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Constructing generalized Heffter arrays via near alternating sign matrices

Lorenzo Mella, Tommaso Traetta

Abstract

Let $S$ be a subset of a group $G$ (not necessarily abelian) such that $S\,\cap -S$ is empty or contains only elements of order $2$, and let $\mathbf{h}=(h_1,\ldots, h_m)\in \mathbb{N}^m$ and $\mathbf{k}=(k_1, \ldots, k_n)\in \mathbb{N}^n$. A generalized Heffter array GHA$^λ_S(m, n; \mathbf{h}, \mathbf{k})$ over $G$ is an $m\times n$ matrix $A=(a_{ij})$ such that: the $i$-th row (resp. $j$-th column) of $A$ contains exactly $h_i$ (resp. $k_j$) nonzero elements, and the list $\{a_{ij}, -a_{ij}\mid a_{ij}\neq 0\}$ equals $λ$ times the set $S\,\cup\, -S$. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of $A$ sums to zero (resp. a nonzero element), with respect to some ordering. In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to $0$ modulo $4$. This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular ($m\neq n$) and with less than $n$ nonzero entries in each row. Furthermore, we build nonzero sum GHA$^λ_S(m, n; \mathbf{h}, \mathbf{k})$ over an arbitrary group $G$ whenever $S$ contains enough noninvolutions, thus extending previous nonconstructive results where $\pm S = G\setminus H$ for some subgroup $H$~of~$G$. Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.

Constructing generalized Heffter arrays via near alternating sign matrices

Abstract

Let be a subset of a group (not necessarily abelian) such that is empty or contains only elements of order , and let and . A generalized Heffter array GHA over is an matrix such that: the -th row (resp. -th column) of contains exactly (resp. ) nonzero elements, and the list equals times the set . We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of sums to zero (resp. a nonzero element), with respect to some ordering. In this paper, we use near alternating sign matrices to build both zero and nonzero sum GHAs, over cyclic groups, having the further strong property of being simple. In particular, we construct zero sum and simple GHAs whose row and column weights are congruent to modulo . This result also provides the first infinite family of simple (classic) Heffter arrays to be rectangular () and with less than nonzero entries in each row. Furthermore, we build nonzero sum GHA over an arbitrary group whenever contains enough noninvolutions, thus extending previous nonconstructive results where for some subgroup ~of~. Finally, we describe how GHAs can be used to build orthogonal decompositions and biembeddings of Cayley graphs (over groups not necessarily abelian) onto orientable surfaces.
Paper Structure (17 sections, 22 theorems, 43 equations)

This paper contains 17 sections, 22 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. Sequences over an arbitrary group $G$
  4. Matrices over an arbitrary group $G$
  5. Near alternating sign matrices
  6. Simple GHAs over a cyclic group
  7. Nonzero sum and simple GHAs
  8. Zero sum and simple GHAs
  9. Nonzero sum GHAs over an arbitrary group
  10. 1. Filling ${\cal B} = skel(A) \setminus ({\cal R} \, \cup \, {\cal C})$.
  11. 2. Filling isolated vertices of $\Phi$.
  12. 3. Filling $W=V(\Phi_z)\setminus V(P_z)$, for every $z \in {\cal L}\setminus {\cal L}'$.
  13. 4. Filling $V(P_z)$, for every $z \in {\cal L}\setminus {\cal L}'$.
  14. GHAs and orthogonal decompositions
  15. Orthogonal decompositions of Cayley (di)graphs
  16. ...and 2 more sections

Key Result

Theorem 1.2

Let $\mathbf h=(h_1, \ldots, h_m)$ and $\mathbf k=(k_1, \ldots,k_n)$ be two sequences of positive integers such that ${\sum_{i=1}^{m} h_i=\sum_{j=1}^{n} k_j}$. There exists an $m\times n$ matrix $A$ over $\mathbb{Z}_2$ with $w_r(A)=\mathbf h$ and $w_c(A)=\mathbf k$ if and only if where $(k'_1, \ldots, k'_n)$ is the decreasing reordering of $\mathbf k$.

Theorems & Definitions (57)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 2.1
  • Definition 3.1
  • ...and 47 more