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o1 = F

o1 : PolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i2 : R = F[x_1..x_4]

o2 = R

o2 : PolynomialRing</code></pre>
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              <pre><code class="language-macaulay2">i3 : conj = map(F,F,{w^2})

o3 = map (F, F, {- w - 1})

o3 : RingMap F &lt;-- F</code></pre>
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              <pre><code class="language-macaulay2">i4 : X = character(R,4,hashTable {(1,{2}) => matrix{{1,1,w,w^2}}})

o4 = Character over R
      
     (1, {2}) => | 1 1 w -w-1 |

o4 : Character</code></pre>
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              <pre><code class="language-macaulay2">i5 : X' = dual(X,conj)

o5 = Character over R
      
     (-1, {-2}) => | 1 1 -w-1 w |

o5 : Character</code></pre>
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          <p>If working over coefficient fields of positive characteristic or if one wishes to avoid defining conjugation, one may replace the second argument by a list containing a permutation $\pi$ of the integers $1,\dots,r$, where $r$ is the number of conjugacy classes of the group. The permutation $\pi$ is defined as follows: if $g$ is an element of the $j$-th conjugacy class, then $g^{-1}$ is an element of the $\pi (j)$-th class.</p>
          <p>In the case of $A_4$, the identity and $(12)(34)$ are their own inverses, while $(123)^{-1} = (132)$. Therefore the permutation $\pi$ is the transposition exchanging 3 and 4. Hence the dual of the character in the example above may also be constructed as follows, with $\pi$ represented in one-line notation by a list passed as the second argument.</p>
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              <pre><code class="language-macaulay2">i6 : perm = {1,2,4,3}

o6 = {1, 2, 4, 3}

o6 : List</code></pre>
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              <pre><code class="language-macaulay2">i7 : dual(X,perm) === X'

o7 = true</code></pre>
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          <p>The page <a title="construct a character table" href="_character__Table.html">characterTable</a> contains some motivation for using conjugation or permutations of conjugacy classes when dealing with characters.</p>
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        <h2>See also</h2>
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          <li><span><a title="construct a character table" href="_character__Table.html">characterTable</a> -- construct a character table</span></li>
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          <h2>Ways to use <span class="tt">dual</span>:</h2>
          <ul>
            <li><span><kbd>dual(BettiTally)</kbd> -- see <span><a title="the class of all Betti tallies" href="../../Macaulay2Doc/html/___Betti__Tally.html">BettiTally</a> -- the class of all Betti tallies</span></span></li>
            <li><kbd>dual(Character,List)</kbd></li>
            <li><kbd>dual(Character,RingMap)</kbd></li>
            <li><span><a title="make the dual of a complex" href="../../Complexes/html/_dual_lp__Complex_rp.html">dual(Complex)</a> -- make the dual of a complex</span></li>
            <li><span><a title="the dual of a map of complexes" href="../../Complexes/html/_dual_lp__Complex__Map_rp.html">dual(ComplexMap)</a> -- the dual of a map of complexes</span></li>
            <li><span><a title="dual of a map" href="../../Macaulay2Doc/html/_dual_lp__Matrix_rp.html">dual(Matrix)</a> -- dual of a map</span></li>
            <li><span><a title="dual module" href="../../Macaulay2Doc/html/_dual_lp__Module_rp.html">dual(Module)</a> -- dual module</span></li>
            <li><span><a title="the Alexander dual of a monomial ideal" href="../../Macaulay2Doc/html/_dual_lp__Monomial__Ideal_rp.html">dual(MonomialIdeal)</a> -- the Alexander dual of a monomial ideal</span></li>
            <li><span><a title="the Alexander dual" href="../../Macaulay2Doc/html/_dual_lp__Monomial__Ideal_cm__List_rp.html">dual(MonomialIdeal,List)</a> -- the Alexander dual</span></li>
            <li><span><a title="the Alexander dual" href="../../Macaulay2Doc/html/_dual_lp__Monomial__Ideal_cm__Ring__Element_rp.html">dual(MonomialIdeal,RingElement)</a> -- the Alexander dual</span></li>
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          <h2>For the programmer</h2>
          <p>The object <a title="dual module or map" href="../../Macaulay2Doc/html/_dual.html">dual</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>
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        <div class="waystouse">
          <p>The source of this document is in <span class="tt">BettiCharacters.m2:2846:0</span>.</p>
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