# Group-to-representation map

This term is related to: linear representation theory
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## Definition

Let $G$ be a finite group and $k$ a field that is sufficiently large for $G$. A group-to-representation map is an expression of $G$ as a disjoint union:

$gtr:G = \bigsqcup_\mu D_\mu \times D_\mu$

where $\mu$ varies over the irreducible representations of $G$, and $D_\mu$ is a set whose size is the degree of $\mu$.

## Properties

We call the group-to-representation map self-adjoint if $gtr(g)$ and $gtr(g^{-1})$ correspond to the same $\mu$ and are simply the same thing with the coordinates flipped.

## Example

The RSK-correspondence gives an example of a group-to-representation map when the group in question is a symmetric group.