A perspective on representations of finite groups

I shall start with some semi-philosophical thoughts on mathematics, and group representations. The new material in the talk is a very systematic construction of representations of a finite solvable group. Briefly, the method is as follows. Let $G$ be a finite solvable group of order $N$, where $N$ is a product of $n$, possibly repeating, primes. Then there exists a maximal subnormal series for $G$, that is, a sequence of subgroups $$ G_0 =\{e\}\lt G_1 \lt G_2 \lt\ldots\lt G_n =G, $$ $G_i$ is normal in $G_{i+1}$, of prime index. Let $x_i,\ i = 1, 2,\ldots, n$, be in $G_i$, whose class modulo $G_{i−1}$ generates $G_i/G_{i−1}$. Let $\rho$ be an (irreducible) representation of $G$. Then one successively constructs actual matrices $\rho(x_i)$, by using the method of induced representations from a subgroup. The method is a modification of the one used by Young who constructed matrices for representations of the symmetric group $S_n$, by using the sequence of subgroups $$ S_1 \lt S_2 \lt S_3 \lt\ldots\lt S_n. $$ But Young’s task was much more difficult since $S_n$, $n \gt 4$ is not solvable.

Ravi Kulkarni