RSK and Matrices It is well known that the RSK algorithm assigns to every square matrix with nonnegative integer entries a pair of semistandard Young Tableaux of same shape. The matrices are here used as just a square array of nonnegative integers. 
I want to know if there are any benefits if we instead treat them as matrices specifically, that is, bring in the algebra of matrices, maybe view them as linear transformations on some modules, find their smith normal form, calculate determinant and stuff like that. How does these reflect on the tableaux? For example the operation transpose interchanges the $P$ and $Q$ tableaux. 
I do not have anything specific in mind, any property that treats them as matrices will be useful to know, and not necessarily having to do with what I mentioned. Apologies if the question is vague.   
 A: The two line arrays in $RSK(p,q)$ for $l(p) = l(q) \leq d$ are exactly those arrays which can be decomposed into at most $d$ non-decreasing subsequences (both index and position must weakly increase). Equivalently (by this question and answer), the arrays which have longest decreasing subsequence (where we must strictly decrease position or increase index after each term) $d.$
One way to see this is via Viennot's geometric construction, for which there is a beautiful animation here. I include a copy below: 
Imagine it in reverse, starting from the tableau and creating the relevant points. Each row of $P,Q$ gives new decreasing sequences (the lines from top left to bottom right in the animation), which are made longer by introducing the next row. So if we have some $d$th row, then we must have a decreasing sequence of length $2$ when we add the next row, and so on until we construct our original matrix with $d$ elements on some decreasing sequence.
I have always thought this should be related to Greene's theorem (Curtis Greene,
Some partitions associated with a partially ordered set,
Journal of Combinatorial Theory, Series A,
Volume 20, Issue 1), but I could not recover the argument after some effort.
