the question may be a little silly but I want to clarify things.
EDIT: Assuming the standard representation is the regular representation then what is the difference between the standard(regular) representation and the permutation representation?
2026-03-29 06:07:59.1774764479
Is the regular representation just another way of saying the standard representation?
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I was wondering the same thing while reading Serre's book. I can't put this response as a comment (don't have a reputation of 50 or higher).
The regular (standard) representation is a special and important type of a permutation representation.
Wikipedia, (Regular Representation), gives the following distinction (in bold)
"To say that G acts on itself by multiplication is tautological. If we consider this action as a permutation representation it is characterised as having a single orbit and stabilizer the identity subgroup {e} of G. The regular representation of G, for a given field K, is the linear representation made by taking this permutation representation as a set of basis vectors of a vector space over K. The significance is that while the permutation representation doesn't decompose - it is transitive - the regular representation in general breaks up into smaller representations. For example, if G is a finite group and K is the complex number field, the regular representation decomposes as a direct sum of irreducible representations, with each irreducible representation appearing in the decomposition with multiplicity its dimension. The number of these irreducibles is equal to the number of conjugacy classes of G."
This post answers your question with a good example: The difference between permutation representation and standard representation of symmetric group S3