New answers tagged characters
0
votes
The number of the double cosets $K\backslash G/H$ equals $ \langle \pi_{G/K} , \pi_{G/H} \rangle$
Since we are finding the number of the $K$-orbits, then
\begin{equation}
\langle Res_K^G(\pi_{G/H}) , 1_K \rangle_K = \langle \pi_{G/H}, Ind_{K}^{G} 1_K\rangle_G = \langle \pi_{G/H}, \pi_{G/K} \...
- 55
0
votes
The number of the double cosets $K\backslash G/H$ equals $ \langle \pi_{G/K} , \pi_{G/H} \rangle$
We have an isomorphism $\mathbb C[K\backslash G] \otimes_{\mathbb C[G]} \mathbb C[G/H]\cong \mathbb C[K\backslash G/H]$ given by $e_{Kg}\otimes e_{g'H}\mapsto e_{K(gg')H}$ Since $\dim \mathbb C[K\...
- 25.1k
0
votes
Group representation over finite fields
Let $G$ be a finite group and let $L/K$ be an extension of finite fields and let $V$ be an absolutely irreducible $L[G]$-module and suppose that $K$ contains the character values of $V$. Let $\...
- 23.6k
1
vote
Prove that there exists a group $G$ of order 24 that is non-abelian and has a faithful two-dimensional complex representation.
The construction you give already gives the two-dimensional complex representation you want:
Your approach is to try to construct a subgroup of the non-zero quaternions of order $24$, starting with ...
- 6,612
0
votes
Accepted
Establishing a unique element in the proof that the dual of $\mathbb{R}$ as an additive group is $\mathbb{R}$.
$y$ exists unique because it's $1/(2\pi\epsilon)$ times the argument of a complex number whose real part is positive. Such an argument would lie in $[-\pi/2, \pi/2]$ modulo $2\pi$, hence the range $[-...
- 6,606
Top 50 recent answers are included