VT Algebra Seminar — Fall 2026

Let \(\Phi\) be a simply-laced root system with Weyl group \(W\), let \(U\) be a set of positive roots in \(\Phi\), and let \(\Omega_U\) be the set of all maximum-cardinality orthogonal subsets of \(U\). Define the generalized Rothe diagram of each \(R\in \Omega_U\) to be the set

\[ D_U(R)=\{\alpha\in U : s_\beta(\alpha)>0 \text{ for all } \beta\in R\}. \]

We show how the set \(\Omega_U\) and the diagrams \(D_U(R)\) provide a root-theoretic framework that recovers many widely studied algebraic and combinatorial objects. For example, for a suitable choice of \(U\) in type \(D_{2k}\), the set \(\Omega_U\) can be identified with the symmetric group \(S_k\), and the diagrams \(D_U(R)\) coincide with the traditional Rothe diagrams of permutations. For another choice in type \(D_{2k}\), the elements of \(\Omega_U\) can be identified with the fixed-point-free involutions in \(S_{2k}\), and the diagrams \(D_U(R)\) recover the involution Rothe diagrams used by Hamaker–Marberg–Pawlowski to obtain their involution Schubert polynomials. Our other examples involve perfect matchings, rook configurations, labelled Fano planes, del Pezzo surfaces, and minuscule representations. The set \(\Omega_U\) also forms a quasiparabolic set in the sense of Rains–Vazirani in all our examples, which endows \(\Omega_U\) with a Bruhat-like partial order and natural connections to the Iwahori–Hecke algebra of \(W\). (Joint work with Richard Green.)