Group-to-representation map

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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.