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authorZhiming Wang <zmwangx@gmail.com>2015-05-04 18:45:17 -0700
committerZhiming Wang <zmwangx@gmail.com>2015-05-04 18:45:17 -0700
commit07bf43a314fe65ccd9c7cb663c3c6134a47cc269 (patch)
tree93104d3f07143b1e62d773d95dd973888cdd7f27 /source/blog/2014-11-19-convolution-of-irreducible-characters.md
parentee374553f2ba385157eec9a816cf9b023fbfb18a (diff)
downloadmy_new_personal_website-07bf43a314fe65ccd9c7cb663c3c6134a47cc269.tar.xz
my_new_personal_website-07bf43a314fe65ccd9c7cb663c3c6134a47cc269.zip
edit posts and (mostly) figured out the theme
Also wrote pyblog that currently can generate parts most of the blog.
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@@ -9,6 +9,6 @@ __*TL; DR:* The actual PDF write-up is [here](https://dl.bintray.com/zmwangx/gen
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 [this post](http://drexel28.wordpress.com/2011/03/02/representation-theory-using-orthogonality-relations-to-compute-convolutions-of-characters-and-matrix-entry-functions/) (in fact, this is just one in a series of posts that lead up to the result). That's a really sad proof.
-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.
+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.
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 [here](https://dl.bintray.com/zmwangx/generic/20141119-convolution-of-irreducible-characters.pdf).