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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 …

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 …

This is answered in Theorem 4 of my paper Comparison of Gelfand-Tsetlin Bases for Alternating and Symmetric Groups, with Geetha Thangavelu, which is published in Algebras and Representation Theory, an …

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 …

The answer is a special case of Young's rule. In my book, I give a very simple method for the slightly easier case where $r=0$. In that case we have: $$ \mathrm{Ind}_{S_k\times S_l}^{S_n} = \bigoplus_ …

These spaces are related to $2$-Sylow subgroups of $S_n$. For example, if $n=2^k$, then $X^n_{k+1}$ is the set of $2$-Sylow subgroups of $S_n$. To see this, note that $S_n$ acts transitively on $X^n_{ …

See Theorem 6.6.1 in Casselman's notes titled "Introduction to the theory of admissible representations of $p$-adic groups", available from https://www.math.ubc.ca/~cass/research/pdf/p-adic-book.pdf

Jeff Adams has comprehensive notes on his website: http://www.math.umd.edu/~jda/characters/characters.pdf The irreducible characters of the groups SL(2), PGL(2), GL(2) and PSL(2) over finite fields a …

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 …

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 …

$K\pi^\lambda K$ has a transitive right action of $K$. The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$. Thus, $K\pi^\lambda K = \coprod_x K\pi^\lambda x$ as $x$ …

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 …

A group is said to be a $Q$-group if the character of any complex representation is rational valued. A well-known internal characterization of $Q$-groups is the following: $G$ is a $Q$-group if an …

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 …

$\newcommand{\GFq}{\mathbf F_q}$ Let $r_n(q)$ denote the number of isomorphism classes of $n$-dimensional modules of the $\GFq$-algebra $\GFq[x,y]$. Is it known whether there exists a polynomial $p_n( …