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      <h1>diagInvariants -- ring of invariants of a diagonalizable group action</h1>
      <ul>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">diagInvariants(T,U,R)</code></dd>
          </dl>
        </li>
        <li>Inputs:          <ul>
            <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose rows are the values of the indeterminates under the torus action</span></li>
            <li><span><span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose rows are the values of the indeterminates under action of the finite group</span></li>
            <li><span><span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>,  the basering</span></li>
          </ul>
        </li>
        <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:          <ul>
            <li><span><span class="tt">allComputations</span> (missing documentation)
<span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span>, <span></span></span></li>
            <li><span><span class="tt">grading</span> (missing documentation)
<span class="tt"> => </span><span class="tt">...</span>, <span>default value {}</span>, <span></span></span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span>a <a title="class of monomial subalgebras" href="___Monomial__Subalgebra.html">monomial subalgebra</a></span>, the ring of invariants</span></li>
          </ul>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
        <p></p>
This function computes the ring of invariants of a diagonalizable group D = T\times G where T is a torus and G is a finite abelian group, both acting diagonally on the polynomial ring K[X_1,\ldots,X_n]. The group actions are specified by the input matrices M and N. The first matrix specifies the torus action, the second the action of the finite group. See <a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</a> or <a title="ring of invariants of a finite group action" href="_finite__Diag__Invariants.html">finiteDiagInvariants</a> for more detail. The output is the monomial subalgebra of invariants.        <p></p>
This method can be used with the options <a href="_all__Computations.html">allComputations</a> and <a href="_grading.html">grading</a>.        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i1 : R=QQ[x,y,z,w];</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : T=matrix({{-1,-1,2,0},{1,1,-2,-1}});

2       4
o2 : Matrix ZZ  &lt;-- ZZ</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : U=matrix{{1,1,1,1,5},{1,0,2,0,7}}

o3 = | 1 1 1 1 5 |
     | 1 0 2 0 7 |

2       5
o3 : Matrix ZZ  &lt;-- ZZ</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i4 : diagInvariants(T,U,R)

o4 = MonomialSubalgebra{cache => CacheTable{...1...}                                                                      }
                                        70 35     19 10   4 6 5   15 5 10   26 4 15   37 3 20   48 2 25   59   30   70 35
                        generators => {y  z  , x*y  z  , x y z , x  y z  , x  y z  , x  y z  , x  y z  , x  y*z  , x  z  }
                        ring => R

o4 : MonomialSubalgebra of R</code></pre>
            </td>
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      <div>
        <h2>See also</h2>
        <ul>
          <li><span><a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</a> -- ring of invariants of torus action</span></li>
          <li><span><a title="ring of invariants of a finite group action" href="_finite__Diag__Invariants.html">finiteDiagInvariants</a> -- ring of invariants of a finite group action</span></li>
        </ul>
      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use <span class="tt">diagInvariants</span>:</h2>
          <ul>
            <li><kbd>diagInvariants(Matrix,Matrix,Ring)</kbd></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="ring of invariants of a diagonalizable group action" href="_diag__Invariants.html">diagInvariants</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Normaliz.m2:1956:0</span>.</p>
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