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Let $\lambda$ and $\mu$ be partitions with at most $n$ parts. Write $\mu = 0^{n_0}1^{n_1}2^{n_2}...$ with $\sum_i n_i = n$, to mean $\mu$ has $n_0$ parts of size $0$, $n_1$ parts of size $1$, and so …

As for the existence question there is an easy way to see this must be the case: Consider the regular representation $k[G]$ there is a unique (up to a scalar) map from the trivial representation to t …

I am aware that classifying all $SL_2(\mathbb{Z})$ representations is more or less completely intractable, but I was wondering what is known about the following simpler question: How do I recognize wh …

Expanding on my comment, here's what the two steps look like combinatorially. Induction from $S_k$ to $B_k$ sends $S^\lambda$ to $\bigoplus_{\mu, \gamma} W(\mu,\gamma)^{\oplus c^\lambda_{\mu,\gamma}}$ …

If G is semisimple and simply connected and the functor is faithful, then Theorem 3.3.1 here: https://arxiv.org/pdf/1911.04303.pdf says the answer is yes. They show that any faithful symmetric tensor …

Let me denote $k = |\lambda|$. We have combinatorial descriptions of the decomposition of $(\mathbb{R}^n)^{\otimes k}$ as an $S_n$ representation - see here for example. We can think of this space as …

$\DeclareMathOperator\SL{SL}$I want to ask if there exists a version of the Jacobson–Morozov theorem for integer matrices. A first approximation would ask: given an integral unipotent matrix $m \in \S …

Well the Okounkov-Vershik approach is close to 20 years old now, and it has been fairly well adopted by the community at large. So I won't say it could never help with the Kronecker coefficient proble …

The branching rule for hook partitions is multiplicity free and pretty easy to describe: $$s_{(k,1^j)} = \sum_{i < k/2}o_{(k-2i,1^j)} + \sum_{i < k/2}o_{(k-2i-1,1^{j-1})}$$ That is: You either remove …

When $\ell = 1$ I know that the smallest non-trivial irreducible complex representations of $SL_n(\mathbb{Z}/p\mathbb{Z})$ has dimension $\frac{p^n - 1}{p-1} - 1$ (with maybe some exceptions for smal …

If you fix a prime $p$ then for $n$ sufficiently large these $n-1$ or $n-2$ dimensional representations are indeed of minimal dimension. For this I believe the right reference is "On the minimal dime …

Here are some of my favorites, some were mentioned in comments already. I'm not going to be too picky about the distinction between conjectures vs open questions. The Saxl Conjecture was already menti …