Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams

Let $U$ be a set of positive roots of type $ADE$, and let ${\mathrm{\Omega }}_U$ be the set of all maximum-cardinality orthogonal subsets of $U$. For each element $R\in {\mathrm{\Omega }}_U$, we define a generalized Rothe diagram whose cardinality we call the level, $\rho (R)$, of $R$. We define the generalized quantum Hafnian of $U$ to be the generating function for $\rho$, regarded as a $q$-polynomial in $U$. In this paper, we study a large number of examples of sets of the form ${\mathrm{\Omega }}_U$, and we explore their connections with a variety of widely studied algebraic and combinatorial objects. One of our motivating examples involves a certain set of $k^2$ roots in type $D_{2k}$, where the elements of ${\mathrm{\Omega }}_U$ can be identified with permutations in $S_k$, the generalized Rothe diagrams are the traditional Rothe diagrams associated to permutations, and the generalized quantum Hafnian is the $q$-permanent. More generally, all our examples in types $A$ and $D$ are closely related to perfect matchings and rook configurations, and our examples in type $E$ have applications to del Pezzo surfaces and labellings of Fano planes. Each of our examples also gives rise to a matroid, and a large family of our examples is in a natural correspondence with the equal-rank simply-laced symmetric pairs.