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$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

Assume that $G$ is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm. We consider $G$ as a sample (probability) space with the normalized counting measure $\mu(A)=\frac{\lvert A\rvert}{\lvert G\rvert}$. So we consider $X$, $Y$ as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$? $$H=\bigl\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigl(\sum_g \phi(g)=0 \land \Cov(X,Y)=0 \bigr ) \bigr \}.$$

Remark 1: Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

Remark 2: What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?

Edit After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the following. In fact the implicit and initial goal of this post was:

Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelation property" (or/and "zero-mean property"). In particular for a finit or LCA group $G$, is the character group $\hat{G}$ maximal with respect to uncorrelation property $\Cov(X,Y)=0$ for all $\phi=X+iY$. In the other words: can one say that there is no any group $H$ with the following property: $\hat{G}\subsetneq H \subset \mathcal{W}$ such that for every $\phi=X+iY \in H$ we have $Cov(X,Y)=0$ where $\mathcal{W}=\{f:G\to \mathbb{T} \bigm|f\quad \text{is continuous}\}$

Please see this related post: https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

Assume that $G$ is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm. We consider $G$ as a sample (probability) space with the normalized counting measure $\mu(A)=\frac{\lvert A\rvert}{\lvert G\rvert}$. So we consider $X$, $Y$ as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$? $$H=\bigl\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigl(\sum_g \phi(g)=0 \land \Cov(X,Y)=0 \bigr ) \bigr \}.$$

Remark 1: Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

Remark 2: What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?

Edit After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the following. In fact the implicit and initial goal of this post was:

Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelation property" (or/and "zero-mean property"). In particular for a finit or LCA group $G$, is the character group $\hat{G}$ maximal with respect to uncorrelation property $\Cov(X,Y)=0$ for all $\phi=X+iY$. In the other words: can one say that there is no any group $H$ with the following property: $\hat{G}\subsetneq H \subset \mathcal{W}$ such that for every $\phi=X+iY \in H$ we have $Cov(X,Y)=0$ where $\mathcal{W}=\{f:G\to \mathbb{T} \bigm|f\quad \text{is continuous}\}$

Please see this related post: https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.

Remark 3: As a generalization of the first lines of this post, the motivation part, one can consider a finite dimensional representation of a group. Then one associate some pair of uncorrelated random variables to this representation. In the motivation part we associate the uncorrelated pair $(X,Y)$

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

Assume that $G$ is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm. We consider $G$ as a sample (probability) space with the normalized counting measure $\mu(A)=\frac{\lvert A\rvert}{\lvert G\rvert}$. So we consider $X$, $Y$ as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$? $$H=\bigl\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigl(\sum_g \phi(g)=0 \land \Cov(X,Y)=0 \bigr ) \bigr \}.$$

Remark 1: Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

Remark 2: What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?

Edit After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the following. In fact the implicit and initial goal of this post was:

Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelation property" (or/and "zero-mean property").

Please see this related post: https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

Assume that $G$ is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm. We consider $G$ as a sample (probability) space with the normalized counting measure $\mu(A)=\frac{\lvert A\rvert}{\lvert G\rvert}$. So we consider $X$, $Y$ as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$? $$H=\bigl\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigl(\sum_g \phi(g)=0 \land \Cov(X,Y)=0 \bigr ) \bigr \}.$$

Remark 1: Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

Remark 2: What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?

Edit After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the following. In fact the implicit and initial goal of this post was:

Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelation property" (or/and "zero-mean property"). In particular for a finit or LCA group $G$, is the character group $\hat{G}$ maximal with respect to uncorrelation property $\Cov(X,Y)=0$ for all $\phi=X+iY$. In the other words: can one say that there is no any group $H$ with the following property: $\hat{G}\subsetneq H \subset \mathcal{W}$ such that for every $\phi=X+iY \in H$ we have $Cov(X,Y)=0$ where $\mathcal{W}=\{f:G\to \mathbb{T} \bigm|f\quad \text{is continuous}\}$

Please see this related post: https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

Assume that $G$ is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm. We consider $G$ as a sample (probability) space with the normalized counting measure $\mu(A)=\frac{\lvert A\rvert}{\lvert G\rvert}$. So we consider $X$, $Y$ as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$? $$H=\bigl\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigl(\sum_g \phi(g)=0 \land \Cov(X,Y)=0 \bigr ) \bigr \}.$$

Remark 1: Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

Remark 2: What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?

Edit After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the following. In fact the implicit and initial goal of this post was:

Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelated" (or/and) "zero-mean property".

Please see this related post: https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.

$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random variables on $G$.

Based on this motivation we consider the following generalization:

Assume that $G$ is a finite group. Let $\phi=X+iY: G\to S^1$ be a complex function of unit norm. We consider $G$ as a sample (probability) space with the normalized counting measure $\mu(A)=\frac{\lvert A\rvert}{\lvert G\rvert}$. So we consider $X$, $Y$ as two random variables. Their covariance is denoted by $\Cov(X,Y)$.

Is the following set $H$ a group with the operation of pointwise multiplication $(\phi\cdot\psi)(x)=\phi(x)\cdot\psi(x)$? $$H=\bigl\{\phi=X+iY:G\to S^1 \bigm| ( \phi\equiv1) \lor \bigl(\sum_g \phi(g)=0 \land \Cov(X,Y)=0 \bigr ) \bigr \}.$$

Remark 1: Every character $\phi:G\to S^1=\mathbb{T}$ obviously belongs to $H$. So this question is an attempt to find a generalization of characters in group theory in the context of probability and statistics.

Remark 2: What about if we remove the condition $\sum_g \phi(g)=0$ from the definition of $H$ above?

Edit After the comment by Yemon Choi I realize the group structure of $G$ does not play a crucial role in the question. We merely use the counting measure or more generally the Haar measure. But we can consider this question as a possible search for the following. In fact the implicit and initial goal of this post was:

Some (maximal) group extension $H$ of character group whose elements satisfy "uncorrelation property" (or/and "zero-mean property").

Please see this related post: https://mathoverflow.net/questions/395254/complex-multiplication-of-two-uncorrelated-pair-of-unit-norm.