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Do we already know the classification of the finite-dimensional irreducible unitary representations of the modular group $PSL(2,\mathbb{Z})=\mathbb{Z}/2*\mathbb{Z}/3$? I'm particularly interested in the following question.

Q: Is there a maximal dimension for the finite-dimensional irreducible unitary representations of $PSL(2,\mathbb{Z})$?

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The answer is No.

There is a mod-$p$ map $f:PSL(2,\mathbb Z) \rightarrow PSL(2,\mathbb F_p)$.

The permutation representation of $PSL(2,\mathbb F_p)$ on the projective line on $\mathbb F_p$ with $p+1$ points splits as a $1$-dimensional and a $p$-dimensional irreducible representation, which can be checked by the character table.

Thus, there is a $p$-dimensional irreducible representation for $PSL(2,\mathbb Z)$. As $p$ can be arbitrarily large, there is no maximal dimension for the finite-dimensional irreducible unitary representations of $PSL(2,\mathbb Z) $.

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