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1d
awarded  Inquisitive
2d
comment Is there such a notion of “expansion” in groups?
Ah - okay! - so thats what I am misreading - so can a finite group has such an exponential expansion property?
2d
comment Is there such a notion of “expansion” in groups?
@DerekHolt But doesn't the wiki link on growth rate (above) say that every free group of rank k > 1 has an exponential growth rate - this seems like a statement for finite groups - right?
2d
comment Is there such a notion of “expansion” in groups?
@Hanno Your link about "Growth Rate" seems to apply to finite groups - right?
2d
asked Is there such a notion of “expansion” in groups?
2d
comment Reference for the proof of interlacing of eigenvalues of submatrices
I can't see it there - any reference?
Jan
23
comment How does one see connectedness of a covering space?
Elencwaig Okay - something along these lines - that for a 2-covering there has to be a homomorphism from he first homology group to $\mathbb{Z}_2$ which satisfies some properties? Like that there should be at least one loop in the base space which is mapped to $-1$ by this? (for n-lifts I think something is required about the map from the first homology group to $\mathbb{Z}_n$ to be surjective?)
Jan
22
comment How does one see connectedness of a covering space?
I have heard stuff like, if you can find a loop in the base space such that it lifts to a loop in a non-trivial way then the cover is connected. Can be stated in some homology terms? Can you explain?
Jan
22
asked How does one see connectedness of a covering space?
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
Let me make my question precise. A 1-dimn representation of $\mathbb{Z}_k$ would be a homomorphism map $\rho : \mathbb{Z}_k \rightarrow GL(\mathbb{C})$ . Are you saying that there are exactly $k$ such maps each corresponding to multiplication by some root of unity?
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
This $\alpha_m$ is a character of which representation?
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
Thanks! Why are these maps $\alpha_m$ (in my notation) called "irreducible characters"? Whats the connection to representation theory here? Characters are trace functions - whats that here.. [...I guess you mean by non-canonical-ness that there is a $k!$ multiplicy of possibilities about the isomorphism map between the dual group of $\{\alpha_m\}$ and the group $\mathbb{Z}_k$ - right?..]
Jan
22
accepted Representation theory of $\mathbb{Z}_k$ and complex roots of unity
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
Now lets say you introduce $\phi_n = e^{\frac{2 \pi i n }{p} }$ as the $p$ $p^{th}$ roots of unity one for each $1\leq n \leq (p-1)$. And the corresponding $pk$ number of characters of $\mathbb{Z}_p \times \mathbb{Z}_k$ are given by the map $\alpha_{(m,n)} (i,j) \rightarrow \xi_m^i \xi_n^j$ - right? And again this set of maps $\{ \alpha_{(m,n)} \}$ form the Pontryagin dual of the group $\mathbb{Z}_k \times \mathbb{Z}_p$ - right?
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
But here we have an extra special feature that is group $\{\alpha_m\}$ (the Pontryagin dual?) is also canonically isomorphic to the original group $\mathbb{Z}_k$ via the map, $\alpha_m \rightarrow \xi_m$ - right?
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
Thanks! Do you have a typo about what is $\xi$? I would have thought that there are $k$ different choices of $\xi$ parametrized as $\xi_m = e^{\frac{2\pi i m }{k}}$ for $0\leq m \leq (k-1)$. Then for every choice of $m$ you have a map, $\alpha_m : \mathbb{Z}_k \rightarrow S^1$ as, $\alpha_m (p) = \xi_m^p$. So if these $\alpha_m$s are called ``irreducible characters" for $\mathbb{Z}_k$ then there are $k$ such irreducible characters - one for each $m$. Now this set of $\{\alpha_m\}$ forms a group on its own under pointwise multiplication on the elements of $\mathbb{Z}_k$.
Jan
22
comment Representation theory of $\mathbb{Z}_k$ and complex roots of unity
But how many irreducible representations are there of Z_k and are they somehow related/parametrized by the k complex roots of unity? Some mapping which can then later be generalized to other groups?
Jan
22
asked Representation theory of $\mathbb{Z}_k$ and complex roots of unity
Jan
18
accepted A question about minors of matrices
Jan
18
comment A question about minors of matrices
Thanks! Another variation of this which I need is the case of comparing $v'^T (B_{\bar{i} \bar{i}})^{-1} v'$ vs $v^T B^{-1} v$ and everything else remaining the same. You have any thoughts about it?