# $U(n)$-submodules of ${\rm SO}(2n)$-modules

Let $\Gamma_{(\lambda_1, \dots, \lambda_{n})}$ denote an irreducible $SO(2n)$-module with highest weight $(\lambda_1, \dots, \lambda_n)$ and let more specifically $X = \Gamma_{(2\lambda, \dots, 0)}$ and $Y = \Gamma_{(2\lambda_1, \dots, 2\lambda_n)}$, where at least one of the $\lambda_j$ with $j>1$ is not equal to $0$. Furthermore, let $W \leq X$ be a (irreducible) $U(n)$-submodule of $X$, and $\tilde{W} \leq Y$ a (irreducible) $U(n)$-submodule of $Y$. My Questions:

1) Is it true that for any $X,Y,W,\tilde{W}$ choosen in such a fashion \begin{align*} W \ncong \tilde{W}? \end{align*}

2) Is there a branching rule for $U(n)$-submodules of $SO(2n)$-modules? (This obviously relates to Question 1)

• @AmirSagiv Maybe it is worth mentioning that some users do not agree to adding MathJax/LaTeX to the titles which are perfectly readable without it. (As far as I can say, there is no clear consensus about this.) See this discussion on meta or this message in chat. (Regardless of this issue, I appreciate efforts to improve posts by editing them, be it for typos, grammar, LaTeX or anything else.) – Martin Sleziak May 25 '16 at 5:50
• I'll read and comment in Meta. Thanks for the heads up. In the post themselves it is a consensus, right? – Amir Sagiv May 25 '16 at 11:48

you can find branching rule ${\rm SO}(2n) \to U(n)$ in the book of Knapp, Lie groups beyond an introduction, and also in some paper of his, and in the book of Zelobenko, compact Lie groups and their rep's, also try the book of Tom Dieck.