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Since it's not clear, I assume the question you're asking is the following. "Fix an $n\geq2$. Which finite groups have a faithful real $n$-dimensional representation? Equivalently, what are the finite …

If you know $\chi$ then you can write down $\det \chi$ using Newton's identities. This is simply the observation that one can express the determinant of a matrix $\rho(g)$ in terms of traces of powers …

I don't think there's any clear cut relationship. In particular, the inequality $e \ge [\mathbb Q(\eta) : K]$ needn't hold. For example, take $G=S_5$, $N=A_5$ and let $\chi$ be the unique irreducible …

This follows from two facts: The complement $E_1^\perp$ of $E_1$ in $\tilde{E}$ is isomorphic to the contragredient of $E/E_1$. If $V$ is admissible and nonzero then $\tilde{V}$ is nonzero (and admi …

I'm reading Borel and Hirzebruch's "Characteristic Classes and Homogeneous Spaces, II" and at one point in the paper (Section 22.4) they obtain Weyl's dimension formula for an irrep of a semisimple co …

There's some terminology here that might be helpful for a literature search: $v$ is said to be semisimple if $Gv$ is closed in $V$ and said to be nilpotent if $v\neq0$ and $Gv$ is not closed in $V\set …

Here's another proof. If $G$ acts transitively on $X$, then the permutation representation of $G$ is induced from the permutation representation of the stabilizer $S$ of an arbitrary point. As a resu …

I'm very skeptical about the possibility of getting the full Borel–Weil–Bott theorem just by studying $G/U \to G/B$. Probably the closest thing I can think of is Bott's original proof of his theorem, …

Lemma 2.1.3 in Rudin's Fourier Analysis on Groups does this for locally compact abelian $G$ and closed $H \leq G$. This might not the best reference (may be a bit too general for a CS readership?), bu …

"Easiest" depends on how you set things up: everything really hinges on how you want to identify $X^\ast(T)$ with $\mathbb Z^n$. It's probably cleanest if you don't work explicitly with $\mathbb Z^n$, …

This is actually a fairly deep question. Your suspicion that there may be multiple answers is correct, but there might be some surprising connections between seemingly unrelated answers. Let me give o …

Actually, my question is a bit more specific: Does every complex semisimple Lie group $G$ admit a faithful finite-dimensional holomorphic representation? [As remarked by Brian Conrad, this is enough t …

I'm not aware of any generalization that is strong enough to compute the cohomology of such bundles completely, but you can at least use the Borel--Weil--Bott theorem to get some vanishing results. Th …

The whole flag variety is definitely not a single $O(n)$-orbit. You can see this already in the case of $n=2$, where the flag variety is just $\mathbb P^1$, and there are two orbits: $\{\pm i\}$ and e …

Is it true that the centralizer $Z_{\frak{g}}(\frak{a})=\{\xi\in\frak{g}:[\xi,\eta]= $0$ \text{ }\forall\eta\in\frak{a}\}$ is trivial (or equivalently, that the trivial one-dimensional representati …