Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$.

Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$.

Assume $\mathbb{F}_{q}$ is a splitting field for $G$ where $q=p^f$ for some positive integer $f$.

Set $r:=|\text{IBr}(G)|$.

Let $\{\rho_1, ..., \rho_r\}$ be the a set of representatives of all simple $\mathbb{F}_{q}G$-modules up to isomorphism.

It is well-known that $\text{IBr}(G)$ is linearly independent.

Now, take the $\mathbb{F}_{q}G$-traces of $\rho_j$ at the $p'$-classes and write the results in a vector, for each $j$.

Does the list of vectors obtained in that way always have the property that there are no repeated rows?

Example:

Doing the computations for $G=A_5$, the alternating group acting on $5$ symbols, for the prime number $p=2$ yields the following:

$[ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ]$

$[ Z(2)^0, 0*Z(2), Z(2^2)^2, Z(2^2) ]$

$[ Z(2)^0, 0*Z(2), Z(2^2), Z(2^2)^2 ]$

$[ Z(2)^0, Z(2)^0, 0*Z(2), 0*Z(2) ]$

$[ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ]$

These vectors are linearly dependent, but still: the list of vectors obtained in that way has the property that no two rows are identical.

Is this always the case (for any finite group G and for any prime number p)?

A reference would be cool.

Let $G$ be a finite group and let $p$ be a prime number such that $p\mid |G|$.

Let $\text{IBr}(G)$ denote the set of irreducible Brauer characters of $G$ for the prime $p$.

Assume $\mathbb{F}_{q}$ is a splitting field for $G$ where $q=p^f$ for some positive integer $f$.

Set $r:=|\text{IBr}(G)|$.

Let $\{\rho_1, ..., \rho_r\}$ be the a set of representatives of all simple $\mathbb{F}_{q}G$-modules up to isomorphism.

It is well-known that $\text{IBr}(G)$ is linearly independent.

Now, take the $\mathbb{F}_{q}G$-traces of $\rho_j$ at the $p'$-classes and write the results in a vector, for each $j$.

Does the list of vectors obtained in that way always have the property that there are no repeated rows?

Example:

Doing the computations for $G=A_5$, the alternating group acting on $5$ symbols, for the prime number $p=2$ yields the following:

$[ Z(2)^0, Z(2)^0, Z(2)^0, Z(2)^0 ]$

$[ 0*Z(2), Z(2)^0, Z(2^2)^2, Z(2^2) ]$

$[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2 ]$

$[ 0*Z(2), Z(2)^0, Z(2)^0, Z(2)^0 ]$

The list of vectors obtained in that way has the property that no two rows are identical.

Is this always the case (for any finite group G and for any prime number p)?

A reference would be cool.