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Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams

R. M. Green, Tianyuan Xu

TL;DR

The paper develops a unified framework for studying orthogonal root configurations in ADE-type root systems through residues, Rothe diagrams, and a level statistic, culminating in the generalized quantum Hafnian $\text{QHf}(U)$. By introducing QP-triples $(W,I,U)$, it shows that many natural choices of $U$ yield quasiparabolic structures on $\Omega_U$, connecting to classical combinatorial objects such as rook configurations and permutation rothe diagrams, and producing factorized Poincaré polynomials into quantum integers. It presents several rich examples, including the D-type $k^2$-root case linking to $S_k$ and $q$-permanents, as well as two deep E-type cases $T(E_8,8)$ and $T(E_7,7)$ tied to minuscule representations and invariant cubic polynomials in $E_6$. The results offer new combinatorial models for geometric and representation-theoretic objects (del Pezzo surfaces, Fano planes, tritangent planes) and open avenues for Hecke-algebra modules built from quasiparabolic sets, with potential extensions to invariant forms and Schubert-theoretic structures.

Abstract

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 $U$. For each element $R \in Ω_U$, we define a generalized Rothe diagram whose cardinality we call the level, $ρ(R)$, of $R$. We define the generalized quantum Hafnian of $U$ to be the generating function of $ρ$, regarded as a $q$-polynomial in $U$. Several widely studied algebraic and combinatorial objects arise as special cases of these constructions, and in many cases, $Ω_U$ has the structure of a graded partially ordered set with rank function $ρ$. A motivating example of the construction involves a certain set of $k^2$ roots in type $D_{2k}$, where the elements of $Ω_U$ correspond to permutations in $S_k$, the generalized Rothe diagrams are the traditional Rothe diagrams associated to permutations, the level of a permutation is its length, the generalized quantum Hafnian is the $q$-permanent, and the partial order is the Bruhat order. We exhibit many other natural examples of this construction, including one involving perfect matchings, two involving labelled Fano planes, and one involving the invariant cubic form in type $E_6$.

Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams

TL;DR

The paper develops a unified framework for studying orthogonal root configurations in ADE-type root systems through residues, Rothe diagrams, and a level statistic, culminating in the generalized quantum Hafnian . By introducing QP-triples , it shows that many natural choices of yield quasiparabolic structures on , connecting to classical combinatorial objects such as rook configurations and permutation rothe diagrams, and producing factorized Poincaré polynomials into quantum integers. It presents several rich examples, including the D-type -root case linking to and -permanents, as well as two deep E-type cases and tied to minuscule representations and invariant cubic polynomials in . The results offer new combinatorial models for geometric and representation-theoretic objects (del Pezzo surfaces, Fano planes, tritangent planes) and open avenues for Hecke-algebra modules built from quasiparabolic sets, with potential extensions to invariant forms and Schubert-theoretic structures.

Abstract

Let be a set of positive roots of type , and let be the set of all maximum cardinality orthogonal subsets of . For each element , we define a generalized Rothe diagram whose cardinality we call the level, , of . We define the generalized quantum Hafnian of to be the generating function of , regarded as a -polynomial in . Several widely studied algebraic and combinatorial objects arise as special cases of these constructions, and in many cases, has the structure of a graded partially ordered set with rank function . A motivating example of the construction involves a certain set of roots in type , where the elements of correspond to permutations in , the generalized Rothe diagrams are the traditional Rothe diagrams associated to permutations, the level of a permutation is its length, the generalized quantum Hafnian is the -permanent, and the partial order is the Bruhat order. We exhibit many other natural examples of this construction, including one involving perfect matchings, two involving labelled Fano planes, and one involving the invariant cubic form in type .

Paper Structure

This paper contains 16 sections, 24 theorems, 56 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Orthogonal roots
  3. Review of root systems
  4. Orthogonal sets of a fixed size
  5. Coplanar quadruples and D4-subsystems
  6. Residues and quasiparabolic sets
  7. Key definitions
  8. Motivating examples
  9. Further examples of QP-triples
  10. Orthogonal bases of roots
  11. An effective construction
  12. The QP-triples T(E8,8) and T(E7,7)
  13. T(E8, 8)
  14. T(E7, 7)
  15. The invariant cubic polynomial in type E6
  16. ...and 1 more sections

Key Result

Lemma 2.3

Let $\Phi$ be an irreducible simply laced root system of rank $n$, and let $W$ be the Weyl group of $\Phi$.

Figures (5)

  • Figure 1: The Rothe diagram of $\underline{3142}\in S_4$, with residues marked by stars
  • Figure 2: Dynkin diagrams of irreducible simply laced Weyl groups
  • Figure 3: Features formed by two blocks $\{a,b\}, \{c,d\}$ in a perfect matching
  • Figure 5: New labelling of the Dynkin diagram of type $E_7$
  • Figure 6: The generalized Rothe diagrams of two elements in $\Omega_{U'}$

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Remark 2.4
  • proof : Proof of Lemma \ref{['lem:nroots']}
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 76 more