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As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than $2$. The Heisenberg group is defined to be $$\left\{\left.\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix}\right|t\in\mathbb{F}_q,\mathbf{x},\mathbf{y}\in\mathbb{F}_q^n \right\}$$ with matrix multiplication. The way to classify them is to use the isomorphism $$\varphi:\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix} \to (t-\frac{1}{2}\mathbf{x}\cdot \mathbf{y},\mathbf{x},\mathbf{y}^T).$$ But in characteristic $2$, the $1/2$ does not make any sense. There's a way to treat the case here, but it seems hard to classify irreducible representation for the group. So my question is:

Are there any references for the classification of irreducible representation of characteristic $2$? Or can you give me some hints of how to do so?

Thanks for your help!

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than $2$. The Heisenberg group is defined to be $$\left\{\left.\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix}\right|t\in\mathbb{F}_q,\mathbf{x},\mathbf{y}\in\mathbb{F}_q^n \right\}$$ with matrix multiplication. The way to classify them is to use the isomorphism $$\varphi:\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix} \to (t-\frac{1}{2}\mathbf{x}\cdot \mathbf{y},\mathbf{x},\mathbf{y}^T).$$ But in characteristic $2$, the $1/2$ does not make any sense. There's a way to treat the case here, but it seems hard to classify irreducible representation for the group. So my question is:

Are there any references for the classification of irreducible representation of characteristic $2$? Or can you give me some hints of how to do so?

Thanks for your help!

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than $2$. The Heisenberg group is defined to be $$\left\{\left.\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix}\right|t\in\mathbb{F}_q,\mathbf{x},\mathbf{y}\in\mathbf{F}_q^n \right\}$$ with matrix multiplication. The way to classify them is to use the isomorphism $$\varphi:\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix} \to (t-\frac{1}{2}\mathbf{x}\cdot \mathbf{y},\mathbf{x},\mathbf{y}^T).$$ But in characteristic $2$, the $1/2$ does not make any sense. There's a way to treat the case here, but it seems hard to classify irreducible representation for the group. So my question is:

Are there any references for the classification of irreducible representation of characteristic $2$? Or can you give me some hints of how to do so?

Thanks for your help!

As we all know that the irreducible representation for Heisenberg group can be classified easily when the group is over a finite field $\mathbb{F}_q$, where $q=p^n$ and $p$ is a prime greater than $2$. The Heisenberg group is defined to be $$\left\{\left.\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix}\right|t\in\mathbb{F}_q,\mathbf{x},\mathbf{y}\in\mathbb{F}_q^n \right\}$$ with matrix multiplication. The way to classify them is to use the isomorphism $$\varphi:\begin{pmatrix}1 & \mathbf{x}& t\\0 & I_n &\mathbf{y} \\ 0 & 0 & 1\end{pmatrix} \to (t-\frac{1}{2}\mathbf{x}\cdot \mathbf{y},\mathbf{x},\mathbf{y}^T).$$ But in characteristic $2$, the $1/2$ does not make any sense. There's a way to treat the case here, but it seems hard to classify irreducible representation for the group. So my question is:

Are there any references for the classification of irreducible representation of characteristic $2$? Or can you give me some hints of how to do so?

Thanks for your help!