# What is the relationship between Mackey's theorem in character theory and Mackey's theorem in transfer theory?

Here are the statements of the two theorems. The first statement I took from a paper I have been reading, but I believe can also be found in Isaacs' Character Theory of Finite Groups as an exercise. The second comes directly from Isaacs' Finite Group Theory.

For both statements, let $X$ be a set of representatives for the $(H,K)$-double cosets in a group $G$, where $H$ and $K$ are subgroups and $G$ is finite.

Mackey's Theorem: Let $\theta$ and $\varphi$ be characters of $H$ and $K$, respectively. Then $$\langle \theta^G, \varphi^G\rangle = \sum_{x \in X}\langle (\theta^x)_{H^x\cap K},\varphi_{H^x\cap K}\rangle.$$

Mackey Transfer: Let $\Lambda:G\rightarrow H$ be a pretransfer map, and for each element $x\in X$, let $W_x:K\rightarrow H^x\cap K$ be a pretransfer map. Then for $k \in K$, we have $$\Lambda(k)\equiv \prod_{x \in X} x W_x(k) x^{-1} \mod{H'}.$$

Here, a "pretransfer map" $\lambda:A\rightarrow B$ refers to a map whose image in the abelianization of $B$ is the transfer homomorphism from $A$ to $B/B'$. The map is given by $\lambda(a)=\prod_{t \in T}ta(t\cdot a)^{-1}$, where $T$ is a right transversal of $B$ in $A$ and $t\cdot a$ is the representative in $T$ for the coset $H(ta)$. This is how Isaacs defines the transfer homomorphism in his book (after which he proves the transfer independent of the choice and order of transversal in the pretransfer).

My question is, how are these two theorems related? This post on MO gives a representation theoretic interpretation of the transfer, so it seems like the first theorem ought to be a stronger version of the second (but I cannot prove it). If this is true, can Mackey transfer be used to talk about irreducibility of induced characters? Or does that require the more general form?

## 1 Answer

What you call Mackey's theorem is a corollary of what is usually called Mackey's theorem about resticting a representation/character induced from one subgroup $H$ up to $G,$ then restricting the result to a (possibly different) subgroup $K.$ The result you state is a corollary of the more general result, together with Frobenius reciprocity. You use the fact that `$$\langle {\rm Ind}^{G}_{H}(\theta), {\rm Ind}_{K}^{G}(\phi) \rangle = \langle {\rm Res}^{G}_{K}({\rm Ind}_{H}^{G}(\theta)), \phi \rangle .$$' You then decompose the leftmost character according to Mackey's "real" formula, and apply Frobenius reciprocity again for each $x \in X.$

If you consider the case of a $1$-dimensional representation induced from $H$, then restricted to $K,$ and you take the determnant, you will get a formula very similar to that which appears in what you call Mackey's transfer formula,