Modular representation theory | Wikipedia audio article
Channel: wikipedia tts
Date: 28/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Modular_representation_theory


00:01:40 1 History
00:02:17 2 Example
00:03:06 3 Ring theory interpretation
00:03:56 4 Brauer characters
00:05:34 5 Reduction (mod ip/i)
00:06:56 6 Number of simple modules
00:07:24 7 Blocks and the structure of the group algebra
00:08:57 8 Projective modules
00:10:54 9 Some orthogonality relations for Brauer characters
00:11:58 10 Decomposition matrix and Cartan matrix
00:13:22 11 Defect groups



Listening is a more natural way of learning, when compared to reading. Written language only began at around 3200 BC, but spoken language has existed long ago.

Learning by listening is a great way to:
- increases imagination and understanding
- improves your listening skills
- improves your own spoken accent
- learn while on the move
- reduce eye strain

Now learn the vast amount of general knowledge available on Wikipedia through audio (audio article). You could even learn subconsciously by playing the audio while you are sleeping! If you are planning to listen a lot, you could try using a bone conduction headphone, or a standard speaker instead of an earphone.

Listen on Google Assistant through Extra Audio:
https://assistant.google.com/services/invoke/uid/0000001a130b3f91
Other Wikipedia audio articles at:
https://www.youtube.com/results?search_query=wikipedia+tts
Upload your own Wikipedia articles through:
https://github.com/nodef/wikipedia-tts
Speaking Rate: 0.9017918738006209
Voice name: en-AU-Wavenet-D


"I cannot teach anybody anything, I can only make them think."
- Socrates


SUMMARY
=======
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive characteristic. As well as having applications to group theory, modular representations arise
naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.
Within finite group theory, character-theoretic results proved
by Richard Brauer using modular representation theory played
an important role in early progress towards the
classification of finite simple groups, especially for simple groups whose characterization was not amenable to purely group-theoretic methods because their Sylow 2 subgroups were too small in an appropriate sense. Also, a general result on embedding of elements of order 2 in finite groups called the Z* theorem, proved by George Glauberman using the theory developed by Brauer, was particularly useful in the classification program.
If the characteristic of K does not divide the order of the group, G, then modular representations are completely reducible, as with ordinary
(characteristic 0) representations, by virtue of  Maschke's theorem. The proof of Maschke's theorem relies on being able to divide by the group order, which is not meaningful when the order of G is divisible by the characteristic of  K. In that case, representations need not be
completely reducible, unlike the ordinary (and the coprime characteristic) case. Much of the discussion below implicitly assumes
that the field K is sufficiently large (for example, K algebraically closed suffices), otherwise some statements need refinement.