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On page 427 of Fulton and Harris's Representation Theory, the authors give the branching rule for the above restriction as $$ \mathrm{Res}_{\mathrm O_m \Bbb C}^{\mathrm{GL}_m \Bbb C} (\Gamma_\lambda) = \bigoplus N_{\lambda \overline \lambda} \Gamma_{\overline \lambda} $$ where $\lambda$ partitions $m$, $\overline \lambda$ denotes the conjugate partition, $\Gamma_\lambda$ is the irrep of $\mathrm{GL}_m$ with highest weight $\lambda$, and $$ N_{\lambda \overline \lambda} = \sum_\delta N_{\delta \overline \lambda \lambda} $$ is the Littlewood-Richardson coefficient and the sum over all $\delta = (\delta_1 \geq \delta_2 \geq \cdots)$ with all $\delta_i$ even.

Question

What if we take $\lambda = (n)$? Then $\Gamma_{(n)} = \mathrm{Sym}^n \Bbb C$ (I believe), and $\overline \lambda = (1, \dots , 1)$. But I am having trouble seeing what $\delta$ will be, or how I will compute this restriction to $\mathrm O_9\Bbb C$.

Peter Patzt
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Moderat
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1 Answers1

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The problem is that $\bar \lambda$ is not the conjugate partition (which is denoted by $\lambda'$ in [FH]) but any partition.

As for your example, if I am not mistaken, $\mathrm{Sym}^n(\mathbb C)$ then restricts to $$\bigoplus_{k\le n, n-k\text{ even}} \Gamma_{(k)}$$ because $\bar \lambda \le \lambda$ and $N_{(n-k),(k),(n)}=1$.

Peter Patzt
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