Partial-Twuality Polynomials of Paired Matrices
Abstract
Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the \emph{partial-twuality polynomials} of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality δ and partial Petrie duality τ, on pairs (M,A), where M is a square matrix whose rows and columns are indexed by a finite set V and A⊆ V, such that δ2=τ2=(δτ)3=id and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying • to (M, A). In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field 𝔾𝔽(2). We prove that there exist two local operations δ and τ on ((M,I|V|),A) satisfying δ2=τ2=(δτ)3=id and P⟨ • ⟩((M,I|V|),z)=P⟨ • ⟩(M,z) for • ∈ {δ, τ, δτ, τδ, δτδ}, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial ⟨δτδ⟩-polynomial with respect to an edge. This recurrence enables the computation of the partial ⟨δτδ⟩-polynomial for certain bouquets, simple graphs, and simple signed graphs.
Related
- Irreducibility of Full Multivariate Partial-Twuality Polynomials
- Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds
- Tridiagonal Matrices Associated with Dual Hahn Polynomials
- Large Growth Happens: Gaussian Elimination with Partial Pivoting on Random Matrices
- The Operator Systems of Hankel Matrices and Complex Polynomials
BibTeX
@misc{yu2026partialtwuality,
title = {{Partial-Twuality Polynomials of Paired Matrices}},
author = {Xiaoxiang Yu and Rong-Xia Hao and Jianbing Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22273},
eprint = {2609.22273},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22273},
}
RIS
TY - UNPB
TI - Partial-Twuality Polynomials of Paired Matrices
AU - Xiaoxiang Yu
AU - Rong-Xia Hao
AU - Jianbing Liu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.22273
UR - https://arxiv.org/abs/2609.22273
AB - Gross, Mansour, and Tucker~[European Journal of Combinatorics, 95 (2021): 103329] introduced the \emph{partial-twuality polynomials} of ribbon graphs. Recently, Deng, Jin, and Yan generalized the partial-twuality polynomials to the framework of matrix algebra and investigated several of their basic properties. They asked whether there exist matrix operations, called partial duality δ and partial Petrie duality τ, on pairs (M,A), where M is a square matrix whose rows and columns are indexed by a finite set V and A⊆ V, such that δ^2=τ^2=(δτ)^3=id and the exponent of the partial-twuality polynomials coincides with some parameter of the matrix obtained by applying • to (M, A). In this paper, we introduce a paired-matrix framework for partial-twuality polynomials over the binary field 𝔾𝔽(2). We prove that there exist two local operations δ and τ on ((M,I_|V|),A) satisfying δ^2=τ^2=(δτ)^3=id and P_⟨ • ⟩((M,I_|V|),z)=P_⟨ • ⟩(M,z) for • ∈ {δ, τ, δτ, τδ, δτδ}, thereby answering their question affirmatively. Finally, we establish a recurrence relation for the partial ⟨δτδ⟩-polynomial with respect to an edge. This recurrence enables the computation of the partial ⟨δτδ⟩-polynomial for certain bouquets, simple graphs, and simple signed graphs.
ER -