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Linear representations of algebras and groups, Lie theory, associative algebras, multilinear algebra.
5
votes
representations of $S_k \times S_j$
If all representations of a group can be realized over $\mathbb{R}$, then $S(G, \mathbb{C})$ equals the number of elements of order 2 in $G$. More generally, the number of solutions of the equation $x …
5
votes
Extending the discussion on "super Catalan" numbers
Let $p\neq 3$ be a prime. Then
\begin{eqnarray*}
\nu_p\left(\frac{(3x)!}{x!^3}\right) & = & \sum_k \left[\frac{3x}{p^k}\right]-3\left[\frac{x}{p^k}\right]\\
& = & \sum_k 3\left\{\frac{x}{p^k}\right\} …
84
votes
Feit-Thompson conjecture
It is not true anymore that a proof of this conjecture would lead to significant simplifications. Peterfalvi proved in 1984 a weaker version of this conjecture, which suffices to get rid of the chapte …