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5

Suppose $G$ is a non-abelian group of order $8$. If its irreps. are of dimensions $n_i$, then from the formula $\sum n_i^2 = 8$ (and remembering that we have at least one $n_i = 1$, since there is the trivial representation, but not all $n_i =1$, since $G$ is non-abelian), we find that the $n_i$ are equal to $1,1,1,1, 4$. The abelianization of $G$ is ...


4

Here are the 101* finite nonabelian groups with 16 conjugacy classes whose order is less than 2000. (*I don't remember if there are any exceptional orders left out of the census.) The names are those given by GAP's structure description, and have not been cleaned up for this answer since there are so many groups. SmallGroup(40,1) = C5 : C8 SmallGroup(40,5) ...


3

Here are the 18 isomorphism classes of finite groups with 8 conjugacy classes: SmallGroup( 20, 1) = $\operatorname{AGL}(1,5)$ SmallGroup( 20, 4) = $D_{10}$ SmallGroup( 24, 13) = $C_2 \times A_4$ SmallGroup( 26, 1) = $D_{13}$ SmallGroup( 48, 3) = $C_3 \ltimes (C_4 \times C_4)$ SmallGroup( 48, 28) = $\operatorname{SL}(2,3) \mathsf{Y} C_4$ SmallGroup( 48, ...


2

One thing I have noticed is that representation theorists love to be loose with notation and definitions (e.g. calling a $G$-invariant subspace a subrepresentation and the restriction of a representation to a $G$-invariant subspace a subrepresentation). It bothers me a lot. This is such an instance, where they have replaced $\rho_1\otimes\rho_2$ with ...


2

I'm assuming here that the question is to show that there does not exist a $W^{0}$ such that $$\mathbb{R}^{2} = W \oplus W^{0}$$ and $W^{0}$ is an invariant subspace of $\mathbb{R}^{2}$. If you did not meant that in the problem, I apologize. So suppose you have a complement $W^{0}$ which is also a subrepresentation of $\mathbb{R}^{2}$. Then, $W^{0}$ is one ...


1

Sym$^2(V)$ decomposes into one dimensional eigenspaces with respect to the given cartan subalgebra and has eigenvalues $$E = \{2L_2,2L_1,2L_3,L_1+L_2,L_2+L_3,L_1+L_3\}$$ The eigenvalues of Sym$^n(V)$ are $E_n = \{a_1+\cdots+a_n\;|\;a_1,\cdots,a_n\in E\}$. The number of times $\mathbb{C}$ appears is the difference between multiplicity of eigenvalue 0 and ...


1

All these computations are happening just on $GL_2(\mathcal O_F)$. (If you like, just extend the functions by zero from $GL_2(\mathcal O_F)$ to $GL_2(F)$.) Then, as the exercise hints, this is just the orthogonality relations for representations of the pro-finite group $GL_2(\mathcal O_F)$. (Note that since only finitely many reps. are involved, they are ...


1

Note that $Sym^2(V) \subset V \otimes V$. A basis of $V \otimes V$ is given by $$(e_1 \otimes e_1, e_1 \otimes e_2, e_2 \otimes e_1, e_2 \otimes e_2) = (f_1,f_2,f_3,f_4).$$ We have a linear map $T:V \otimes V \rightarrow V \otimes V$ given on generators by $$a \otimes b \rightarrow a \otimes b + b \otimes a$$ (so the image of a sum $a_1 \otimes b_1 + \dots ...


1

I do not think it is full. For example let $M = \mathtt{Com}$ the commutative operad defined by $M(n) = k$ (with the trivial $\mathbb{S}_n$ action) for all $n > 0$ and $M(0) = 0$. The associated Schur functor is the free symmetric algebra functor $F = \tilde{M}$. There is a natural transformation $\alpha : F \to F$, $\alpha_V : F(V) \to F(V)$ given by ...



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