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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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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117 views

Problem concerning Haar measures on locally compact Hausdorff groups [migrated]

Before I state what my problem is I first want to give some context. A Haar measure is a measure on the Borel subsets of a locally compact Hausdorff group $X$. The Haar measure is inner regular on ...
somethingsomething69's user avatar
2 votes
1 answer
137 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
91 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
0 votes
0 answers
60 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
49 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
216 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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93 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
89 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
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2 votes
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57 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
67 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
222 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
234 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
107 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
77 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
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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
390 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
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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
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1 vote
1 answer
152 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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7 votes
0 answers
106 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 invariant? (And we can restrict to Moufang loops if ...
Harry Altman's user avatar
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20 votes
0 answers
274 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
211 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
95 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
134 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
148 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
215 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
136 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
428 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
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4 votes
1 answer
147 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
144 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
328 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
  • 701
5 votes
1 answer
236 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
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 ...
Claude's user avatar
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5 votes
0 answers
180 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
156 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
255 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
84 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
94 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
  • 670
2 votes
0 answers
125 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
  • 1,443
7 votes
1 answer
257 views

A group where the Weil topology induced by the Haar measure does not coincide with the original topology

Let $(G,\tau)$ be a locally compact Hausdorff topological group that is $\sigma$-finite with respect to the Haar measure $\mu:\mathcal{B}(G)\to[0,\infty]$ ($\mathcal{B}(G)$ is the Borel $\sigma$-...
Saúl Pilatowsky-Cameo's user avatar
1 vote
1 answer
284 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 ...
Andrew Musso's user avatar
3 votes
0 answers
108 views

Is this a lattice?

Let $R$ be a locally compact ring (commutative with unit) and let $D\subset R$ be a discrete cocompact subring (cocompact means the additive group $R/D$ is compact). Let $G$ be a semisimple linear ...
user avatar
0 votes
0 answers
45 views

How to prove invariance implies quadrature about Haar measure?

I'm reading paper, which contains a lemma named as 'Invariance Implies Quadrature'. The lemma is stated as follows. Lemma Let $f$ be a function from $O(d)$ to $\mathbb{R}$ and let $H$ be a finite ...
Nate's user avatar
  • 131
6 votes
1 answer
205 views

Comparing $X+Y$ and $X-Y$ for independent random variables with values in an abelian locally compact group

Let $G$ be an abelian locally (separable?) compact group with Haar measure $\mu$. Inspired by the interesting proof of A sum of two binomial random variables : Let $X$ and $Y$ be $G$-valued ...
Dieter Kadelka's user avatar
1 vote
0 answers
175 views

L2 norm of the diagonal entries of a random rotation of a fixed matrix?

Let $X\in\mathbb{R}^{d\times d}$ be the diagonal matrix with $d/2$ entries equal to $1$ and $d/2$ entries equal to $-1$. Let $F_U \triangleq \frac{1}{d}\|\operatorname{diag}(U^{\dagger}XU)\|^2_F$ ...
Sitan Chen's user avatar
3 votes
1 answer
260 views

Asymptotic analysis using the p-adic Mellin Transform?

In ordinary analysis, given a sufficiently nice $f:\left[0,\infty\right)\rightarrow\mathbb{C}$, if we can compute the Mellin transform: $$\mathscr{M}\left\{ f\right\} \left(s\right)=\int_{0}^{\infty}x^...
MCS's user avatar
  • 1,256
0 votes
0 answers
96 views

How large this subset is to say that it should equal the group?

Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set $$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
Meisam Soleimani Malekan's user avatar
4 votes
1 answer
305 views

Measure of subsets of profinite groups

Let $G$ be an infinite profinite group, so $$G=\lim_{\longleftarrow}G/N$$ where $N$ runs through the open normal subgroups. I have two questions: Is $G$ of Haar measure zero in the compact group $\...
Meisam Soleimani Malekan's user avatar
8 votes
2 answers
588 views

Average of the maximum matrix element over the Haar measure

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 ...
user149918's user avatar
2 votes
0 answers
58 views

Approximating a volume along a submersion

Here is the setup. I have a submersion $f:X \to T$, where $X$ is manifold and $T$ is a torus (you can chose a circle for the beginning if it is simpler). The manifold $X$ has a volume form $\alpha$ ...
Sovy's user avatar
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