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<div id="archival-notice">This blog has been archived.<br/>Visit my home page at <a href="https://zhimingwang.org">zhimingwang.org</a>.</div>
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<h1 class="article-title">Convolution of irreducible characters</h1>
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<time class="article-timestamp" datetime="2014-11-19T20:40:37-0800">November 19, 2014</time>
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<p><strong><em>TL; DR:</em> The actual PDF write-up is <a href="https://dl.bintray.com/zmwangx/generic/20141119-convolution-of-irreducible-characters.pdf">here</a>.</strong></p>
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<p>Yesterday I was trying to establish the formula for orthogonal primitive central idempotents of a group ring. It is possible to establish the result through the convolution of irreducible characters. However, I stuck quite a while on trying to work out the convolutions themselves. For a formidable and unenlightening proof using "matrix entry functions" (i.e., fix a basis, induce a matrix representation, and explicitly expand everything in matrix elements), see <a href="http://drexel28.wordpress.com/2011/03/02/representation-theory-using-orthogonality-relations-to-compute-convolutions-of-characters-and-matrix-entry-functions/">this post</a> (in fact, this is just one in a series of posts that lead up to the result). That's a really sad proof.</p>
<p>It turns out that I really should have been working the other way round — first establish the orthogonal idempotents (the proof of which is really simple and elegant, I was just trapped in a single thread of thought), then use that to compute the convolution of irreducible characters.</p>
<p>I feel like this is worth presenting (as the only proof I saw online is the really sad one above), so I TeX'ed it up. I tried to convert to MathJax HTML but eventually gave up (that's the story for another post). So, the write-up is in good ol' PDF, available <a href="https://dl.bintray.com/zmwangx/generic/20141119-convolution-of-irreducible-characters.pdf">here</a>.</p>
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