Would someone please explain what a standard representation is? Not all groups seem to have a standard representation, and those I have seen are related to the symmetric group. However, I can't seem to find a clear explanation.
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1$\begingroup$ I don't think the word "standard" is standard, except in the case of $S_n$. See this: groupprops.subwiki.org/wiki/Standard_representation $\endgroup$– Prahlad VaidyanathanApr 30, 2017 at 2:34
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$\begingroup$ The context of my question comes from Michael Artins book Algebra, chapter 10.1. He states that the 'standard representation' of the dihedral group and symmetric group of order 3 are two 2x2 matricies: the rotation matrix and ((1,0),(0,-1)). He continues to use this definition in his problems as well. $\endgroup$– nobodyApr 30, 2017 at 2:45
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$\begingroup$ In the same section, he states: 'the elements of a finite rotation group are rotations of a 3 dimensional Euclidean space V without reference to a basis, and these orthogonal operators give us what we call the standard representation of the group.' $\endgroup$– nobodyApr 30, 2017 at 2:56
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1$\begingroup$ A standard representation is a representation that is built into (at least one) definition of the group itself. For instance, $SO(3)$ acts on $\Bbb R^3$. If you define dihedral groups as symmetries of a regular polygon, then they are already linear operators acting on $\Bbb R^2$. $\endgroup$– arctic ternApr 30, 2017 at 3:16
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$\begingroup$ Some authors use the term "standard representation of $S_n$" refers to the representation $\rho:S_n\to GL(\mathbb{C}^n)$, $\rho(\sigma)(e_i)=e_{\sigma(i)}$. See Example 3.1.9 in Steinberg's Representation Theory of Finite Groups $\endgroup$– bfhahaApr 14, 2018 at 18:45
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1 Answer
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The group $S_n$ acts on $\mathbb{C}^n$ by permuting basis vectors: this defines a representation. This representation has a 1-dimensional invariant subspace, spanned by the vector $e_1 + e_2 + \cdots + e_n$, which is the trivial representation. A complementary subspace to this is $$ V = \{ a_1 e_1 + \cdots + a_n e_n \mid a_1 + \cdots + a_n = 0\} $$ $V$ is what many authors would call "the standard representation" of $S_n$. This is irreducible and faithful.
As was pointed out in the comments, a "standard representation" is one coming very naturally from the definition of the group. Permutation matrices are very natural linearisation of the symmetric group. For groups sitting naturally in $\mathrm{GL}_n$, like the dihedral group of any order sits in $\mathrm{GL}_2$, the natural representation here is just acting on $\mathbb{C}^n$.
