\mathbb{C} \right) \) acts on \(\mathfrak{R}_{r}\) by conjugation. For \(\rho\in \mathfrak{R}_{r}\) we let \(\left[ \rho\right] \) denote its orbit closure, and say \(\rho_{1}\sim\rho_{2}\) if \(\left[ \rho_{1}\right] \cap\left[ \rho_{2}\right] \) is nonempty. This is an equivalence relation, and we let \(\mathfrak{X}_{r}\) be the set of equivalence classes \(\overline{\left[ \rho\right] }.\)<br class="zbmathjax-paragraph">In the work under review, the authors give minimal generators of the coordinate ring of \(\mathfrak{X}_{r}.\) There are \(N_{r}=\left( r/240\right) \left( 396+65r^{2}-5r^{3}+19r^{4}+5r^{5}\right) \) such generators. An explicit theorem describes how \(\mathbb{C}\left[ \mathfrak{R}_{r} //\text{SL}\left( 3,\mathbb{C}\right) \right]\) is generated by invariants of the form tr\(\left( \pmb{\Pi}\right) ,\) where \(\pmb{\Pi}\) is a product of (up to six) words, obtained by replacing words in \(F_{r}\) with the corresponding product of generic matrices. For each \(\pmb{\Pi}\) the number of invariants in the generating set are given, for example the generating set has \(\binom{r}{1}\) invariants of the form tr\(\left( {\mathbf{X}}\right) ,\) \(\binom{r}{1}\) invariants of the form tr\(\left( {\mathbf{X}}^{-1}\right) ,\) and \(2\binom{r}{3}\) invariants of the form tr\(\left( {\mathbf{XYZ}}\right) .\) Note that not all products of up to six words need to be considered due to cyclic equivalence, e.g. tr\(\left( {\mathbf{XY}}^{-1}\right) =\)tr\(\left( {\mathbf{Y}}^{-1}{\mathbf{X}}\right) .\) There is a considerable amount of counting needed to prove this theorem &ndash; much of the latter half of the paper is devoted to this task. We have that \(\mathfrak{X}_{r}=\)Spec\(_{\max}\left( \mathbb{C}\left[ t_{1},\dots,t_{N_{r}}\right] /\mathcal{J}\right) \) for some ideal \(\mathcal{J}.\)<br class="zbmathjax-paragraph">Thus there is an affine embedding \(\mathfrak{X}_{r}\rightarrow\mathbb{C} ^{N_{r}},\) and \(N_{r}\) is the smallest exponent for which an embedding exists, giving the minimality. A minimal generating set for \(\mathbb{C}\left[ \mathfrak{X}_{r}\right] \) is thus given by a set of words \(\left\{ w_{1},\dots,w_{N_{r}}\right\} \) in \(F_{r}.\) Then the map \(\overline{\left[ \rho\right] }\mapsto\left( \text{tr}\left( \rho\left( w_{1}\right) \right) ,\dots,\text{tr}\left( \rho\left( w_{N_{r}}\right) \right) \right) \) provides the embedding. The choice of the \(w_{i}\) is not unique, even up to cyclic equivalence (for example tr\(\left( {\mathbf{UX}} ^{2}{\mathbf{V}}\right) =\)tr\(\left( {\mathbf{UX}}^{-1}{\mathbf{V}}\right) \)). However, formulas are given to change the variables and thus the coordinate system, corresponding to the induced polynomial mapping \(\mathbb{C}^{N_{r} }\rightarrow\mathbb{C}^{N_{r}.}.\)<div class="reviewer"> Reviewer:&nbsp;<a href="/authors/?q=rv%3A9334">Alan Koch (Decatur)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5380153">Cited in <strong>2</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A5380153">Cited in <strong>14</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A14L24" title="MSC2020">14L24</a> </td> <td class="space"> Geometric invariant theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Acharacter+variety">character variety</a>; <a href="/?q=ut%3Afree+group">free group</a>; <a href="/?q=ut%3Aminimal+generators">minimal generators</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">&times;</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1157.14030" data-ciurl="/ci/05380153" data-biburl="/bibtex/05380153.bib" data-amsurl="/amsrefs/05380153.bib" data-xmlurl="/xml/05380153.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05380153.pdf" title="Zbl 1157.14030 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jalgebra.2008.06.031" aria-label="DOI for “Minimal affine coordinates for SL\((3,\mathbb C)\) character varieties of free groups”" title="10.1016/j.jalgebra.2008.06.031">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/0709.4403"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Nagata, M., Invariants of a group in an affine ring, J. 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