On reduction modulo p of a group representation.
于靖教授台灣大學 & 中研院院士 下午 3:30–5:00 鴻經館 107 室Abstract：
Given a finite group G, any finite-dimensional complex representation is equivalent to a representation of G by matrices with entries from a number field (in fact a cyclotomic field). Thus we may consider reduction of modulo given prime number p of these matrices. Algebra tells us that the number of inequivalent irreducible complex representations of G is the same as the number of conjugacy classes inside G. On the other hand, with prime p fixed, the number of inequivalent mod p irreducible vrepresentations of G is the same as the number of p-regular conjugacy classes of G, i.e. conjugacy classes consisting of elements of order prime to p. When p is prime to |G|, every conjugacy class is p-regular, one has "good" reduction of all the complex representations of G. We are interested in the situation concerning possible "bad" reductions when p divides |G|. Usually given complex representation mod such p will no longer be "semi-simple". It cannot be decomposed into direct sum of irreducible subrepresentations. One has to consider the Jordan-Holder composition series of the mod p representation in question, the mod p invariants are the irreducible representations which come up as "simple" factors of the composition series. I will report on the developments originated from my fall 2016 group representation course, in particular experiences from using the online platform Overleaf collaborating with my students.