https://www.chenk.top/en/tags/representation-theory/Recent content in Representation-Theory on Chen Kai BlogHugoenSun, 19 Sep 2021 09:00:00 +0000https://www.chenk.top/en/abstract-algebra/10-representation-theory/Sun, 19 Sep 2021 09:00:00 +0000https://www.chenk.top/en/abstract-algebra/10-representation-theory/<p><figure class="article-figure"> <img src="https://blog-pic-ck.oss-cn-beijing.aliyuncs.com/posts/en/abstract-algebra/figures/10_maschke.png" alt="Maschke&rsquo;s theorem: complete reducibility" loading="lazy" decoding="async" class="content-image"> </figure> </p> <p>An abstract group is a set with a binary operation satisfying certain axioms. This is elegant but sometimes hard to work with — how do you compute with elements of a group defined by generators and relations, or extract numerical invariants from a multiplication table? The solution, going back to Frobenius and Burnside over a century ago, is to represent group elements as matrices. Matrices are concrete: you can multiply them, take traces, compute determinants, decompose them into eigenspaces. Representation theory is the systematic study of this idea, and it has become one of the most powerful tools in modern algebra, number theory, and mathematical physics.</p>