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seen Oct 4 at 17:36

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awarded  Popular Question
Aug
18
comment The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$
What you say is not equivalent to what I say. What I say allows more functions. I think that the space contains functions without "radius of constancy". For example, for each $n$, choose one coset of $1+p^n\mathbb{Z}_p$, such that the chosen cosets are pairwise disjoint, then take the characteristic function of the union of these sets. I hope what you say is true and that I am wrong. I was hoping exactly for "radius of constancy".
Aug
16
asked Strong Approximation for adelic quaternionic groups
Aug
3
asked The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$
Jul
2
awarded  Curious
Jul
2
awarded  Inquisitive
Apr
25
asked The embedding $L^2(\Gamma(N)\backslash\text{SL}_2(\mathbb{R})) \hookrightarrow L^2(\text{SL}_2(\mathbb{Q})\backslash \text{SL}_2(\mathbb{A})))$?
Jan
2
comment irrep over $\mathbb R$
And by the way, if you write '@' before my name, I will get a notification that you're saying something to me and will see it more quickly.
Jan
2
comment irrep over $\mathbb R$
I don't exactly understand what you're saying there. Anyway, there are only $8$ real irreps of $C_2\times C_2\times C_2$, and they are all $1$-dimensional.
Jan
2
comment irrep over $\mathbb R$
I am glad to see you wrote your answer. Let's start with $C_2\times C_2\times C_2$. I suggest you start by finiding the complex irreps of this group first. You will see that there are $8$ of them, all one-dimensional, and in fact, they are real representations.
Jan
2
comment irrep over $\mathbb R$
You are welcome. You may consider writing an answer to your own question here on the website, if you have the time.
Jan
2
comment irrep over $\mathbb R$
As for other abelian groups of order $8$? What are all of these groups? Find all of them. One is $C_2\times C_2\times C_2$. For this particular one, start by finding its complex irreps.
Jan
2
comment irrep over $\mathbb R$
Ok. Great. Indeed, there are five nonequivalent irreps of $C_8$: Two one dimensional irreps, and three two dimensional irreps (given by rotation by $1/8,2/8,3/8$ times $2\pi$).
Jan
2
comment irrep over $\mathbb R$
I suggest you actually try to construct the irreps, and not use these counting considerations. You said correctly that the irreps are $1$ or $2$ dimensional. Try to construct a $2$ dimensional one. The vector space will be $\mathbb{R}^2$. Let $x$ be a generator of the cyclic group of order $8$. A representation is determined by the linear operator (=matrix) mapped to by $x$. You are not free to choose any opearator you like. For example, you must send $x$ to a matrix $A$ such that $A^8=I$ (why?). Do you know such matrices?
Jan
2
comment irrep over $\mathbb R$
This formula works for complex reps, but usually not for real reps. Here it doesn't work.
Jan
1
comment irrep over $\mathbb R$
The sum of dimensions is too large. Maybe some of your irreps are in fact isomorphic?
Jan
1
comment irrep over $\mathbb R$
(except for the trivial one)
Jan
1
comment irrep over $\mathbb R$
Try an example. Can you construct a specific real irrep of, say, the cyclic group of order $8$?
Dec
2
comment Limit of $\ln(1\cdot\ln(2\cdot\ln(3\cdot\ln(4\cdots))))$
@HaraldHanche-Olsen: You over-fixed it. You changed the * in the code too.
Nov
15
awarded  Yearling