$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sp{Sp}$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).

  • $\begingroup$ 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? $\endgroup$ Dec 3 '20 at 16:17
  • $\begingroup$ (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/…) $\endgroup$ Dec 3 '20 at 17:53

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