Generalized Rothe diagrams for orthogonal roots
Let $U$ be a set of positive roots of type $ADE$, and let $Ω_U$ be the set of all maximum-cardinality orthogonal subsets of $U$. We associate a generalized Rothe diagram to each element $R\in Ω_U$ as a broad, root-theoretic generalization of the traditional Rothe diagrams of permutations, and we use the generalized Rothe diagrams to define a $q$-polynomial in $U$ that we call the generalized quantum Hafnian of $U$. We study a large number of examples where these constructions recover a variety of widely studied algebraic and combinatorial objects. One of our motivating examples involves a certain set $U$ of $k^2$ roots in type $D_{2k}$, where the elements of $Ω_U$ can be identified with permutations in $S_k$, the generalized Rothe diagrams are the traditional Rothe diagrams of permutations, and the generalized quantum Hafnian is the $q$-permanent. In another example, the generalized quantum Hafnian gives a non-recursive method to compute the 45 terms of a well-known invariant cubic polynomial of type $E_6$. 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 labelled Fano planes, del Pezzo surfaces, and minuscule representations. Each of our examples also gives rise to a matroid, and many of our examples have an associated equal-rank simply-laced symmetric pair.