{"title":"通过近交替符号矩阵构建广义赫夫特阵列","authors":"L. Mella , T. Traetta","doi":"10.1016/j.jcta.2024.105873","DOIUrl":null,"url":null,"abstract":"<div><p>Let <em>S</em> be a subset of a group <em>G</em> (not necessarily abelian) such that <span><math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math></span> is empty or contains only elements of order 2, and let <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <span><math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A <em>generalized Heffter array</em> GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over <em>G</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> such that: the <em>i</em>-th row (resp. <em>j</em>-th column) of <em>A</em> contains exactly <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) nonzero elements, and the list <span><math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math></span> equals <em>λ</em> times the set <span><math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math></span>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of <em>A</em> sums to zero (resp. a nonzero element), with respect to some ordering.</p><p>In this paper, we use <em>near alternating sign matrices</em> 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 (<span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>) and with less than <em>n</em> nonzero entries in each row. Furthermore, we build nonzero sum GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over an arbitrary group <em>G</em> whenever <em>S</em> contains enough noninvolutions, thus extending previous nonconstructive results where <span><math><mo>±</mo><mi>S</mi><mo>=</mo><mi>G</mi><mo>∖</mo><mi>H</mi></math></span> for some subgroup <em>H</em> of <em>G</em>.</p><p>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.</p></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Constructing generalized Heffter arrays via near alternating sign matrices\",\"authors\":\"L. Mella , T. Traetta\",\"doi\":\"10.1016/j.jcta.2024.105873\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <em>S</em> be a subset of a group <em>G</em> (not necessarily abelian) such that <span><math><mi>S</mi><mspace></mspace><mo>∩</mo><mo>−</mo><mi>S</mi></math></span> is empty or contains only elements of order 2, and let <span><math><mi>h</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>h</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <span><math><mi>k</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A <em>generalized Heffter array</em> GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over <em>G</em> is an <span><math><mi>m</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi><mo>=</mo><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></math></span> such that: the <em>i</em>-th row (resp. <em>j</em>-th column) of <em>A</em> contains exactly <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> (resp. <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>) nonzero elements, and the list <span><math><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>,</mo><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>|</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≠</mo><mn>0</mn><mo>}</mo></math></span> equals <em>λ</em> times the set <span><math><mi>S</mi><mspace></mspace><mo>∪</mo><mspace></mspace><mo>−</mo><mi>S</mi></math></span>. We speak of a zero sum (resp. nonzero sum) GHA if each row and each column of <em>A</em> sums to zero (resp. a nonzero element), with respect to some ordering.</p><p>In this paper, we use <em>near alternating sign matrices</em> 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 (<span><math><mi>m</mi><mo>≠</mo><mi>n</mi></math></span>) and with less than <em>n</em> nonzero entries in each row. Furthermore, we build nonzero sum GHA<span><math><msubsup><mrow></mrow><mrow><mi>S</mi></mrow><mrow><mi>λ</mi></mrow></msubsup><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>;</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math></span> over an arbitrary group <em>G</em> whenever <em>S</em> contains enough noninvolutions, thus extending previous nonconstructive results where <span><math><mo>±</mo><mi>S</mi><mo>=</mo><mi>G</mi><mo>∖</mo><mi>H</mi></math></span> for some subgroup <em>H</em> of <em>G</em>.</p><p>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.</p></div>\",\"PeriodicalId\":50230,\"journal\":{\"name\":\"Journal of Combinatorial Theory Series A\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-02-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Theory Series A\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0097316524000128\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316524000128","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
设 S 是一个群 G 的子集(不一定是非良性的),使得 S∩-S 是空的或只包含阶数为 2 的元素,并且设 h=(h1,...,hm)∈Nm,k=(k1,...,kn )∈Nn。G 上的广义赫夫特数组 GHASλ(m,n;h,k) 是一个 m×n 矩阵 A=(aij),使得:A 的第 i 行(或第 j 列)恰好包含 hi(或 kj)个非零元素,且列表 {aij,-aij|aij≠0} 等于集合 S∪-S 的 λ 倍。在本文中,我们使用近交替符号矩阵来构建循环群上的零和(或非零和)GHA,它还具有简单的强性质。特别是,我们构建的零和简单 GHA,其行权重和列权重同余为 0 modulo 4。这一结果还提供了第一个矩形(m≠n)且每行非零条目少于 n 个的简单(经典)赫夫特数组无穷族。此外,只要 S 包含足够多的非卷积,我们就能在任意群 G 上建立非零和 GHASλ(m,n;h,k),从而扩展了之前的非构造性结果,即对于 G 的某个子群 H,±S=G∖H。最后,我们描述了如何利用 GHA 来建立 Cayley 图(在不一定是无性的群上)到可定向曲面上的正交分解和双嵌套。
Constructing generalized Heffter arrays via near alternating sign matrices
Let S be a subset of a group G (not necessarily abelian) such that is empty or contains only elements of order 2, and let and . A generalized Heffter array GHA over G is an matrix such that: the i-th row (resp. j-th column) of A contains exactly (resp. ) nonzero elements, and the list equals λ times the set . 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 () and with less than n nonzero entries in each row. Furthermore, we build nonzero sum GHA over an arbitrary group G whenever S contains enough noninvolutions, thus extending previous nonconstructive results where 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.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.