Representation Theory

Searcing for solution to part c) (pieri s rule is attached last,where the sum is taken over all λ obtained from ν by removing one box )

By the Artin - Wedderburn theorem, we know $\mathbb{C}[S_3] \cong \mathbb{C}\oplus \mathbb{C}\oplus Mat_{2x2}(\mathbb{C}) $, and the isomorphism is given by $g \mapsto (1, sgn(g), \rho(g))$, where $ \...

Consider the obvious inclusion $S_3 \subset S_4$ and let $V$ be the permutation representation of $S_3$. Compute the character of $Ind_{S_3}^{S_4} (V )$ (Induced representation) and decompose it as di...

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...