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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 …

The irreducible complex representations of the simply connected simple group $G=Sp_{r,{\mathbb C}}$ of type $C_r$, for $r>1$, of dimension $n<{\rm dim}\ G$ are listed in the paper of Andreev, Vinberg …

$ \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 …

This question is a duplicate of that 2010 MO question. I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$. Clearly, a …

This is a version of this question of Klim Efremenko. Let $r>2$ be a natural number, say $r=3$ or $r=10$. Let $G$ be a finite group and $\rho$ be an irreducible complex representation of $G$. We con …

Irreducible real representations of complex type of a compact group correspond to irreducible complex representations that do not admit an invariant bilinear form. Irreducible real representations of …

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 …

See Appendix A in M. Borovoi and D. A. Timashev, Galois cohomology and component group of a real reductive group.

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Aut{Aut}$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 …

First, let $A$ be a finitely generated free abelian group, and $s$ an automorphism of order $2$ of $A$. Set $G=\{1,s\}$. Then we know that $A$ is a sum of indecomposable $G$-lattices $A_i$, where eac …

$\newcommand\Alt{\bigwedge\nolimits}$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 $R_p=S^p R$ …

I answer the second question. First I classify the connected $\mathbb{R}$-subgroups of full (absolute) rank of the compact $\mathbb{R}$-group $SO(p,\mathbb{R})$ that are $\mathbb{R}$-irreducible in t …

See V.L. Popov, E.B. Vinberg, Invariant Theory, Encyclopaedia of Mathematical Sciences, Vol. 55, Algebraic Geometry IV, Springer-Verlag, Berlin, 1994. You can find the invariants of a binary quartic …

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 …

We need some notation. I write $V$ for your $\Bbb C^{2nN}$, $T$ for your $T^n$, and $\rho\colon T\to GL(V)$ for the representation of $T$ in $V$. I define the character $\chi_j$ of $T$ by $$\chi_j( …