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diff --git a/build/blog/2014-11-19-convolution-of-irreducible-characters.html b/build/blog/2014-11-19-convolution-of-irreducible-characters.html new file mode 100644 index 00000000..5b1a90f3 --- /dev/null +++ b/build/blog/2014-11-19-convolution-of-irreducible-characters.html @@ -0,0 +1,44 @@ +<!DOCTYPE html> +<html> +<head> +<meta charset="utf-8"/> +<meta content="pandoc" name="generator"/> +<meta content="Zhiming Wang" name="author"/> +<meta content="2014-11-19T20:40:37-0800" name="date"/> +<title>Convolution of irreducible characters</title> +<link href="/img/apple-touch-icon-152.png" rel="apple-touch-icon-precomposed"/> +<meta content="#FFFFFF" name="msapplication-TileColor"/> +<meta content="/img/favicon-144.png" name="msapplication-TileImage"/> +<meta content="width=device-width, initial-scale=1" name="viewport"/> +<link href="/css/normalize.min.css" media="all" rel="stylesheet" type="text/css"/> +<link href="/css/theme.css" media="all" rel="stylesheet" type="text/css"/> +</head> +<body> +<div id="archival-notice">This blog has been archived.<br/>Visit my home page at <a href="https://zhimingwang.org">zhimingwang.org</a>.</div> +<nav class="nav"> +<a class="nav-icon" href="/" title="Home"><!--blog icon--></a> +<a class="nav-title" href="/"><!--blog title--></a> +<a class="nav-author" href="https://github.com/zmwangx" target="_blank"><!--blog author--></a> +</nav> +<article class="content"> +<header class="article-header"> +<h1 class="article-title">Convolution of irreducible characters</h1> +<div class="article-metadata"> +<time class="article-timestamp" datetime="2014-11-19T20:40:37-0800">November 19, 2014</time> +</div> +</header> +<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> +<hr/> +<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> +</article> +<hr class="content-separator"/> +<footer class="footer"> +<span class="rfooter"> +<a class="rss-icon" href="/rss.xml" target="_blank" title="RSS feed"><!--RSS feed icon--></a><a class="atom-icon" href="/atom.xml" target="_blank" title="Atom feed"><!--Atom feed icon--></a><a class="cc-icon" href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Released under the Creative Commons Attribution 4.0 International license."><!--CC icon--></a> +<a href="https://github.com/zmwangx" target="_blank">Zhiming Wang</a> +</span> +</footer> +</body> +</html> |