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March 4, 2025

How Good are Permutation Represesentations?

Posted by John Baez

Any action of a finite group GG on a finite set XX gives a linear representation of GG on the vector space with basis XX. This is called a ‘permutation representation’. And this raises a natural question: how many representations of finite groups are permutation representations?

Most representations are not permutation representations, since every permutation representation has a vector fixed by all elements of GG, namely the vector that’s the sum of all elements of XX. In other words, every permutation representation has a 1-dimensional trivial rep sitting inside it.

But what if we could ‘subtract off’ this trivial representation?

There are different levels of subtlety with which we can do this. For example, we can decategorify, and let:

  • the Burnside ring of GG be the ring A(G)A(G) of formal differences of isomorphism classes of actions of GG on finite sets;

  • the representation ring of GG be the ring R(G)R(G) of formal differences of isomorphism classes of finite-dimensional representations of GG.

In either of these rings, we can subtract.

There’s an obvious map β:A(G)→R(G)\beta : A(G) \to R(G) , since any action of GG on a finite set gives a permutation representation of GG on the vector space with basis XX.

So I asked on MathOverflow: is β\beta typically surjective, or typically not surjective?

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