Is there a formula for the determinant of an induced representation, e.g. in the fashion of the Frobenius character formula.


I would hope for something:



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> $$ det \; Ind\_H^G \rho(g) = (-1)^\alpha \prod\limits\_{\gamma, \gamma\_1 \in G/H \atop \gamma^{-1}g\gamma\_1 \in H} det \rho(\gamma^{-1}g\gamma\_1).$$ 
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> 
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If you have a reference or a quick proof, that would be most helpful.



Frobenius formula for the determinant
Well, yes there is, but it's slightly complicated by the fact that the permutation action of $G$ on the (say right) cosets of $H$ introduces a sign. Also, you have to worry about "non-diagonal" blocks. Hence, if we let $T$ be a complete set of representatives for the right cosets of $H$ in $G,$ then ${\rm det} {\rm Ind}\_{H}^{G}(\rho)[g] = \prod\_{s,t \in T: sgt^{-1} \in H} {\rm det}(\rho(sgt^{-1}))$ if $\rho(1)$ is even, but is ${\rm sign}\_{H}(g)\prod\_{s,t \in T :sgt^{-1} \in H} {\rm det}(\rho(sgt^{-1})$ if $\rho(1)$ is odd, where ${\rm sign}\_{H}(g)$ denotes the sign of the permutation of the right cosets of $H$ in $G$ induced by right multiplication by $g.$ In the 1970's, T. Yoshida did some work on "character-theoretic transfer", which exploited the determinant of the induced representation, especially when $\rho$ was linear. In some situations, Mackey type formulae for the determinant of the induced representation can simplify calculations.
Later edit: Perhaps a word about the use of Mackey type formulae. It can simplify things to look at the orbits of $\langle g \rangle$ on right cosets of $H$ in $G,$ in other words, to group contributions from the $(H, \langle g \rangle)$-double cosets. The factor ${\rm sign}\_{H}(g)$, can be accounted for as above: suppose that there are $k$ of these double cosets, and that $t\_1,t\_2, \ldots, t\_k$ are representatives, where the double coset of $t\_i$ consists of $m\_i$ right cosets of $H.$ Then the formula for the determinant of ${\rm Ind}\_{H}^{G}(\rho)[g]$ may be expressed as ${\rm sign}\_{H}(g)^{\rho(1)} \prod\_{i=1}^{k} {\rm det}(\rho( t\_{i}g^{m\_i}t\_{i}^{-1})).$ 


