# Interplay between two definitions of the transfer homomorphism.

The transfer homomorphism can be defined in a couple of ways. For the purposes of this question, assume that all groups are finite.

Defintion 1. Let $1\leqslant K'\leqslant L \leqslant K\leqslant G$ and $T_K^G$ be a right transversal of $K$ in $G$. Define the pretransfer $\Lambda:G\rightarrow K$ by $\Lambda(g)=\prod_{t\in T_K^G}tg(t\cdot g)^{-1}$, where the $\cdot$ action is given by $t\cdot g$ is the element of $T_K^G$ associated with the coset $K(tg)$. Then $\lambda:G\rightarrow K/L$ given by $\lambda(g)=(\Lambda(g))L$ is the transfer from $G$ to $K/L$ - that is, the image of $\Lambda$ in the factor group.

Definition 2. Consider the canonical homomorphism $\mu:K\rightarrow K/L$ as the rank one representation of $K$ over the group ring $\mathbb{F}_2 [K/L]$. Let $n=[G:H]$ be finite, and consider the induced representation $\mu^G$ from $G$ to $GL_n(R)$. Then the transfer from $G$ to $K/L$ is the determinant of $\mu^G$.

These definitions seem to be, for lack of a better way of saying it, the same thing from opposite directions. This motivates the question of whether one can use one definition to learn things about the other.

First, I have been trying to chase down the kernel of $\lambda$ (as in def. 1). Using only the first definition the best I can do is $\{g\in G:\Lambda(g)\in L\}$ (not a compelling revelation). In def. 2 the inducing part would seem to imply that the kernel of $\lambda$ has to do with the $G$-core of $L$, but I don't know how to reconcile this with the determinant.

Question 1: What do we know about $\mbox{Ker}\lambda$ from definition 2?

Second, what I am more interested in knowing is: given transfers to two different groups (as given in def. 1), is any way of telling if $\mu_1^G$ and $\mu_2^G$ (as given in def. 2) are isomorphic representations? It's like I want to "un-determinant" the $\lambda$ formulation for each transfer and see if I can get the same thing. Would this be equivalent to the statement that $\Lambda_1$ and $\Lambda_2$ are the same homomorphism, perhaps by an appropriate choice of transversals?

Question 2: Is there a formulation of the statement that $\mu_1^G = \mu_2^G$ in language similar to definition 1?

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Very related to mathoverflow.net/questions/83790/… –  Benjamin Steinberg Aug 5 '12 at 14:39