27
votes
Accepted
Is there a non-constant function on the sphere that diagonalizes all rotations simultaneously?
For $\mathrm{C}^1$-functions, the argument can be reduced to the circle case: Write the sphere as a union of circles meeting at the poles. On each circle the considered function is equivariant under ...
- 1,942
18
votes
Accepted
Finite-dimensional faithful unitary representations of SL(2,Z)
Here a non-explicit proof of the existence of a faithful representation of $\mathrm{SL}_2(\mathbf{Z})$ in $\mathrm{SU}(2)$, using basic algebraic geometry and topology, and relying on the amalgam ...
- 63.9k
12
votes
Good source for representation of GL(n) over finite fields?
All finite dimensional complex representations of finite groups are equivalent to unitary representations, so the requirement that the representations be unitary is not really a restriction.
The 1955 ...
- 44.4k
12
votes
Is there a non-constant function on the sphere that diagonalizes all rotations simultaneously?
For a non-abelian group $G$ acting on a space $X$, such a function would have to be invariant under the commutator subgroup $[G,G]$ of $G$, and thus constant on the orbits of the commutator subgroup.
...
- 149k
11
votes
Accepted
Average of the maximum matrix element over the Haar measure
The answer to the question as stated (maximum of row elements) has been solved in Extreme statistics of complex random and quantum chaotic states, see also this MO posting:
$$\int dU \max_j |U_{1,j}|...
- 188k
10
votes
Accepted
Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation
You can just use the embedding $f\colon G/H=U(n)/(U(k)\times U(n-k))\to M_n(\mathbb{C})$ given by $f(gH)=gpg^{-1}$, where $p=1_k\oplus 0_{n-k}$. This gives a homeomorphism from $G/H$ to the space
$$ ...
- 56.9k
8
votes
Good source for representation of GL(n) over finite fields?
Especially for combinatorialists, I found the book "Representations of finite classical groups. A Hopf algebra approach. Lecture Notes in Mathematics, 869" by Andrei Zelevinsky useful. It devolops a ...
- 15.5k
8
votes
Accepted
Uniform Roe algebra of virtually abelian group is type I C*-algebra?
It is not Type I in general. Probably it is not Type I whenever $G$ is infinite. Here is an argument when $G=\mathbb{Z}.$
Consider the projections in $\ell^\infty(\mathbb{Z})$ defined by ...
- 2,689
8
votes
Accepted
Kazhdan's property (T) for $\tilde{C}_2$-lattices
I don't have access to Zuk's note, but I remember finding an error in it when I read it (so this could be the same problem you found). He did improve on Garland in terms of thickness by taking average ...
- 1,163
8
votes
Accepted
Existence of 'maximal' finite permutation groups?
The standard representation of Sn+1 is faithful and n-dimensional. We may also assume it preserves a Hermitian inner product. When restricted to a standard copy of Sn, it becomes isomorphic to the ...
- 8,866
8
votes
Accepted
Is the left-regular representation of a locally compact group a homeomorphism onto its image?
Yes. It's more generally true for every faithful $C^0$ unitary representation $\pi$ of $G$. (Recall that a unitary representation $\pi$ is $C^0$ if for all $v,w$ in the Hilbert space, one has $\langle ...
- 63.9k
8
votes
Is there a non-constant function on the sphere that diagonalizes all rotations simultaneously?
As requested by the OP, I am making an answer from my comments to the question.
This is all standard material but I agree with him that it might be useful for somebody interested in this material.
...
- 18.4k
8
votes
Accepted
Irreducible unitary representation of PSL(2,Z)
The answer is No.
There is a mod-$p$ map $f:PSL(2,\mathbb Z) \rightarrow PSL(2,\mathbb F_p)$.
The permutation representation of $PSL(2,\mathbb F_p)$ on the projective line on $\mathbb F_p$ with $p+1$ ...
- 9,501
7
votes
Accepted
Definition of unitary representation of $\mathbf G(\mathbb A_k)$
It is absolutely essential that the space of (bounded/continuous) operators be given the "strong" operator topology (strictly weaker than the norm topology), and the map $G\times V\to V$ to be jointly ...
- 23k
7
votes
Accepted
Schur positivity of a polynomial
Given $f_1,\dots,f_p$ and $d\geq \max f_i$, a necessary and sufficient condition is that all zeros of the polynomial $\sum x^{f_j}$ are real. See Enumerative Combinatorics, vol. 2, Exercise 7.91. Note....
- 50.8k
7
votes
Accepted
Characterize this subspace of the bounded operators on $ L^2(\mathbb{R}) $
Any linear combination $L$ of $U_{a,b}$'s can be written $(L\psi)(x) = \sum_{k=1}^n \alpha_ke^{ib_kx}\psi^{\to a_k}(x)$, where $\psi^{\to a_k}(x) = \psi(x + a_k)$. Fix $L$.
Let $N \in \mathbb{N}$ be ...
- 42.8k
6
votes
Good source for representation of GL(n) over finite fields?
I would highly recommend "Complex Representations of GL(2,K) for Finite Fields K" by Piatetski-Shapiro. (I know you're interested in GL(n), but this book is a great place to start.)
- 11.1k
6
votes
Good source for representation of GL(n) over finite fields?
I think the standard reference for representations over finite fields still is
J. L. Alperin, "Local Representation Theory" (1986)
I you want a much briefer introduction, the last chapters of Serre'...
- 17.6k
6
votes
Systems of imprimitivity for unitary representations - reference request
This is not an answer, but here is another proof in the same spirit as yours.
Write $(-\vert -)$ for the canonical scalar product of ${\mathbb C}^d$.
Since the $G$-representation ${\mathbb C}^d$ is ...
- 6,349
6
votes
Accepted
Unitary dual of the motion group $M(n)$, for $n> 2$
This is done by Mackey theory and discussed in many places, e.g. see Lipsman (1974, page 72) for an explicit list. In short, there are two series:
Your sought $\chi$’s: all (finite dimensional) ...
- 31.5k
5
votes
Accepted
Is the linear span of irrep matrices a complete matrix basis?
This is known as Burnside's theorem. Nowadays people formulate it as any algebra of matrices over an algebraically closed field acting irreduciblly is the whole matrix algebra.
5
votes
Accepted
Unitary representation is strictly continuous
As you note, on bounded sets, the strict topology and the strong-$\ast$ topology agree on bounded sets. As the set of unitary operators is bounded, we can just work with the strong-$\ast$ topology. ...
- 18.7k
5
votes
Accepted
Are generalized symmetric groups maximal finite groups (in a certain sense)?
I think the answer is "yes" when $m >6$. By arguments along the lines of Frobenius, Schur and Blichfeldt, if we set $G = \langle M(m,n), U^{\prime} \rangle $ and assume that $G$ is finite,...
- 44.4k
5
votes
Book on Hilbert spaces, including non-separable
Halmos's Introduction to Hilbert space and the theory of spectral multiplicity is what you are looking for. Since it's published by Dover, the price is very reasonable.
4
votes
Accepted
Principal series representations of $SL(2,\mathbb{R})$: introductory textbooks
As an introduction I can recommand the following two books:
Taylor, Michael Eugene. Noncommutative harmonic analysis. No. 22. American Mathematical Soc., 1986.
In this book he discusses the unitary ...
- 126
4
votes
Is a reductive adelic group a Type I group?
Freitag and van Dijk proved that the adelic points of a reductive group over a global field is trace class (Theorem 2.3), while every trace class group is of type I (Theorem 1.7). So the answer is yes ...
- 105k
4
votes
Accepted
Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
You can find such a formula with indices here:
https://en.wikipedia.org/wiki/Wigner_D-matrix#Kronecker_product_of_Wigner_D-matrices,_Clebsch-Gordan_series
4
votes
Accepted
tensor product of massless Poincare representations
I think the answer is given in the paper https://aip.scitation.org/doi/10.1063/1.1703659 (Decomposition of Direct Products of Representations of the Inhomogeneous Lorentz Group, by J. S. Lomont).
- 16.5k
4
votes
Definition of unitary representation of $\mathbf G(\mathbb A_k)$
Considering a topological group $G$, a Hilbert space $V$ and a corresponding unitary representation, that is a homomorphism $\pi:G\to U(V)$, the following are equivalent:
$\pi$ is continuous when $U(...
- 11.6k
4
votes
Accepted
When can an $\mathfrak{S}_n$-equivariant map be extended to an $\textrm{O}(n)$-equivariant map?
If I have understood the problem correctly, the map $\Phi$ deterimines $\varphi = \Phi \circ d$, so the question amounts to classifying possible compositions $\Phi \circ d$, where $d$ is the "...
- 1,276
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