# Does a symplectic group act on a tensor power of a spin representation?

$$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$$More specifically, let $$S_k$$ be the spin representation of $$\Spin(2k+1)$$. Then is there are action of $$\Sp(2r-2)$$ on $$\bigotimes^{2r}S_k$$ which commutes with the action of $$\Spin(2k+1)$$?

I am not asking for a duality. That is, I do not require that each isotypic vector space is an irreducible representation of $$\Sp(2r-2)$$.

The motivation is that the space of invariant tensors in $$\bigotimes^{2r}S_k$$ looks like the irreducible representation of $$\Sp(2r-2)$$ with highest weight $$k\omega_{r-1}$$. For instance, for $$k=1$$, both vector spaces have dimension given by the Catalan numbers $$C_r = \frac{1}{r+1}\binom{2r}{r}$$. More generally, for arbitrary $$k$$ the dimension of both of these spaces is $$\prod_{1\leq i \leq j \leq r-1} \frac{i+j+2k}{i+j}$$ which is the number of so-called "$$k$$-fans of nested Dyck paths of semilength $$r$$" (see, e.g., Section 3.1.6 of Federico Ardila - Algebraic and geometric methods in enumerative combinatorics).

• As mentioned in the bounty message, I would be interested in any algebraic explanation of the numerical coincidence of the dimensions of the spaces discussed in Bruce's question. Does some version of Howe duality apply here? Dec 3 '20 at 16:17
• (For an explanation of a similar, well-known numerical "coincidence," but for $SL_n$ and $GL_m$, using Howe duality, see: sbseminar.wordpress.com/2007/08/10/the-ubiquity-of-howe-duality/…) Dec 3 '20 at 17:53