Character Tables for S4 and A4
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange users link it as a worked example when others are stuck on character tables for S4 and A4 or on how characters relate to tetrahedral symmetries.
Recommended by 2 people across 2 different months, 2015 to 2022.
What it is
A lecture-style video by MathDoctorBob on the representation theory of finite groups. It builds the character tables of S4 and A4 from scratch, then uses irreducible characters to decompose a tensor product.
What people said
Recommends Math137 · 2015-01 · Mathematics Stack Exchange · on “Character Table S4”
This video [this video] might help you. By identity you mean trivial representation? For the other one dimensional, we have $\chi_{sgn}(\rho)=trace(sng(\rho))$
Recommends Wasradin · 2022-04 · Mathematics Stack Exchange · on “Understanding permutations of $S_4$ which correspond to symmetries of a regular tetrahedron”
I'm trying to understand values of the character of the representation of $S_4$ corresponding to the symmetries of a regular tetrahedron (whew, that's a mouthful!). One illustrative video is found here [this video] , where the different conjugacy classes of $S_4$ acting on the regular tetrahedron are discussed. What I'm struggling with is understanding how you can deduce how the $x, y, z$ values are mapped to each other in the case of a single two cycle $(1, 2)$ , two two cycles $(1, 2)(3, 4)$ and the four cycle $(1, 2, 3, 4)$ . Let's consider for example the single two cycle $(1, 2)$ : We'd like to flip (I'm referring to the illustrations of the video) the swap the vertices of the lower right ed
Cites ushham · 2017-02 · Mathematics Stack Exchange · on “Identify irreducible representations of $A_4$”
I have been watching the linked video on character tables of $S_4$ and $A_4$. I understand the process used in $S_4$, but I am having difficulty with that of $A_4$. The conjugacy classes of $S_4$ are given in the video by $\{id, [(12)], [(123)], [(12)(34)], [(1234)]\}$, this I understand. However when moving to $A_4$ the classes are given by $\{id, [(123)], [(132)], [(12)(34)]\}$. Why are the three cycles now represented by two conjugacy classes? I do not see how this result is arrived at. Video: [this video]