Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams

TL;DR

This paper introduces a new algebraic and combinatorial framework involving orthogonal roots, quantum Hafnians, and generalized Rothe diagrams, revealing connections to permutations, matchings, and geometric objects across types A, D, and E.

Contribution

It defines the generalized quantum Hafnian and explores its properties and connections with various algebraic and combinatorial structures in types A, D, and E.

Findings

Generalized quantum Hafnian relates to $q$-permanent and permutation enumeration.

Examples connect to perfect matchings, rook configurations, and geometric objects.

Identifies correspondences with matroids and symmetric pairs.

Abstract

Let $U$ be a set of positive roots of type $ADE$, and let $\Omega_U$ be the set of all maximum-cardinality orthogonal subsets of $U$. For each element $R \in \Omega_U$, 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 $\Omega_U$, 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 $\Omega_U$ 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.