Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
We should have something on the relationship between hook lengths and rep dimensions. So for $Sym(n)$, the dimension of the rep corresponding to YT $\lambda$ is
$\frac{n!}{\prod_{u: \lambda} h_{\lambda(u)}}.$From a nice efficient set of notes here:
I have now added statement of and references for the actual/standard form of the hook-content formula:
$\left\vert ssYTableaux_\lambda(N)\right\vert \;=\; s_{\lambda} \big( x_1 \!=\! 1, \cdots, x_N \!=\! 1 \big) \;\; = \;\; \underset{ (i,j) }{\prod} \frac{ N + content(i,j) }{ \ell hook_\lambda(i,j) } \,.$I am indebted to Abdelmalek Abdesselam for hints. (The combinatorics literature is a bit weird about this.)
1 to 5 of 5