Abstract: The Weil representation is relatively well understood for local fields or finite fields of odd characteristics. In characteristic two, the situation is quite different. In this talk, we will present three explicit constructions of the Weil representation for a finite field of characteristic two. The first one is related to Weil’s pseudosymplectic group and is valid for an arbitrary finite field of characteristic two. The two others aim to provide Weil representations related to the symplectic group for $\mathbb{F}_2$ and $\mathbb{Z}/4\mathbb{Z}$. We obtain explicit formulas for these representations, their characters and the associated cocycles. We examine the relations among these constructions, and show how they naturally arise in Quantum Information Theory. All along, we will illustrate our results with the example of a two-dimensional vector space over $\mathbb{F}_2$.
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