Decomposition of symmetric powers of the fundamental representation of $\text{Sp}(2n,\mathbb{C})$

Question

Let $(k,0,...,0)$ denote the highest weights vector of an irreducible representation of $\text{Sp}(2n,\mathbb{C})$. I read in Fulton-Harris, that this representation may be obtained as a direct summand of $\text{Sym}^kV$, where $V$ is the fundamental representation of $\text{Sp}(2n,\mathbb{C})$.

Is there a simple way to describe $(k,0,...,0)$ (as a subrepresentation of $\text{Sym}^kV$) as the kernel of some explicit map ($\text{Sym}^kV\longrightarrow *$)? I read that the $(0,...,0,1,0,...,0)$ representations could be constructed from the $k$'th alternating product of $V$ using the contraction maps associated to the symplectic form. I am particularly interested to know if there are known constructions of an infinite, symplectic family of irreducible representations of $\text{Sp}(2n,\mathbb{C})$ (direct summands of $V^{\otimes k}$ for $k$ odd), which project as a vector space to $\text{Sym}^{k'}V$ for some arbitrarily large $k' < k$.

A reference would also be appreciated.

Answer 1

Yes, it is just $\operatorname{Sym}^k(V)$ itself.

Specifically: Let $V = \mathbb{C}^{2n}$ be the natural module for $Sp(2n, \mathbb{C})$. Then for all $k \geq 1$, the symmetric power $\operatorname{Sym}^k(V)$ is an irreducible $Sp(2n,\mathbb{C})$-module of highest weight $(k,0,\ldots,0) = k \varpi_1$.

See for example §24.2, p. 406 in Fulton-Harris.

Some alternative references, which are also relevant when you look at the same question for $Sp(2n,K)$ with $K$ an algebraically closed field of positive characteristic.

• II.2.17 in: Jantzen, Jens Carsten Representations of algebraic groups. Second edition. Mathematical Surveys and Monographs, 107. American Mathematical Society, Providence, RI, 2003. xiv+576 pp. ISBN: 0-8218-3527-0.
• 8.1(c) in: Seitz, Gary M. The maximal subgroups of classical algebraic groups. Mem. Amer. Math. Soc. 67 (1987), no. 365, iv+286 pp.
• Suprunenko, I. D. Conditions for the irreducibility of the restrictions of irreducible representations of the group SL(n,K) to connected algebraic subgroups. Dokl. Akad. Nauk BSSR 30 (1986), no. 3, 204-207, 284.

Answer 2

As testaccount says, it is just $\mathrm{Sym}^k(V)$. One way to see it is the Weyl dimension formula. The positive roots of type $C_n$ are $\{2\epsilon_i:1\le i\le n\}\cup\{\epsilon_i\pm\epsilon_j:1\le i<j\le n\}$. Weyl dimension formula tells us $$\dim(V_{k\epsilon_1})=\prod_{\alpha\in\Delta_+}\frac{(k\epsilon_1+\rho,\alpha)}{(\rho,\alpha)},$$ but the only $\alpha\in\Delta_+$ that contribute to the product are those with nontrivial $\epsilon_1$-component, i.e., $\{2\epsilon_1\}\cup\{\epsilon_1\pm\epsilon_j:1<j\le n\}$.

Now noting that $\rho=n\epsilon_1+\cdots+\epsilon_n$, $$\begin{align*} \dim(V_{k\epsilon_1})&=\frac{(k\epsilon_1+\rho,2\epsilon_1)}{(\rho,2\epsilon_1)}\prod_{j=2}^n\frac{(k\epsilon_1+\rho,\epsilon_1+\epsilon_j)}{(\rho,\epsilon_1+\epsilon_j)}\cdot\prod_{j=2}^n\frac{(k\epsilon_1+\rho,\epsilon_1-\epsilon_j)}{(\rho,\epsilon_1-\epsilon_j)}\\ &=\frac{n+k}n\prod_{j=2}^n\frac{2n+k+1-j}{2n+1-j}\cdot\prod_{j=2}^n\frac{j+k-1}{j-1}\\ &=\frac{(k+1)\cdots(2n+k-1)}{1\cdots (2n-1)}\\ &={2n+k-1\choose k}. \end{align*}$$ But this is just the dimension of $\mathrm{Sym}^k(V)$ so the natural inclusion $V_{k\epsilon_1}\hookrightarrow\mathrm{Sym}^k(V)$ is an isomorphism!