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from now on let $\mathbb{C}$ denote the complex number field, $G$ a finite group , and $T$ be a irreducible representation of G over $\mathbb{C}$ whose character is $\chi$.

Frobenius - Schur indicator is given by

$ \mu = \frac{1}{|G|} \sum_{g\in G} \chi(g^2) $

if $\mu$ =1 ,then $\chi$ can be realized over the real number field $\mathbb{R}$. if $\mu$ =0 , then $\chi$ is not real. if $\mu$ =-1 , then $\chi$ is real, but $T$ cannot be realized over $\mathbb{R}$.

also we know there are three different division algebra (finite dimension) over $\mathbb{R}$ : $\mathbb{R}$ , $\mathbb{C}$ and $\mathbb{H}$ (Hamilton quternions) , so the ring of endomorphisms commuting with the group action can be isomorphic only to either the real numbers, or the complex numbers, or the quaternions.

So I believe there are some connections between the three possible values of $\mu$ and the three kind of division algebra over $\mathbb{R}$. But I do not know why. Can anyone explain the connections between them?

additionally, these three cases corresponde to where there is a symmetrc/skew symmetric $G$ -invariant bilinear form in V . Though I can go through the proofs, but I don't konw why the indicator can be controled by the existence of a bilinear form of a particular kind. Is there something essential lying behind?

So far we have

indicator--bilinear form--division algebra.

Also, for a given field $F$, and a division algebra $D$ over $F$, can we assign $D$ a “number invariant” $\mu_D$ such that $\mu_D$ completely determines $D$ (under isomorphism) ?

from now on let $\mathbb{C}$ denote the complex number field, $G$ a finite group , and $T$ be a irreducible representation of G over $\mathbb{C}$ whose character is $\chi$.

Frobenius - Schur indicator is given by

$ \mu = \frac{1}{|G|} \sum_{g\in G} \chi(g^2) $

if $\mu$ =1 ,then $\chi$ and $T$ can both be realized over the real number field $\mathbb{R}$. if $\mu$ =0 , then $\chi$ is not real. if $\mu$ =-1 , then $\chi$ is real, but $T$ cannot be realized over $\mathbb{R}$.

also we know there are three different division algebra (finite dimension) over $\mathbb{R}$ : $\mathbb{R}$ , $\mathbb{C}$ and $\mathbb{H}$ (Hamilton quternions) , so the ring of endomorphisms commuting with the group action can be isomorphic only to either the real numbers, or the complex numbers, or the quaternions.

So I believe there are some connections between the three possible values of $\mu$ and the three kind of division algebra over $\mathbb{R}$. But I do not know why. Can anyone explain the connections between them?

additionally, these three cases corresponde to where there is a symmetrc/skew symmetric $G$ -invariant bilinear form in V . Though I can go through the proofs, but I don't konw why the indicator can be controled by the existence of a bilinear form of a particular kind. Is there something essential lying behind?

So far we have

indicator--bilinear form--division algebra.

Also, for a given field $F$, and a division algebra $D$ over $F$, can we assign $D$ a “number invariant” $\mu_D$ such that $\mu_D$ completely determines $D$ (under isomorphism) ?

from now on let $\mathbb{C}$ denote the complex number field, $G$ a finite group , and $T$ be a irreducible representation of G over $\mathbb{C}$ whose character is $\chi$.

Frobenius - Schur indicator is given by

$ \mu = \frac{1}{|G|} \sum_{g\in G} \chi(g^2) $

if $\mu$ =1 ,then $\chi$ can be realized over the real number field $\mathbb{R}$. if $\mu$ =0 , then $\chi$ is not real. if $\mu$ =-1 , then $\chi$ is real, but $T$ cannot be realized over $\mathbb{R}$.

also we know there are three different division algebra (finite dimension) over $\mathbb{R}$ : $\mathbb{R}$ , $\mathbb{C}$ and $\mathbb{H}$ (Hamilton quternions) , so the ring of endomorphisms commuting with the group action can be isomorphic only to either the real numbers, or the complex numbers, or the quaternions.

So I believe there are some connections between the three possible values of $\mu$ and the three kind of division algebra over $\mathbb{R}$. But I do not know why. Can anyone explain the connections between them?

Also, for a given field $F$, and a division algebra $D$ over $F$, can we assign $D$ a “number invariant” $\mu_D$ such that $\mu_D$ completely determines $D$ (under isomorphism) ?

from now on let $\mathbb{C}$ denote the complex number field, $G$ a finite group , and $T$ be a irreducible representation of G over $\mathbb{C}$ whose character is $\chi$.

Frobenius - Schur indicator is given by

$ \mu = \frac{1}{|G|} \sum_{g\in G} \chi(g^2) $

if $\mu$ =1 ,then $\chi$ can be realized over the real number field $\mathbb{R}$. if $\mu$ =0 , then $\chi$ is not real. if $\mu$ =-1 , then $\chi$ is real, but $T$ cannot be realized over $\mathbb{R}$.

also we know there are three different division algebra (finite dimension) over $\mathbb{R}$ : $\mathbb{R}$ , $\mathbb{C}$ and $\mathbb{H}$ (Hamilton quternions) , so the ring of endomorphisms commuting with the group action can be isomorphic only to either the real numbers, or the complex numbers, or the quaternions.

So I believe there are some connections between the three possible values of $\mu$ and the three kind of division algebra over $\mathbb{R}$. But I do not know why. Can anyone explain the connections between them?

additionally, these three cases corresponde to where there is a symmetrc/skew symmetric $G$ -invariant bilinear form in V . Though I can go through the proofs, but I don't konw why the indicator can be controled by the existence of a bilinear form of a particular kind. Is there something essential lying behind?

So far we have

indicator--bilinear form--division algebra.

Also, for a given field $F$, and a division algebra $D$ over $F$, can we assign $D$ a “number invariant” $\mu_D$ such that $\mu_D$ completely determines $D$ (under isomorphism) ?

from now on let $\mathbb{C}$ denote the complex number field, $G$ a finite group , and $T$ be a irreducible representation of G over $\mathbb{C}$ whose character is $\chi$.

Frobenius - Schur indicator is given by

$ \mu = \frac{1}{|G|} \sum_{g\in G} \chi(g^2) $

if $\mu$ =1 ,then $\chi$ can be realized over the real number field $\mathbb{R}$. if $\mu$ =0 , then $\chi$ is not real. if $\mu$ =-1 , then $\chi$ is real, but $T$ cannot be realized over $\mathbb{R}$.

also we know there are three different division algebra over $\mathbb{R}$ : $\mathbb{R}$ , $\mathbb{C}$ and $\mathbb{H}$ (Hamilton quternions) , so the ring of endomorphisms commuting with the group action can be isomorphic only to either the real numbers, or the complex numbers, or the quaternions.

So I believe there are some connections between the three possible values of $\mu$ and the three kind of division algebra over $\mathbb{R}$. But I do not know why. Can anyone explain the connections between them?

Also, for a given field $F$, and a division algebra $D$ over $F$, can we assign $D$ a “number invariant” $\mu_D$ such that $\mu_D$ completely determines $D$ (under isomorphism) ?

from now on let $\mathbb{C}$ denote the complex number field, $G$ a finite group , and $T$ be a irreducible representation of G over $\mathbb{C}$ whose character is $\chi$.

Frobenius - Schur indicator is given by

$ \mu = \frac{1}{|G|} \sum_{g\in G} \chi(g^2) $

if $\mu$ =1 ,then $\chi$ can be realized over the real number field $\mathbb{R}$. if $\mu$ =0 , then $\chi$ is not real. if $\mu$ =-1 , then $\chi$ is real, but $T$ cannot be realized over $\mathbb{R}$.

also we know there are three different division algebra (finite dimension) over $\mathbb{R}$ : $\mathbb{R}$ , $\mathbb{C}$ and $\mathbb{H}$ (Hamilton quternions) , so the ring of endomorphisms commuting with the group action can be isomorphic only to either the real numbers, or the complex numbers, or the quaternions.

So I believe there are some connections between the three possible values of $\mu$ and the three kind of division algebra over $\mathbb{R}$. But I do not know why. Can anyone explain the connections between them?

Also, for a given field $F$, and a division algebra $D$ over $F$, can we assign $D$ a “number invariant” $\mu_D$ such that $\mu_D$ completely determines $D$ (under isomorphism) ?