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We show that for every pair of matrices (S,P), having the closed symmetrizedbidisc $\Gamma$ as a spectral set, there is a one dimensional complex algebraicvariety $\Lambda$ in $\Gamma$ such that for every matrix valued polynomial f,the norm of f(S,P) is less then the sup norm of f on $\Lambda$.  The variety $\Lambda$ is shown to have a particular determinantalrepresentation, related to the so-called "fundamental operator" of the pair(S,P).  When (S,P) is a strict $\Gamma$-contraction, then $\Lambda$ is adistinguished variety in the symmetrized bidisc, i.e., a one dimensionalalgebraic variety that exits the symmetrized bidisc through its distinguishedboundary. We characterize all distinguished varieties of the symmetrized bidiscby a determinantal representation as above.

Spectral sets and distinguished varieties in the symmetrized bidisc