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      <h1>schurBetti -- a hash table for Schur module decomposition of Veronese Betti tables</h1>
      <ul>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">schurBetti(d,n,b)</code></dd>
          </dl>
        </li>
        <li>Inputs:          <ul>
            <li><span><span class="tt">d</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, </span></li>
            <li><span><span class="tt">n</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, </span></li>
            <li><span><span class="tt">b</span>, <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, </span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span>a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, </span></li>
          </ul>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
        <div>
          <p>This function returns a hash table with the Schur functor decompositions of the syzygies of $\mathcal{O}(b)$ under the embedding by $\mathcal{O}(d)$. The keys of the hash table $H$ are pairs $(p,q)$ where $H#(p,q)$ gives the Schur functor decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$. $\mathcal{O}(b)$ the Schur functor decomposition as a list of tuples $(\{a_1,a_2,a_3\},b)$ where $\{a_1,a_2,a_3\}$ specifies the weight of the Schur functor and $m$ the multiplicity with which that particular Schur functor appears in the decomposition of $K_{p,q}(\mathbb{P}^n, d;b)$.</p>
          <p>Some tables are incomplete and we mark unknown entries with ({0,0,0},infinity).</p>
          <p></p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i1 : schurBetti(3,2,0)

o1 = HashTable{(0, 0) => {({0, 0, 0}, 1)}                                                                                                }
               (0, 1) => {}
               (0, 2) => {}
               (1, 0) => {}
               (1, 1) => {({4, 2, 0}, 1)}
               (1, 2) => {}
               (2, 0) => {}
               (2, 1) => {({6, 2, 1}, 1), ({5, 4, 0}, 1), ({5, 3, 1}, 1), ({4, 3, 2}, 1)}
               (2, 2) => {}
               (3, 0) => {}
               (3, 1) => {({7, 4, 1}, 1), ({7, 3, 2}, 1), ({6, 5, 1}, 1), ({6, 4, 2}, 1), ({6, 3, 3}, 1), ({5, 5, 2}, 1), ({5, 4, 3}, 1)}
               (3, 2) => {}
               (4, 0) => {}
               (4, 1) => {({8, 5, 2}, 1), ({8, 4, 3}, 1), ({7, 6, 2}, 1), ({7, 5, 3}, 1), ({7, 4, 4}, 1), ({6, 6, 3}, 1), ({6, 5, 4}, 1)}
               (4, 2) => {}
               (5, 0) => {}
               (5, 1) => {({9, 5, 4}, 1), ({8, 7, 3}, 1), ({8, 6, 4}, 1), ({7, 6, 5}, 1)}
               (5, 2) => {}
               (6, 0) => {}
               (6, 1) => {({9, 7, 5}, 1)}
               (6, 2) => {}
               (7, 0) => {}
               (7, 1) => {}
               (7, 2) => {({9, 9, 9}, 1)}

o1 : HashTable</code></pre>
            </td>
          </tr>
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      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use <span class="tt">schurBetti</span>:</h2>
          <ul>
            <li><kbd>schurBetti(ZZ,ZZ,ZZ)</kbd></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="a hash table for Schur module decomposition of Veronese Betti tables" href="_schur__Betti.html">schurBetti</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">SchurVeronese/doc.m2:180:0</span>.</p>
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