Let $USp(2n)$ be the compact symplectic group of size $2n$, $dA$ its Haar measure
of total mass one, and $\det(1−A)$ being computed for the standard representation of
$A\in USp(2n)$ as a matrix of size $2n$. Also, $C\_n=\frac1{n+1}\binom{2n}n$ be Catalan number.


This paper entitled [A note on random matrix integrals, moment identities, and Catalan numbers](https://web.math.princeton.edu/%7Enmk/catalan11.pdf) (page 5) lists the equality
$$\int\_{USp(2n)}\det(1+A)^rdA=\det(C\_{r+1+i+j})\_{i,j=0}^{n-1} \tag1$$
whose justification is solely based on just **equality as numbers**.



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> **QUESTION.** Is there a conceptual or combinatorial reason for (1)?
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Moment integrals and determinants
Yes, there is a combinatorial way to see this.


Firstly, it is much easier to show a baby version of this type of phenomenon. A while ago I read a nice [blog post](https://qchu.wordpress.com/2010/03/07/walks-on-graphs-and-tensor-products/) by Qiaochu explaining the identity
$$\int\_{0}^1 (2\cos \pi x)^n (2\sin^2\pi x)dx=\begin{cases} 0 &\text{ if n is odd} \\
C\_{n/2} & \text{ if n is even} \end{cases}$$
by interpreting this integral as $\int\_{SU(2)}\chi\_V(g)^nd\mu$, with respect to the normalized Haar measure, where $V$ is the defining representation. This in turn is the multiplicity of the trivial representation in $V^{\otimes n}$. Next, you make a graph with vertices corresponding to the irreducible representations of $SU(2)$, and add as many edges from $A$ to $B$ as the multiplicity of $A$ in $B\otimes V$. The irreducible representations of $SU(2)$ are the symmetric powers $\operatorname{Sym}^n(V)$, thus they are in bijection with the natural numbers. As a corollary of the fact that
$$\operatorname{Sym}^n(V)\otimes V\cong \operatorname{Sym}^{n-1}(V)\otimes \operatorname{Sym}^{n+1}(V)$$
we obtain that the graph is the half infinite line
$$0\leftrightarrow 1\leftrightarrow 2\leftrightarrow \cdots$$
and the desired multiplicity is the number of walks from zero to itself of length $n$, thus giving the answer in terms of the Catalan numbers.




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Now for your question, the representation of $USp(2n)$ under consideration is the exterior algebra of the standard representation, $W$. So $V=\bigoplus\_{k=0}^{2n}\Lambda^{k}(W)$ and your integral measures the multiplicity of the trivial representation in the representation $V^{\otimes r}$. A general combinatorial set up that keeps track of these types of multiplicities is that of [crystal bases](https://en.wikipedia.org/wiki/Crystal_base). It is much more technical but it is what ultimately brings the result in terms of enumerations of certain types of tableaux or non-intersecting lattice paths. Fortunately, the details of this specific problem are all worked out in the paper



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> Se-jin Oh, Travis Scrimshaw "Identities from representation theory" Discrete Math., 342(9):2493–2541, 2019 ([arxiv](https://arxiv.org/abs/1805.00113))
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