# Questions tagged [unitary-representations]

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### Irreducible group representation(algebraic and topological irreducibility)

In page 280 of "C^* algebra" by Dixmier, in the context of group representation, it is written 'We never encounter the concept of algebraic irreducibility except in finite dimensional ...
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### Bounding the dimensions of faithful representations of a quotient group

For $G$ a compact Lie group, let $\operatorname{mdfr}(G)$ be the minimum dimension of a faithful complex representation of $G$. Is there a bound on $\operatorname{mdfr}(N(H)/H)$ for $H$ a subgroup of ...
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### Questions on the group $\mathrm{GL}(H)$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\U{U}$Let $H$ be an infinite dimensional complex Hilbert space. Consider the group $\GL(H)$ of bounded invertible operators on $H$. Question 1. I've ...
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### Zero entropy and the Koopman representation

Let $T$ be a measure preserving bijection of a probability space $(X,\nu)$. Consider the Koopman representation of $\mathbb{Z}$ on $L^2(X,\nu)$ given by $[z.f](x) = f(T^{-z}(x))$. The question is: can ...
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$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$The compact simple Lie groups $\SO_8(\mathbb{R})$ and $\SO_9(\mathbb{R})$ both have rank 4. The group $$G=\SU_3 \times \SU_2 \times \... 2 votes 1 answer 279 views ### Continuity of left regular representation on space of continuous functions I am teaching a course on locally compact groups and their representation theory and I am at a point where I would like to introduce continuous representations as generally as possible and provide ... 4 votes 1 answer 740 views ### Find unitary transformation between two sets of matrices that represent group generators I have a set of matrices A_i that represent the generators of a finite group within a certain basis, and B_i represent the same operators in a different basis. How can I find a unitary ... 3 votes 0 answers 36 views ### Generating K-types of a (\mathfrak g,K)-module for K disconnected Let G be a real reductive Lie group, let K be a maximal compact subgroup of G, and let V be a (\mathfrak g,K)-module. For \sigma\in\widehat{K} we denote the \sigma-isotypic component of ... 4 votes 2 answers 218 views ### Schur positivity of a polynomial Suppose a polynomial of the form$$\prod_i^d \sum_j^p x_i^{f_j}$$clearly symmetric, where f_j\in \mathbb{N}. There is a way to find the set of f numbers such that this polynomial is Schur ... 1 vote 1 answer 241 views ### Understanding the regular representation of an LCA group as a 'direct integral' The reference for what I'm asking is page 107 from Folland's harmonic analysis. G is a locally compact abelian group with dual \hat{G}. Let H denote the Hilbert space L^2(G). I'm trying to ... 7 votes 1 answer 294 views ### Induction and restriction of unitary representations \DeclareMathOperator\Rep{Rep}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Res{Res}Given a locally compact group G and a closed subgroup H\subset G, let \Rep(G) and \Rep(H) denote their ... 3 votes 1 answer 244 views ### Eigenvalues of product of unitaries Consider d\times d unitary matrices U, \, V, \, W such that$$ W=UV. $$Suppose that the eigenvalues of U and V are (e^{i\theta_1},\cdots,e^{i\theta_d}) and (e^{i\phi_1},\cdots,e^{i\phi_d})... 1 vote 1 answer 463 views ### Haar measure coming from Pontryagin duality v/s Fourier inversion Not research but advertising this question from mse in case someone wants to answer. I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ... 1 vote 0 answers 156 views ### A p-adic analogue of a result due to Kirillov Let k be a non-Archimedean local field with char(k)=0. Let N be the group of k-rational points of a unipotent algebraic group defined over k. It is known that N is a locally compact and ... 1 vote 1 answer 124 views ### Induced representations: space of continuous functions on G to a Hilbert space This question was asked in math stack exchange but received no replies: https://math.stackexchange.com/questions/4000655/induced-representations-space-of-continuous-functions-on-g-to-a-hilbert-space ... 4 votes 1 answer 136 views ### Explicit example of an equivariant embedding of U(n)/( U(k) \times U(n-k)) into a finite dimensional U(n)-representation We know that if H is a closed subgroup of a compact Lie group G one can find a finite dimensional G-representation V and an element v_0 \in V such that \textrm{Stab}(v_0)= H. This gives a ... 9 votes 1 answer 207 views ### Kazhdan's property (T) for \tilde{C}_2-lattices It is known that higher rank lattices have property (T) and also that lattices on 2-dimensional Euclidean buildings have property (T) provided the thickness q+1 of the building is large enough (... 4 votes 0 answers 209 views ### Which tensor power of a given representation contains the trivial one? If R is an irreducible representation of a simple Lie-groups G I assume there is always a lowest integer n such that the tensor product representation R \otimes R \otimes \ldots \otimes R (n ... 7 votes 1 answer 147 views ### Unitary representation is strictly continuous Let G be a compact group and u: G \to B(H) be a strongly continuous unitary representation on the Hilbert space H. Then is u: G \to B(H) strictly continuous? That is, give B(H) the topology ... 2 votes 0 answers 151 views ### Explicit tensor product decomposition for the representations of PSL(2,q) \DeclareMathOperator\PSL{PSL}Let the type of the character theory of a finite group G be the list [[d_1,n_1], \dotsc, [d_k,n_k]] with 1=d_1 < \dotsb < d_k and n_i the number of ... 6 votes 3 answers 386 views ### When can an \mathfrak{S}_n-equivariant map be extended to an \textrm{O}(n)-equivariant map? The symmetric group \mathfrak{S}_n can be regarded as a subgroup of the orthogonal group \textrm{O}(n) via the permutation matrices. Let V be a finite dimensional \textrm{O}(n)-module and \... 4 votes 1 answer 249 views ### Properties of the spectrum of the Koopman representation Let G be a discrete countable infinite group acting on a compact metric space X via homeomorphisms preserving a probability measure \mu. A function \lambda\colon G\to \mathbb C is an ... 2 votes 0 answers 59 views ### Realization of limit of discrete series using Dirac operators I wonder if there is a geometric realization of limit of discrete series in the flavor of Atiyah-Schmid or Parthasarathy realizing discrete series using Dirac operators on G/K. I know you can see ... 3 votes 0 answers 239 views ### Kazhdan Property T of semisimple Lie groups I am reading the paper [Margulis, G. A.; Nevo, A.; Stein, E. M., Analogs of Wiener's ergodic theorems for semisimple Lie groups. II. Duke Math. J. 103 (2000), no. 2, 233–259] (MSN). I want to ... 2 votes 0 answers 94 views ### Examples of groups admitting a proper 1-cocyle for a bounded representation A representation \pi: G \to B(H) of a group G on a Hilbert space H is called bounded iff \sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty. A 1-cocycle with respect to the representation \... 4 votes 0 answers 114 views ### Systems of imprimitivity for irreducible subgroup of GU(n,q) My question is similar to this one but about finite field case. So, the set up is the following: Let G be GU_n(q) acting on unitary space (V, {\bf f}), where V=\mathbb{F}_{q^2}^n and {\bf f}... 5 votes 3 answers 367 views ### Parametrization of real-valued SU(N) I want to construct a SU(N) matrix V, with the following property: All the elements of the first row are given, i.e. V_{1,j}=a_i (with \sum_i a_i^2=1) All matrix elements are real, i.e. V_{i,... 3 votes 0 answers 167 views ### Baker–Campbell–Hausdorff formula for exponential of general Hermitian operators Let A and B be two anti-Hermitian operators on a finite-dimensional Hilbert space. BCH formula gives an explicit expression for e^A e^B as e^C=e^A e^B, for C in the Lie algebra generated by ... 1 vote 0 answers 114 views ### Commutation fo a self-adjoint operator with a unitary operator Let A be a selfadjoint bounded operator on a Hilbert space. Let M be another bounded selfadjoint operator. Let me assume the commutation property$$ [A, e^{iM}]=0. \tag 1$$Does (1) imply that$$...
Let $U$ be a $d\times d$ unitary matrix, and $U_{i,j}$ be its matrix elements. I am interested in the following quantity $$\int dU \max_j |U_{1,j}|^2 \ ,$$ where $dU$ is the uniform Haar measure over ...
I came across the following while doing some related proof; It seems easy to prove. $\quad$ We are in ${\mathbb{M}}_n(\mathbb{C})$, $n>1$: $1$) Given a unitary $n\times n$ matrix $U$, there is ...