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This is the Bruhat order on $S_n / (S_k \times S_{n-k})$, which models the inclusion relations of Schubert varieties on the Grassmannian $Gr(k,n)$. The elements of $S_n / (S_k \times S_{n-k})$ are ge …

You need to use the fact that $x'_j = M x_q + M'$, where $M$ and $M'$ are Laurent monomials in the remaining variables. Their observation is that if $x'_j$ factored, it would be as $x'_j = (P_1 x_q + …

Discrete difference equations generalize differential equations. In a similar spirit, divided difference operators generalize partial differentiation operators. Though such operators go back to Newt …

One interesting fact in symmetric function theory is that the power symmetric function $p_n$ can be written as an alternating sum of hook Schur functions $s_{\lambda}$: $$ p_n = \sum_{k+\ell = n} (-1) …

This is a manifestation of the Bruhat order on the Grassmannian $Gr(k,n)$ of $k$-planes in an $n$-dimensional vector space. In terms of partitions (Young diagrams): Given a set $X$ as above, let $\la …

While the paper of Richardson-Springer does study the weak order, it also has useful results on the usual (strong) Bruhat order. In particular, Theorem 7.11 says that Bruhat order is characterized as …

There is a detailed analysis in Chapters 7 and 8 of Bump and Schilling's Crystal Bases. They work through the connection between RSK and crystals in careful detail, though I don't recall how much deta …

Neither of these suggestions seem to exactly fit the level the OP was aiming for, but I add them for others who come across this thread with a different group of students in mind: (1) For a gentle, p …

This is discussed in Stanley's paper Some combinatorial aspects of the Schubert calculus. Corollary 3.7 says that under the natural isomorphism given by the Borel presentation of $H^*(G/P)$ which send …