# Sum of subdeterminants of an orthogonal matrix

Let $$A$$ be an $$n \times n$$ real orthogonal matrix, $$A^T A = I$$. For subsets $$S, S'$$ of $$\{1,\dots, n\}$$, let $$A_{S,S'}$$ denote the restriction of $$A$$ to rows indexed by $$S$$ and columns indexed by $$S'$$. I'm wondering if there is a general formula or bound for (the absolute value of) $$\sum_{S' \subseteq\{1,\dots, n\}:|S'| = |S|} \text{det}(A_{S,S'})$$ for a given $$S \subseteq \{1,\dots, n\}$$?

For instance, in the special case where $$A$$ is not only orthogonal, but also a permutation matrix, this sum evaluates to $$+1$$ or $$-1$$, since for fixed $$S$$, there is exactly one $$S'$$ for which $$A_{S,S'}$$ has a nonzero entry in each row (and for that $$S'$$, $$A_{S,S'}$$ is itself a permutation matrix). But more generally, for orthogonal $$A$$, I'm interested in whether (the absolute value of) this sum can be bounded (as a function of $$n$$ and/or $$|S|$$). Is there a known reference for this problem? Even just knowing whether it's polynomially or exponentially increasing in $$n$$ (and/or $$|S|$$) would suffice as a starting point.

When $$|S|=1$$ we are asking for the upper bound of $$x_1+x_2+\dots+x_n$$ subject to $$x_1^2+\dots+x_n^2=1$$. This is achieved when the $$x_i$$ are all equal to $$\frac{1}{\sqrt{n}}$$: and so the upper bound is $$\sqrt{n}$$.
Now note that when $$A$$ is orthogonal so too is the matrix of $$r\times r$$ cofactors $$A^{(r)}$$: this follows from the general facts that $$(XY)^{(r)}=X^{(r)}Y^{(r)}$$, $$(X^{T})^{(r)}=(X^{(r)})^T$$ and $$I^{(r)}=I$$.
We can therefore apply the result for $$|S|=1$$ to the $${n\choose |S|}\times {n\choose |S|}$$ orthogonal matrix $$A^{({n\choose |S|})}$$ and obtain the result that for arbitrary $$|S|$$ the upper bound is $$\sqrt{n\choose |S|}$$.