# Character of 2-dimensional irreducible representation of $S_4$

Consider the 2-dimensional irreducible representation of $S_4$, obtained by composing the standard 2-dimensional irreducible representation of $S_3$ with the surjective homomorphism $f:S_4 \rightarrow S_3, (1 2) \mapsto (1 2), (2 3) \mapsto (1 3), (3 4) \mapsto (1 2)$. Please compute its character directly, without using character table orthogonality relations. Thanks!

• This is not my area of expertise, but your deadline seems too short for the level of your question and the offered bounty.

• Your text has typos, it’s not clear what you mean. You want the 3d rep. of. S3 Not S4. The result is then not a 2d rep but a 3d rep, is this what you mean?

• @Dynkin, sorry yes, I edited the text

• Hey there, I hope you're alright! I was wondering if you could help me with some questions this Friday, 10a.m. GMT. I will have around a 2-2.5 hour window and I'm willing to pay $60 per question if you're online and able to answer within the time limit, would you be interested? • Sorry, it is actually May 20, 10 a.m. GMT +1. Please let me know if you see this and are interested. • Sure, I‘ll keep an eye out, but I can‘t guarantee that I’ll have time. The answer is accepted. • answered • 248 views •$15.00