Group-to-representation map

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.