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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3 votes
1 answer
172 views

Anti-concentration of polynomials on Haar measure

Let $X\in\mathbb{R}^n$ follow the Haar measure (i.e. uniformly distributed over the unit sphere), and $P$ be a degree-$d$ polynomial such that $\mathrm{Var}[P(X)]=1$. Are there constants $c(n,d)>0$ ...
1 vote
1 answer
67 views

Calculating an integral involving Haar measure on orthogonal projections

Let $\sigma_{n, m}$ denote the uniform measure over matrices $U \in \mathbb{R}^{n \times m}$ satisfying $U^T U = I_m$. Let $1_k \in \mathbb{R}^k$ denote the vector with all entries equal to $1$. I am ...
2 votes
1 answer
157 views

Integral over the space of $p$-adic matrices

$\DeclareMathOperator\Mat{Mat}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers with a uniformizer $\pi$. Let $|\cdot|\colon \mathbb{F}\to \mathbb{R}$ be ...
6 votes
0 answers
87 views

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 $E' \subset E$...
0 votes
0 answers
44 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 ...
0 votes
0 answers
31 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 ...
1 vote
1 answer
391 views

Pushforward of measure under surjective map

Let $X, Y, Z$ be measurable spaces with measures $\mu_X, \mu_Y, \mu_Z$ respectively. Let $\pi_Y : Y \times Z \rightarrow Y$ be the projection on $Y$ and $\pi_Z : Y \times Z \rightarrow Z$ the ...
12 votes
2 answers
1k views

Entropy for Haar measure on $O(n)$

Let $G$ be a locally compact group. A measure $\mu$ is the right-Haar measure on $G$ if for every $g\in G$ and $E\subseteq G$ Borel set $\mu(Eg)=\mu(E)$. It is known that every locally compact group ...
6 votes
2 answers
276 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 ...
1 vote
1 answer
74 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.
3 votes
1 answer
121 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: \...
1 vote
0 answers
265 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 ...
-3 votes
1 answer
266 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 $d$-...
0 votes
1 answer
163 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} (...
0 votes
0 answers
70 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 ...
12 votes
0 answers
200 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 ...
3 votes
2 answers
688 views

Measures and structure on conjugacy classes

Given a locally compact group $G$, does there exist a measure $\nu$ on the conjugacy classes $conj(G)$ such that for $f \in C_c(G)$ $$ \int_G f(g) d \mu_G(g) = \int_{conj(H)} \int_{G / G_\gamma} f(g\...
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 ...
3 votes
0 answers
112 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 ...
12 votes
3 answers
680 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 ...
3 votes
1 answer
124 views

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 ...
2 votes
1 answer
214 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 ...
6 votes
1 answer
240 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 $G(\mathbb{A})^1_\gamma \backslash G(\mathbb{A})^1$...
0 votes
0 answers
78 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 ...
2 votes
0 answers
109 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\...
0 votes
0 answers
116 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 ...
1 vote
1 answer
158 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 ...
2 votes
0 answers
91 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{...
1 vote
0 answers
167 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{\...
6 votes
0 answers
239 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,...
19 votes
9 answers
6k views

Haar measure on a quotient, References for

I remember reading Weil's "Basic Number Theory" and giving up after a while. Now I find myself thinking of it (thanks to some comments by Ben Linowitz). Right from the very beginning, Weil ...
1 vote
1 answer
131 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 ...
7 votes
4 answers
2k views

Measure on real Grassmannians

OK, so I'm reading about this nice measure you can define on a (real) Grassmannian on Wikipedia. Basically, and to save you the trip through the link, consider the Haar measure $\theta$ on $O(n)$, fix ...
2 votes
1 answer
334 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 $V$, $...
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}...
1 vote
1 answer
116 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 $...
0 votes
2 answers
418 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 \| ...
8 votes
2 answers
357 views

Extending the product measure on $2^\omega$

Consider the standard completed product measure $P$ on $\Omega=\{0,1\}^\omega$ corresponding to an i.i.d. sequence of fair coin-flips. Given $n\in\omega$, let $\rho_n$ be the bijection of $\Omega$ ...
2 votes
1 answer
141 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) ...
1 vote
1 answer
214 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 ...
20 votes
0 answers
301 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 ...
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 ...
4 votes
1 answer
139 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_{...
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 $G$ ...
0 votes
1 answer
182 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 ...
4 votes
1 answer
162 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 ...
4 votes
1 answer
304 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 ...
15 votes
1 answer
460 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 \...
19 votes
2 answers
1k views

Haar measures in Solovay's model

Haar measure is a measure on locally compact abelian groups which is invariants to translations. For example, the Lebesgue measure on the reals is such measure. It can be shown without the use of the ...
5 votes
1 answer
168 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 ...