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The complex Clifford algebra $A$ of a complex, non-degenerate quadratic space $(V,q)$ of odd dimension $2k+1$ admits up to isomorphism exactly two non-trivial, irreducible and finite-dimensional representations on a complex vector space $S$. If $\Phi:A\to \mathrm{End} (S)$ represents one, then $\Phi\circ\chi$ represents the other, where $\chi: A\to A$ is the automorphism defined by $\chi(v)=-v$ for $v\in V$.

My question is now: how can one decide whether two generic (irreducible, non-zero, finite-dimensional complex) representations $\Phi_1,\Phi_2$ of $A$ belong to the same isomorphism class? From what I have seen so far, I suspect that it should have something to do with the volume element of $A$...?

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    You are looking for a method to determine whether or not two representations are isomorphic? Did you intend to assume any conditions on them at all? Are we assuming the form is nondegenerate?2012-09-10
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    Oh yes, $q$ is non-degenerate. The representations $\Phi_1,\Phi_2$ under consideration are required to be non-zero, irreducible and finite-dimensional.2012-09-10
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    Put all your requirements in the question, please.2012-09-10

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