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If you like combinatorics, you may enjoy learning about the representations of $S_n$ by reading Chapter 7 of Stanley's Enumerative Combinatorics, Volume 2.

The space that you seek is the two-sided ideal in $\mathbb C[G]$ generated by the character of $\pi$ (see details below). This follows from the explicit Wedderburn decomposition of $\mathbb C[G]$. If …

Here is just one example (I know there are others too): Just as representations of the Hecke algebra associated to a Weyl group correspond to representations of a finite group of Lie type which are i …

If you take the space $X_\lambda$ of flags of shape $\lambda$ (here $\lambda$ is a partition of $n$, and a flag of shape $\lambda$ is one where the $i$th subspace has dimension $\lambda_1+\dotsc+\lamb …

Victor's answer shows that it is important to understand the action of $Out(G)$ on the conjugacy classes of $G$. This can be interesting even in the abelian case, where the problem amounts to calculat …

For a field, this is given by the rational canonical form (see Section 7.2 of Hoffman and Kunze's Linear Algebra, for example). Even in this case, the trace and characteristic polynomial are quite wea …

This answer essentially summarizes information from the other answers, hopefully, making the whole picture clear. For each self-conjugate partition $\lambda$ (i.e., $\lambda=\lambda'$) of $n$, the irr …

According to Djokovic and Maizan, the Specht module $V_{(3, 1, 1)}$ of $S_5$ is monomial. This is a representation of dimension $6$, induced from a representation of dimension $3$ of $A_5$. Since $A_5 …

If $m=1$ and $n>1$, then the decomposition is multiplicity-free and has $q$ irreducible representations. The way to see this is the following: The representation of $GL_n(\mathbf F_q)$ that you are lo …

This question got answered by Gjergji Zaimi and Richard Stanley in the comments. I simply reproduce their comments here as an answer: A very simple explanation for this identity comes from the theory …

The proof of Mackey's theorem on intertwiners actually tells you how to construct the endomorphism algebra of an induced representation, not just its dimension. So, if you work a little harder, you ma …

Firstly, about "known classes" of examples. Most obviously, if $X$ itself has irreducible characteristic polynomial, in which case it does not admit invariant subspaces. A slightly more interesting e …

To a finite $p$-group, we can associate two vectors $(v_0,v_1,\dotsc)$: The class vector - $v_i$ is the number of conjugacy classes of order $p^i$. The character vector - $v_i$ is the number of comp …

I found the paper of Chriss and Khuri-Makdisi (Chriss, Neil; Khuri-Makdisi, Kamal. On the Iwahori-Hecke algebra of a $p$-adic group. Internat. Math. Res. Notices 1998, no. 2, 85--100.) quite helpful. …

It is not difficult to show that the number of conjugacy classes in the alternating group $A_n$ is given by classes in the alternating group = no. of even partitions + no. of self-transpose partit …