Locally conjugate representations

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Definition

Symbol-free definition

Two linear representations of a group over a field are said to be locally conjugate if for any element of the group, its image under the two representations defines equivalent linear transformations (in particular, if we identify the vector spaces on which the representations are there, its images must be conjugate).

Definition with symbols

Let $G$ be a group and $k$ a field.

Two linear representations $\rho _{1}:G\to GL(V)$ and $\rho _{2}$ of $G$ over $k$ are locally conjugate if for any $g\in G$ , there exists $\sigma :V\to W$ such that:

$\rho _{2}(g)=\sigma \circ \rho _{1}(g)\circ \sigma ^{-1}$

Definition using the L-map

Let $G$ be a group and $k$ a field. Let $C(G)$ denote the set of conjugacy classes of $G$ and $I(G)$ the set of indecomposable linear representations of $G$ over $k$ .

Let $L(c,\rho )$ denote the conjugacy class of $\rho (g)$ in the general linear group for any $g\in c$ (they are all in the same conjugacy class). $L$ is thus a map:

$L:C(G)\times I(G)\to C(GL)$

where $C(GL)$ is the union of the set of conjugacy classes in $GL(V)$ for all vector spaces $V$ over $k$ .

Then, we say that two representations $\rho _{1}$ and $\rho _{2}$ are locally conjugate with respect to $k$ if for any $\rho \in I(G)$ :

$L(c,\rho _{1})=L(c_{2},\rho _{2})$

Note that though the above is described in terms of indecomposable linear representation, we can define the $L$ -map for arbitrary linear representations, and the same definition can thus be used to call two arbitrary linear representations locally conjugate.

Related notions