Questions tagged [haar-measure]

Everything the deals with properties and definitions of Haar measure, as well as related fields when the question relies heavily on the notion of haar measure - group harmonic analysis, group ergodic theory etc.

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The uniform odd and even subgraph of $\mathbb{Z}^2$

Given a (first finite and later infinite) graph $G =(V,E)$ the uniform even graph is the uniform probability measure on the set of spanning even subgraphs. That is subgraphs (V, E') with $E' \subset E$...
Frederik Ravn Klausen's user avatar
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41 views

Conditional distributions of random orthogonal projection matrix

I have encountered a rather curious question. Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n$-dimensional ...
Landon Carter's user avatar
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29 views

Are closures of products of unimodular subgroups unimodular?

Let $G$ be a locally compact group, $N \subset G$ a unimodular normal subgroup, and $H \subset G$ a discrete (hence unimodular) subgroup. Does it follow that the closure $\overline{NH} \subset G$ is ...
Kim's user avatar
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1 vote
1 answer
72 views

Intergral over Haar random unitary

Given matrices $A_i$, and map $F(X)=A_nXA_{n-1}\cdots A_1$. How to compute $$\int_U F(U)\otimes F(U)^{*} dU$$ where $X^*$ denotes the complex conjugate transpose and $dU$ denotes the Haar measure.
gondolf's user avatar
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1 vote
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260 views

Using the von Neumann crossed product to introduce a measure on the orbit space?

Suppose we're given an action (possibly: ergodic) of a group G (say, $\mathbb{R}$) on a measure space $(X, \mu)$ (possibly: a standard probability space). Question: is there a natural way of using the ...
Stepan Plyushkin's user avatar
-3 votes
1 answer
245 views

On Haar measure and Spherical measure [closed]

Let $d$-dimensional complex sphere be $$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$ We can define the Haar measure on this sphere by regarding the unitary group $U(d)$. We can regard the $d$-...
gondolf's user avatar
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0 votes
1 answer
155 views

Action of the Haar measure on the Heisenberg group

The Heisenberg group is $\mathbb{H}^N=\mathbb{R}^{2N+1} = \left\{ (x,y, \tau ) \right\}\in \mathbb{R}^N \times \mathbb{R}^N \times \mathbb{R}$ equipped with the group operation \begin{equation} (...
Ilovemath's user avatar
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6 votes
2 answers
264 views

Uniqueness of left-invariant Borel probability measure on compact groups

On a compact topological group, consider two left-invariant probability measures $\mu$ and $\nu$ defined on the Borel sigma-algebra. Is it true that they coincide? It is classical that the Haar ...
Sebastien Gouezel's user avatar
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0 answers
69 views

Uniform distribution on pairs of unitary matrices

This question has two parts. In Part 1, I would like to know if the following distribution on pairs of $d$-dimensional unitary matrices has popped up in the literature: "Uniform distribution on ...
user3826143's user avatar
3 votes
1 answer
114 views

Existence of a bounded measurable subset of $\text{SL}(d,\mathbb R)$ that is Borel isomorphic to $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$?

$\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$ has two interesting properties: on one hand it is non-compact, but on the other hand it admits a unique $\text{SL}(d,\mathbb R)$-invariant finite ...
user506835's user avatar
3 votes
0 answers
110 views

Integrating over a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action and the choice of the fundamental domain

Let $\mathcal{F}$ be a fundamental domain in $\text{SL}(d,\mathbb R)$ under $\text{SL}(d,\mathbb Z)$ action. It is well known that there exists a unique $\text{SL}(d,\mathbb R)$-invariant probability ...
user506835's user avatar
12 votes
3 answers
668 views

If $A, B$ is a non-trivial partition of $S^1$, is it possible that $R_\theta(A) \cap B$ has measure zero for all rotations $R_\theta$?

This was previously posted to Math StackExchange. I was originally unsure whether it is suitable for posting here, but I've yet to get an answer there, so I'm just trying to see if people here can ...
David Gao's user avatar
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Asymptotics of Haar moments on general Lie groups

I am trying to understand the asymptotics of Haar Moments on general compact Lie groups (in particular, subgroups of $\mathrm{SU}(n)$). I have learned that closed form formulae for these moments are ...
dylan7's user avatar
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2 votes
1 answer
207 views

Haar measures of compact subgroups

Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$: $$ \mu_K(K)=1. $$ Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
Sergei Akbarov's user avatar
3 votes
1 answer
120 views

Maximum norm within a random subspace intersected with an ellipsoid

Let $d < n$, and let $G_n(d)$ denote the space of all $d$-dimensional subspaces of $\mathbb{R}^n$. Let $a = (a_1,\dots, a_n)$ denote a positive sequence, and define $U(a) = \{u \in \mathbb{R}^n: \...
Drew Brady's user avatar
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0 answers
76 views

Determinant of SU(N) elements, and radius of associated manifold

I'm wondering if the fact $SU(2)$ group elements have $det = 1$ is connected with the radius of the unitary $S^{3}$ manifold associated. The context is demonstration of dU being an Haar invariant ...
Matteo's user avatar
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2 votes
0 answers
92 views

Double quotient integral formula on $\Gamma \backslash G /K$

Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...
taylor's user avatar
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6 votes
1 answer
236 views

Divergence of integrals in the trace formula

I am trying to understand the following situation for $G=GL(2)$, when going from the compact trace formula to the non-compact case. The integral over $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
TheStudent's user avatar
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0 answers
111 views

Calculation of first correction to Selberg type integral

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\Tr{Tr}\DeclareMathOperator\arcsinh{arcsinh}$Let $U \in G$, where $G$ is $\SU(N)$ matrix. $\Tr U$ will denote the character ...
Sergii Voloshyn's user avatar
1 vote
1 answer
154 views

Is svd of a Gaussian iid matrix corresponds to Haar measure on the Stiefel manifold?

I want to draw a matrix $A\in \mathbb{R}^{n\times k}$ uniformally at random from the Stiefel manifold $\mathbb{V}_k(\mathbb{R}^n)$, that is from the collection of all $n\times k$ matrices $A$ such ...
Student88's user avatar
  • 503
2 votes
0 answers
84 views

Formulation of $p$-adic Haar measure decomposition

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\vol{vol}\DeclareMathOperator\diag{diag}$Suppose: $F$ is a non-archimedean local field, $\mathcal{O} \subset F$ its ring of integers, $\pi \in \mathcal{...
Maty Mangoo's user avatar
1 vote
0 answers
158 views

Integration with respect to Haar measures normalised over a subspace

Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work $\int_{\mathcal{U}(d)} \frac{\...
N A McMahon's user avatar
6 votes
0 answers
236 views

Direct proof that $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic

I wonder if there are any direct proof that $g_t=\operatorname{diag}(e^{t/m}I_m,e^{-t/n}I_n)$'s action on $\operatorname{SL}(d,\mathbb R)/{\operatorname{SL}(d,\mathbb Z)}$ is ergodic (or even stronger,...
No One's user avatar
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2 votes
1 answer
322 views

Question about squaring a Haar random unitary

Consider an $n \times n$ unitary $U$, drawn from the Haar measure. I'm trying to find the distribution for $U^{2}$. Is it true that $U^{2}$ is also Haar random? Note that for any fixed unitary $V$, $...
RandomMatrices's user avatar
1 vote
1 answer
125 views

Question about the inverse operator on PSL(2,R) with respect to Liouville measure

In GTM 259 chapter 9 and Katok Hasselbaltt Introduction to Modern Theory of Dynamical System chapter 5 (using the Iwasawa KAN decomposition) we see the Unit Tangent bundle of Hyperbolic half plane is ...
WaoaoaoTTTT's user avatar
2 votes
0 answers
80 views

Biased ensemble in the unitary group

I am interested in studying the ensemble of unitary random matrices in $U(L)$ made as follows $$ \mu(U)=\frac{1}{\mathcal{Z}[\omega]}\mu_{\rm Haar}(U) e^{-\sum_{k=1}^L \sum_{l=1}^N \omega_k |U_{kl}|^2}...
abenassen's user avatar
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1 vote
1 answer
115 views

Distribution of top left block from unitary symmetric matrices

If $U$ is a $N\times N$ random unitary matrix uniformly distributed with respect to Haar measure, a $M\times M$ block $A$ from it has distribution given by $$ \det(1-AA^\dagger)^{N-2M}.$$ If $O$ is a $...
Marcel's user avatar
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0 votes
2 answers
416 views

The (last step of the) proof that the set of badly approximable matrices has measure zero

An $m \times n$ matrix $A$ is called badly approximable if there exists $c > 0$ such that for all integer vectors $p \in \mathbb Z^m$ and $q \in \mathbb Z^n-\{0\}$ we have $$ \|A q + p \| \ge c \| ...
No One's user avatar
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2 votes
1 answer
138 views

Convergence of measure of products of random unitaries

I'm trying to read Convergence conditions for random quantum circuits by Emerson, Livine, Llyod (https://doi.org/10.1103/PhysRevA.72.060302), arXiv version: (https://arxiv.org/abs/quant-ph/0503210) ...
nervxxx's user avatar
  • 207
1 vote
1 answer
208 views

Integrals of the type $\delta(g^{n})$ on $\mathrm{SU}(2)$

I posted this question previously to MathSE. However, I have still not solved it, so lets try to ask it here. When doing some calculations with spin-foam models for 3d quantum gravity for some ...
B.Hueber's user avatar
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12 votes
0 answers
199 views

Do compact inverse-property loops (or just compact Moufang loops) have bi-invariant Haar measure?

So, the overall question is in the title: Does a compact topological loop with the inverse property have a Haar measure that is simultaneously left and right invariant? (And we can restrict to ...
Harry Altman's user avatar
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20 votes
0 answers
298 views

Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
Calculix's user avatar
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4 votes
2 answers
225 views

A 'projective' property of the Haar U(n) measure

Let $U(n)$ be the compact manifold of unitary $(n \times n)$-matrices and let $\mu_n$ denote the Haar-probability measure on $U(n)$. For $m < n$ does there exists a measurable (maybe even ...
Tardis's user avatar
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1 vote
0 answers
97 views

Only Zariski-closed subsets of compact Lie groups with nonempty interior have nonzero measure

In this question, the following fact was used by the respondent A Zariski-closed subset of a compact Lie group $G$ with nonzero Haar measure contains a coset of $G^0$, the connected component of $G$ ...
Meisam Soleimani Malekan's user avatar
4 votes
1 answer
138 views

Can the degree of $k$-nilpotence of a simple simply connected compact Lie group be in $(0,1)$?

Let $G$ be a simple (i.e. every proper normal subgroup is discrete) simply connected compact Lie group. Define the degree of $k$-nilpotence of $G$ to be the Haar measure of the set $$\{(x_1,\dotsc,x_{...
Meisam Soleimani Malekan's user avatar
0 votes
1 answer
177 views

When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist?

When does the measure integral of the form $\int_{\log(S)} f d \mu$ exist ? Here $\mu$ can be any measure (Lebesgue, Borel, Haar etc), $f$ is a measurable function, $S$ is any measurable set with ...
MAS's user avatar
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4 votes
1 answer
295 views

Haar integral of rational function of unitaries

I'm trying to compute the following Haar integral over the unitary group: $$ \int\limits_{\mathbb{U}(d)}\dfrac{1}{\sum_{k,l=1}^d u_{ik}\overline{u_{il}}c_{kl}}dU. $$ Is there anything known about the ...
TheBluegrassMathematician's user avatar
4 votes
1 answer
160 views

Invariant measure on affine charts of complex Grassmannian

Consider the complex Grassmannian $U(n)/U(k)\times U(n-k)$ with it's $U(n)$-invariant measure. The affine chart corresponding to $i_1, \ldots, i_k$ is given by $n\times k$ matrices for which the ...
Vít Tuček's user avatar
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15 votes
1 answer
455 views

For what LCH groups is the Haar measure $\mu(U x U)$ bounded?

Let $G$ be a locally compact Hausdorff (LCH) topological group with left Haar measure $\mu$. Given a compact unit neighborhood $U$, consider the function $$ \Phi: \quad G \to (0,\infty), \quad x \...
PhoemueX's user avatar
  • 754
5 votes
1 answer
167 views

Hopf algebra that is unimodular and counimodular but not involutory

I'm looking for a finite-dimensional Hopf algebra (over any field) that is unimodular, has unimodular dual, but is not involutory. Is there such a thing? Here's what I know: By Radford's formula, the ...
Tobias Fritz's user avatar
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2 votes
0 answers
148 views

About the probability of satisfying in a commutator type equation

Let $N$ be a closed normal subgroup of a compact group $G$. We denote the unique Haar measure of a compact group $A$ by $\mathbf m_A$, and drop $A$ if there exists no ambiguity. Fix $y\in G$. For $g\...
Meisam Soleimani Malekan's user avatar
4 votes
1 answer
481 views

Formula involving Wigner's 3j symbols and integration over irreducible representations of SU(2)

In some calculations, I saw the following formula $$\int_{\mathrm{SU}(2)}\,\mathrm{d}g\,D^{j_{1}}_{m_{1}n_{1}}(g)D^{j_{2}}_{m_{2}n_{2}}(g)D^{j_{3}}_{m_{3}n_{3}}(g)=(-1)^{j_{1}+j_{2}+j_{3}}\begin{...
B.Hueber's user avatar
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5 votes
1 answer
349 views

When does Haar measure decompose into products of such?

We get a "nice" Haar measure on $G=SL(2,R)$ in Iwasawa coordinates $G=NAK$ as follows: $dg=dx {dy\over y^2} dk$. Here $N=\{ n_x\}$, $A=\{a_y\}$ and $K=SO(2)$. Note that $dg=dn\, da\, dk$ is ...
Alex Kontorovich's user avatar
1 vote
1 answer
538 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 ...
Chertopkhanov on Malek Adel's user avatar
5 votes
0 answers
186 views

Haar mesure on $\mathrm{GL}_{d}(F)$

$\DeclareMathOperator\GL{GL}$Let $F$ be a $\mathfrak{p}$-adic field and $\mathscr{O}_{F}$ its valuation ring. For any measurable subset of $M_{d}(F)$ such as $$ A= \left( \begin{array}{ccc} a_{11}+t^{\...
M masa's user avatar
  • 479
0 votes
1 answer
162 views

Haar measure on ${\cal P}(\omega)$

First, we note that there is a natural bijection ${\cal P}(\omega) \to \{0,1\}^\omega$ and endow the latter with the product topology (where $\{0,1\}$ carries the discrete topology). So we get a ...
Dominic van der Zypen's user avatar
2 votes
2 answers
274 views

"Haar-like" measure on $S_\omega$

Let $S_\omega$ be the collection of bijections $f:\omega\to \omega$. Endow $\omega$ with the discrete topology and let $S_\omega$ be endowed with the subspace topology of $\omega^\omega$, where $\...
Dominic van der Zypen's user avatar
4 votes
0 answers
94 views

Dot product of functions on cosets

Some time ago I asked this same question at Math Stackexchange, because I thought that the question is nearly elementary. To my surprise, it was never answered. So I am elevating it to MathOverflow. I ...
Michael_1812's user avatar
1 vote
0 answers
110 views

Haar measure decomposition using orbital integrals

Let $G$ be a unimodular locally compact group, $N,A \le G$ be unimodular closed subgroups. Suppose that $A$ normalizes $N$. Let $N_0 \le N$ be a compact open subgroup. Suppose that a function $f : N \...
darkl's user avatar
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3 votes
0 answers
140 views

Haar measure and Integral

I am wondering whether the following integral over Haar measure has explicit form(edit: say $U$ is $d\times d$ unitary, orthogonal or symplectic matrix) $$ \int dU [(U\otimes U^*)X(U^{+}\otimes U^T)]^{...
gondolf's user avatar
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