  Let $\mathbf{x}_{n \times n}$ be an $n \times n$ matrix of variables and let
$\mathbb{F}[\mathbf{x}_{n \times n}]$ be the polynomial ring in these variables
over a field $\mathbb{F}$. We study the ideal $I_n \subseteq
\mathbb{F}[\mathbf{x}_{n \times n}]$ generated by all row and column variable
sums and all products of two variables drawn from the same row or column. We
show that the quotient $\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ admits a
standard monomial basis determined by Viennot's shadow line avatar of the
Schensted correspondence. As a corollary, the Hilbert series of
$\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ is the generating function of
permutations in $\mathfrak{S}_n$ by the length of their longest increasing
subsequence. Along the way, we describe a `shadow junta' basis of the vector
space of $k$-local permutation statistics. We also calculate the structure of
$\mathbb{F}[\mathbf{x}_{n \times n}]/I_n$ as a graded $\mathfrak{S}_n \times
\mathfrak{S}_n$-module.
