Skip to main content
MathOverflow

Return to Question

replaced http://math.stackexchange.com/ with https://math.stackexchange.com/
Source Link
Community Bot
Community Bot
  • 1
  • 2
  • 3

This is a crosspost from math.SEmath.SE. Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form $$ \bigoplus_i V_i \otimes W_i, $$ where $V_i, W_i$ are reps of $G$ and $H$, respectively?

I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case?

I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.

This is a crosspost from math.SE. Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form $$ \bigoplus_i V_i \otimes W_i, $$ where $V_i, W_i$ are reps of $G$ and $H$, respectively?

I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case?

I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.

This is a crosspost from math.SE. Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form $$ \bigoplus_i V_i \otimes W_i, $$ where $V_i, W_i$ are reps of $G$ and $H$, respectively?

I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case?

I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.

Source Link
Eric O. Korman
Eric O. Korman
  • 3.2k
  • 1
  • 24
  • 35

Are all representations of $G\times H$ induced from representations of $G$ and $H$?

This is a crosspost from math.SE. Suppose $G$ and $H$ are discrete groups. Is it always the case that any finite dimensional complex representation of $G\times H$ is of the form $$ \bigoplus_i V_i \otimes W_i, $$ where $V_i, W_i$ are reps of $G$ and $H$, respectively?

I know this is true when $G$ and $H$ are finite and when the representation of $G\times H$ is completely reducible, but is there a simple counterexample to the general case?

I'm also curious if it is ``usually true," in some sense, that any rep of $G\times H$ has the above form.