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Let us write the Frobenius-Schur indicator of the representation $V$ with highest weight $\lambda$ as $\mathrm{FS}(\lambda)$. Let $R$ denote the root system, and let $\Pi=\{\alpha_1,\dots,\alpha_l\}$ …

$\newcommand\Alt{\bigwedge\nolimits}$Let $G=\operatorname{SL}(2,\Bbb C)$, and let $R$ denote the natural 2-dimensional representation of $G$ in ${\Bbb C}^2$. For an integer $p\ge 0$, write $R_p=S^p R$ …

Let $G$ be a compact (anisotropic) real algebraic group. Let $\rho\colon G\to {\rm GL}(n,{\mathbb{C}})$ be a complex linear (polynomial) representation of $G$. Following OP, we say that $\rho$ is pse …

I am looking for a reference where the relative root system, the relative system of simple roots, and parabolic $\Bbb R$-subgroups for the real algebraic group ${\rm SU}(p,q)$ are explicitly computed. …

$ \newcommand{\g}{{\mathfrak g}} \newcommand{\h}{{\mathfrak h}} \newcommand{\t}{{\mathfrak t}} \newcommand{\C}{{\mathbb C}} \newcommand{\Ad}{{\rm Ad}} \newcommand{\ad}{{\rm ad}} $Theorem. Let $G$ be …

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$EDITED Let $G=\SL_{n,{\mathbb{C}}}$, the special linear group over ${\mathbb{C}}$. Let $H\subset G$ be a finite subgroup. Set $X=G/H$ be the c …

Let ${\frak g}$ be a simple Lie algebra over $\Bbb C$, and let $\theta$ be an inner involution of ${\frak g}$, that is, an inner automorphism of ${\frak g}$ of order dividing 2. Such automorphisms are …

As I have written in a comment, the answer is YES (any abstract homomorphism into an abelian group is trivial) when $G$ is an isotropic, simply connected, simple algebraic group over a nonarchmedean l …

A subgroup of maximal rank of maximal dimension is certainly a maximal subgroup of maximal rank. Maximal connected subgroups of maximal rank in $Spin(n)$ correspond to maximal reductive Lie subalgebra …

A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, Corollary 3.11. In positive characteristic $p$: see loc. cit., Theorem 3.14. It says that if (and only if) $ …

If you are interested in algebraic groups over complex and real numbers only, try Onishchik and Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag 1990. This book contains also representation t …

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. M …

Let $O$ denote the division algebra of octonions over $\Bbb R$, and write $V$ for the 7-dimensional quotient space $O/{\Bbb R}$. The compact group $G_2:={\rm Aut}(O)$ naturally acts on $V$, and clearl …

You can find the decomposition of $S^kV$ and $\Lambda^k V$ into a direct sum of irreducible representations in Table 5 in the book: Onishchik and Vinberg, Lie Groups and Algebraic Groups, Springer-Ver …

Let $G$ be a (connected) semisimple algebraic group over an algebraically closed field $k$ of characteristic 0. We consider the adjoint representation $$ {\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$ …