<!doctype html>
<html lang="en">
    <head>
        <meta charset="utf-8">
        <meta http-equiv="x-ua-compatible" content="ie=edge">
        <meta name="viewport" content="width=device-width, initial-scale=1">
        <title>Math190-Tufts University - Representations and the symmetric group - Diaconis data</title>
	<script>
	  window.MathJax = {
	    options: {
	      ignoreHtmlClass: 'tex2jax_ignore',
	      processHtmlClass: 'tex2jax_process'
	    },
	    tex: {
	      autoload: {
		color: [],
		colorv2: ['color']
	      },
	      packages: {'[+]': ['noerrors']}
	    },
	    loader: {
	      load: ['input/asciimath', '[tex]/noerrors']
	    }
	  };
	</script>
	<script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" id="MathJax-script"></script>
        <link rel="stylesheet" href="../css/default.css" />
	<link rel="stylesheet" href="../css/haddock.css" />
	
	<link href="https://fonts.googleapis.com/css?family=Lato:400,700" type="text/css" rel="stylesheet" media="screen" />
	<link href="https://fonts.googleapis.com/css?family=Arimo:400,700" type="text/css" rel="stylesheet" media="screen" />
	<link href="https://fonts.googleapis.com/css?family=Roboto:400,700" type="text/css" rel="stylesheet" media="screen" />
	<link href="https://fonts.googleapis.com/css?family=Inconsolata:400,700" type="text/css" rel="stylesheet" media="screen" />
	<link href="https://use.fontawesome.com/releases/v5.6.3/css/all.css" type="text/css" rel="stylesheet" media="screen" integrity="sha384-UHRtZLI+pbxtHCWp1t77Bi1L4ZtiqrqD80Kn4Z8NTSRyMA2Fd33n5dQ8lWUE00s/" crossorigin="anonymous" />			
    </head>
    <body>
        <header>
          <div class="logo">
            <a href="../">
	      Math190 - Spring 2024 - Tufts Univ</a>
          </div>
          <nav>
            <a href="../about.html">About</a>
	    <a href="../course-pages/Math190--course-info.html">Info</a>	    
            <a href="../archive.html">Archive</a>
          </nav>
        </header>

        <main role="main">
            <h1>Representations and the symmetric group - Diaconis data</h1>
            <article>
    <section class="header">
        Posted on 2024-02-07
        
    </section>
    <section>
        <h1 id="character-table-of-s_5">character table of <span class="math inline">$S_5$</span></h1>
<p>Let’s use <code>GAP</code> to find the character table of <span class="math inline">$S_5$</span>.</p>
<pre><code>gap&gt; G:=SymmetricGroup(5);
Sym( [ 1 .. 5 ] )
gap&gt; tab:=CharacterTable(G);
CharacterTable( Sym( [ 1 .. 5 ] ) )
gap&gt; Display(tab);
CT1

     2  3  2  3  1  1  2  .
     3  1  1  .  1  1  .  .
     5  1  .  .  .  .  .  1

       1a 2a 2b 3a 6a 4a 5a
    2P 1a 1a 1a 3a 3a 2b 5a
    3P 1a 2a 2b 1a 2a 4a 5a
    5P 1a 2a 2b 3a 6a 4a 1a

X.1     1 -1  1  1 -1 -1  1
X.2     4 -2  .  1  1  . -1
X.3     5 -1  1 -1 -1  1  .
X.4     6  . -2  .  .  .  1
X.5     5  1  1 -1  1 -1  .
X.6     4  2  .  1 -1  . -1
X.7     1  1  1  1  1  1  1
gap&gt; </code></pre>
<h1 id="diaconis-example-survey-data">Diaconis example – survey data</h1>
<p>This data is taken from the paper <span class="citation" data-cites="diaconisGeneralizationSpectralAnalysis1989">(Diaconis 1989)</span></p>
<p>It describes 5,738 completed ballots rank-ordering 5 candidates.</p>
<p>View a rank-ordered ballot as an element of the symmetric group <span class="math inline">$S_5$</span>; we want
to study the <em>frequency function</em> <span class="math inline">$f$</span>.</p>
<h2 id="first-ranking-table">first ranking table</h2>
<p><img src="../course-assets/images/diaconis-1.png" /></p>
<h2 id="the-regular-representation">the regular representation</h2>
<p><img src="../course-assets/images/diaconis-2.png" /></p>
<p>This diagram shows the decomposition of the regular representation into
<em>isotypic components</em>.</p>
<p>Be careful: the notation Diaconis is using here does not match that
used by <code>GAP</code> above. For example, the representation Diaconis writes
as <span class="math inline">$V_3$</span> is the isotypic component determined by the irreducible
representation labeled <code>X.5</code> by <code>GAP</code>.</p>
<p>The second row reflects the decomposition of the frequency function <span class="math inline">$f$</span>. Namely,
write
<span class="math display">\[f = \sum_{i=1}^7 f_i \quad \text{with $f_i \in V_i$}.\]</span></p>
<p>The second row entries are the “sums of squares” <span class="math inline">$\langle f_i,f_i \rangle$</span>.</p>
<p>Remember that we can compute the <span class="math inline">$f_i$</span> using the <em>idempotents</em> in
<span class="math inline">$\mathbb{C}[G]$</span>.</p>
<p>For example,</p>
<p><span class="math display">\[f_1 = \dfrac{1}{5!} \sum_{\sigma \in S_5} \sigma.f\]</span></p>
<p>More generally, if <span class="math inline">$\chi_i$</span> denotes the character of the irreducible
representation <span class="math inline">$L_i$</span> with <span class="math inline">$V_i = \mathbb{C}[G]_{(L_i)}$</span> then
<span class="math display">\[f_i = \dfrac{1}{5!} \sum_{\sigma \in S_5} \chi_i(\sigma^{-1}) \sigma.f\]</span></p>
<p>Note that <span class="math inline">$\langle f_3,f_3 \rangle = 459$</span> is relatively large
(ignoring <span class="math inline">$\langle f_1,f_1 \rangle$</span> since <span class="math inline">$f_1$</span> is trivial).</p>
<h2 id="normalizing-the-first-order-data">normalizing the first-order data</h2>
<p><img src="../course-assets/images/diaconis-2.5.png" /></p>
<p>THe <span class="math inline">$i,j$</span> entry in this table is the number of votes ranking candidate <span class="math inline">$i$</span> in the <span class="math inline">$j$</span>-th position,
minus the sample size over 5.</p>
<p>In particular, rows and columns sum to 0.</p>
<p>This normalization can also be achieved as follows:</p>
<p>Let <span class="math inline">$f_2$</span> be the projection on <span class="math inline">$V_2$</span>, and consider the functions
<span class="math display">\[\sigma \mapsto \delta_{i,\sigma(j)}.\]</span></p>
<p>The <span class="math inline">$i,j$</span> entry of the preceding table is <span class="math inline">$\langle f_2 , \delta_{i,\sigma(j)} \rangle$</span></p>
<p><img src="../course-assets/images/diaconis-3.png" /></p>
<p>Interpretation in this last table:</p>
<p>Compute the projection <span class="math inline">$f_3$</span> of <span class="math inline">$f$</span> into the component <span class="math inline">$V_3$</span> of <span class="math inline">$M = \mathbb{C}[S_5]$</span>.</p>
<p>Now, consider the easily understood functions
<span class="math display">\[\sigma \mapsto \delta_{\{i,i'\},\{\sigma(j),\sigma(j')\}}\]</span>
in <span class="math inline">$\mathbb{C}[S_5]$</span> for distinct <span class="math inline">$i,i'$</span> and distinct <span class="math inline">$j,j'$</span>.</p>
<p>The space of these functions is a 100 dimensional subspace of <span class="math inline">$W$</span> <span class="math inline">$\mathbb{C}[G]$</span>.</p>
<p>The entries in the table are the <em>inner products</em>
<span class="math display">\[\langle f_3 , \delta_{\{i,i'\},\{\sigma(j),\sigma(j')\}} \rangle\]</span></p>
<h2 id="summary-observations">Summary observations</h2>
<p>The data were to elect a president for the American Psychological
Association. Candidates 1 and 3 were <code>clinicians</code> while candidates 4
and 5 were <code>academicians</code>, two groups within the association with
somewhat divergent perspectives.</p>
<p>In the second-order table, we see a preference for candidates 1 &amp; 3 witnessed
by the entry 376 corresponding to the entry for candidates <span class="math inline">$\{1,3\}$</span> and ranks <span class="math inline">$\{1,2\}$</span>.</p>
<p>And we see a (slightly smaller) preference for candidates 4 and 5 witnessed by the entry
296 corresponding to the entry for candidates <span class="math inline">$\{4,5\}$</span> and ranks <span class="math inline">$\{1,2\}$</span>.</p>
<h1 id="bibliography">Bibliography</h1>
<div class="refs">

</div>
<div id="refs" class="references csl-bib-body hanging-indent" role="list">
<div id="ref-diaconisGeneralizationSpectralAnalysis1989" class="csl-entry" role="listitem">
Diaconis, Persi. 1989. <span>“A <span>Generalization</span> of <span>Spectral Analysis</span> with <span>Application</span> to <span>Ranked Data</span>.”</span> <em>The Annals of Statistics</em> 17 (3): 949–79. <a href="https://www.jstor.org/stable/2241705">https://www.jstor.org/stable/2241705</a>.
</div>
</div>
    </section>
</article>

        </main>

        <footer>
	  <nav>
	    <a href="http://www.tufts.edu">
	      <i class="fas fa-chalkboard"></i>
	      Tufts
	    </a>	    
	    <a href="http://math.tufts.edu">
	      <i class="fas fa-chalkboard-teacher"></i>
	      Tufts Math
	    </a>
	    <a href="https://gmcninch-tufts.github.io/math">
	      <i class="fas fa-coffee"></i>
	      G McNinch
	    </a>
	    <a href="https://github.com/gmcninch-tufts">
	      <i class="fab fa-github"></i>
	      GitHub
	    </a>	    
	  </nav>
        </footer>
    </body>
</html>


Collaborative Calculation and Data Science

colby

sagemath

Product

Resources

Company