Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908886753083392 |
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| author | Green, R. M. Xu, Tianyuan |
| author_facet | Green, R. M. Xu, Tianyuan |
| contents | 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 for $ρ$, regarded as a $q$-polynomial in $U$. In this paper, we study a large number of examples of sets of the form $Ω_U$, 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 $Ω_U$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams Green, R. M. Xu, Tianyuan Combinatorics Group Theory Quantum Algebra Primary: 17B22, Secondary: 05E18, 20F55 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 for $ρ$, regarded as a $q$-polynomial in $U$. In this paper, we study a large number of examples of sets of the form $Ω_U$, 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 $Ω_U$ 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. |
| title | Orthogonal roots, quantum Hafnians, and generalized Rothe diagrams |
| topic | Combinatorics Group Theory Quantum Algebra Primary: 17B22, Secondary: 05E18, 20F55 |
| url | https://arxiv.org/abs/2510.25041 |