The minimal dimension of a faithful linear representation of all finite subgroups of the two-dimensional Cremona group is 8 when the field contains a primitive cube root of unity, 6 otherwise, and infinite in positive characteristic.
Maximal Representation Dimensions of Algebraic Tori of Fixed Dimension Over Arbitrary Fields
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abstract
We define the representation dimension of an algebraic torus $T$ to be the minimal positive integer $r$ such that there exists a faithful embedding $T \hookrightarrow \operatorname{GL}_r$. Given a positive integer $n$, there exists a maximal representation dimension of all $n$-dimensional algebraic tori over all fields. In this paper, we use the theory of group actions on lattices to find lower bounds on this maximum for all $n$. Further, we find the exact maximum value for irreducible tori for all $n \in \left\lbrace 1, 2, \dots, 10, 11, 13, 17, 19, 23\right\rbrace$ and conjecturally infinitely many primes $n$.
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Representations of finite subgroups of Cremona groups
The minimal dimension of a faithful linear representation of all finite subgroups of the two-dimensional Cremona group is 8 when the field contains a primitive cube root of unity, 6 otherwise, and infinite in positive characteristic.