Let G be a (finite or infinite) group such that G/Z(G) is not simple. The non-commuting, non-generating graph Ξ(G) of G has vertex set G∖Z(G), with vertices x and y adjacent whenever [x,y]≠1 and 〈x,y〉≠G. We investigate the relationship between the structure of G and the connectedness and diameter of Ξ(G). In particular, we prove that the graph either: (i) is connected with diameter at most 4; (ii) consists of isolated vertices and a connected component of diameter at most 4; or (iii) is the union of two connected components of diameter 2. We also describe in detail the finite groups with graphs of type (iii). In the companion paper [17], we consider the case where G/Z(G) is finite and simple.
{"title":"The non-commuting, non-generating graph of a non-simple group","authors":"Saul D. Freedman","doi":"10.5802/alco.305","DOIUrl":"https://doi.org/10.5802/alco.305","url":null,"abstract":"Let G be a (finite or infinite) group such that G/Z(G) is not simple. The non-commuting, non-generating graph Ξ(G) of G has vertex set G∖Z(G), with vertices x and y adjacent whenever [x,y]≠1 and 〈x,y〉≠G. We investigate the relationship between the structure of G and the connectedness and diameter of Ξ(G). In particular, we prove that the graph either: (i) is connected with diameter at most 4; (ii) consists of isolated vertices and a connected component of diameter at most 4; or (iii) is the union of two connected components of diameter 2. We also describe in detail the finite groups with graphs of type (iii). In the companion paper [17], we consider the case where G/Z(G) is finite and simple.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ademir Hujdurović, Đorđe Mitrović, Dave Witte Morris
A graph X is said to be unstable if the direct product X×K 2 (also called the canonical double cover of X) has automorphisms that do not come from automorphisms of its factors X and K 2 . It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors.
{"title":"Automorphisms of the double cover of a circulant graph of valency at most 7","authors":"Ademir Hujdurović, Đorđe Mitrović, Dave Witte Morris","doi":"10.5802/alco.303","DOIUrl":"https://doi.org/10.5802/alco.303","url":null,"abstract":"A graph X is said to be unstable if the direct product X×K 2 (also called the canonical double cover of X) has automorphisms that do not come from automorphisms of its factors X and K 2 . It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135431087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A PSCA(v,t,λ) is a multiset of permutations of the v-element alphabet {0,⋯,v-1} such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly λ permutations. For v⩾t, let g(v,t) be the smallest positive integer λ such that a PSCA(v,t,λ) exists. Kuperberg, Lovett and Peled proved g(v,t)=O(v t ) using probabilistic methods. We present an explicit construction that proves g(v,t)=O(v t(t-2) ) for fixed t⩾4. The method of construction involves taking a permutation representation of the group of projectivities of a suitable projective space of dimension t-2 and deleting all but a certain number of symbols from each permutation. In the case that this space is a Desarguesian projective plane, we also show that there exists a permutation representation of the group of projectivities of the plane that covers the vast majority of 4-sequences of its points the same number of times.
{"title":"A polynomial construction of perfect sequence covering arrays","authors":"Aidan R. Gentle","doi":"10.5802/alco.308","DOIUrl":"https://doi.org/10.5802/alco.308","url":null,"abstract":"A PSCA(v,t,λ) is a multiset of permutations of the v-element alphabet {0,⋯,v-1} such that every sequence of t distinct elements of the alphabet appears in the specified order in exactly λ permutations. For v⩾t, let g(v,t) be the smallest positive integer λ such that a PSCA(v,t,λ) exists. Kuperberg, Lovett and Peled proved g(v,t)=O(v t ) using probabilistic methods. We present an explicit construction that proves g(v,t)=O(v t(t-2) ) for fixed t⩾4. The method of construction involves taking a permutation representation of the group of projectivities of a suitable projective space of dimension t-2 and deleting all but a certain number of symbols from each permutation. In the case that this space is a Desarguesian projective plane, we also show that there exists a permutation representation of the group of projectivities of the plane that covers the vast majority of 4-sequences of its points the same number of times.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate Voronoi diagrams with respect to an asymmetric tropical distance function, in particular for infinite point sets. These Voronoi diagrams turn out to be much better behaved than those arising from the standard tropical distance, which is symmetric. In particular, we show that the asymmetric tropical Voronoi diagrams may be seen as tropicalizations of power diagrams over fields of real Puiseux series. Our results are then applied to rational lattices and Laurent monomial modules.
{"title":"Asymmetric tropical distances and power diagrams","authors":"Andrei Comăneci, Michael Joswig","doi":"10.5802/alco.306","DOIUrl":"https://doi.org/10.5802/alco.306","url":null,"abstract":"We investigate Voronoi diagrams with respect to an asymmetric tropical distance function, in particular for infinite point sets. These Voronoi diagrams turn out to be much better behaved than those arising from the standard tropical distance, which is symmetric. In particular, we show that the asymmetric tropical Voronoi diagrams may be seen as tropicalizations of power diagrams over fields of real Puiseux series. Our results are then applied to rational lattices and Laurent monomial modules.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135433008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We compare the canonical bases of level-1 quantised Fock spaces in affine types A (1) and A (2) , showing how to derive the canonical basis in type A 2n (2) from the the canonical basis in type A n (1) in certain weight spaces. In particular, we derive an explicit formula for the canonical basis in extremal weight spaces, which correspond to RoCK blocks of double covers of symmetric groups. In a forthcoming paper with Kleshchev and Morotti we will use this formula to find the decomposition numbers for RoCK blocks of double covers with abelian defect.
{"title":"Comparing Fock spaces in types A (1) and A (2) ","authors":"Matthew Fayers","doi":"10.5802/alco.300","DOIUrl":"https://doi.org/10.5802/alco.300","url":null,"abstract":"We compare the canonical bases of level-1 quantised Fock spaces in affine types A (1) and A (2) , showing how to derive the canonical basis in type A 2n (2) from the the canonical basis in type A n (1) in certain weight spaces. In particular, we derive an explicit formula for the canonical basis in extremal weight spaces, which correspond to RoCK blocks of double covers of symmetric groups. In a forthcoming paper with Kleshchev and Morotti we will use this formula to find the decomposition numbers for RoCK blocks of double covers with abelian defect.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the open subvariety Gr 0 (3,8) of the Grassmannian Gr(3,8) determined by the nonvanishing of all Plücker coordinates is schön, i.e. all of its initial degenerations are smooth. Furthermore, we find an initial degeneration that has two connected components, and show that the remaining initial degenerations, up to symmetry, are irreducible. As an application, we prove that the Chow quotient of Gr(3,8) by the diagonal torus of PGL(8) is the log canonical compactification of the moduli space of 8 lines in ℙ 2 , resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes.
{"title":"The Grassmannian of 3-planes in ℂ 8 is schön","authors":"Daniel Corey, Dante Luber","doi":"10.5802/alco.302","DOIUrl":"https://doi.org/10.5802/alco.302","url":null,"abstract":"We prove that the open subvariety Gr 0 (3,8) of the Grassmannian Gr(3,8) determined by the nonvanishing of all Plücker coordinates is schön, i.e. all of its initial degenerations are smooth. Furthermore, we find an initial degeneration that has two connected components, and show that the remaining initial degenerations, up to symmetry, are irreducible. As an application, we prove that the Chow quotient of Gr(3,8) by the diagonal torus of PGL(8) is the log canonical compactification of the moduli space of 8 lines in ℙ 2 , resolving a conjecture of Hacking, Keel, and Tevelev. Along the way we develop various techniques to study finite inverse limits of schemes.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135540530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field ℚ. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.
{"title":"On the anisotropy theorem of Papadakis and Petrotou","authors":"Kalle Karu, Elizabeth Xiao","doi":"10.5802/alco.298","DOIUrl":"https://doi.org/10.5802/alco.298","url":null,"abstract":"We study the anisotropy theorem for Stanley-Reisner rings of simplicial homology spheres in characteristic 2 by Papadakis and Petrotou. This theorem implies the Hard Lefschetz theorem as well as McMullen’s g-conjecture for such spheres. Our first result is an explicit description of the quadratic form. We use this description to prove a conjecture stated by Papadakis and Petrotou. All anisotropy theorems for homology spheres and pseudo-manifolds in characteristic 2 follow from this conjecture. Using a specialization argument, we prove anisotropy for certain homology spheres over the field ℚ. These results provide another self-contained proof of the g-conjecture for homology spheres in characteristic 2.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135540335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let ℓ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when n is sufficiently large, the ℓ-adic valuation of the number of spanning trees at the nth layer of a ℤ ℓ d -tower of graphs is given by a polynomial in ℓ n and n with rational coefficients of total degree at most d and of degree in n at most one.
{"title":"On ℤ ℓ d -towers of graphs","authors":"Sage DuBose, Daniel Vallières","doi":"10.5802/alco.304","DOIUrl":"https://doi.org/10.5802/alco.304","url":null,"abstract":"Let ℓ be a rational prime. We show that an analogue of a conjecture of Greenberg in graph theory holds true. More precisely, we show that when n is sufficiently large, the ℓ-adic valuation of the number of spanning trees at the nth layer of a ℤ ℓ d -tower of graphs is given by a polynomial in ℓ n and n with rational coefficients of total degree at most d and of degree in n at most one.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let r≤n be nonnegative integers, and let N=n r-1. For a matroid M of rank r on the finite set E=[n] and a partial field k in the sense of Semple–Whittle, it is known that the following are equivalent: (a) M is representable over k; (b) there is a point p=(p J )∈P N (k) with support M (meaning that Supp(p):={J∈E r|p J ≠0} of p is the set of bases of M) satisfying the Grassmann-Plücker equations; and (c) there is a point p=(p J )∈P N (k) with support M satisfying just the 3-term Grassmann-Plücker equations. Moreover, by a theorem of P. Nelson, almost all matroids (meaning asymptotically 100%) are not representable over any partial field. We prove analogues of these facts for Lagrangian orthogonal matroids in the sense of Gelfand–Serganova, which are equivalent to even Delta-matroids in the sense of Bouchet.
{"title":"Representability of orthogonal matroids over partial fields","authors":"Matthew Baker, Tong Jin","doi":"10.5802/alco.301","DOIUrl":"https://doi.org/10.5802/alco.301","url":null,"abstract":"Let r≤n be nonnegative integers, and let N=n r-1. For a matroid M of rank r on the finite set E=[n] and a partial field k in the sense of Semple–Whittle, it is known that the following are equivalent: (a) M is representable over k; (b) there is a point p=(p J )∈P N (k) with support M (meaning that Supp(p):={J∈E r|p J ≠0} of p is the set of bases of M) satisfying the Grassmann-Plücker equations; and (c) there is a point p=(p J )∈P N (k) with support M satisfying just the 3-term Grassmann-Plücker equations. Moreover, by a theorem of P. Nelson, almost all matroids (meaning asymptotically 100%) are not representable over any partial field. We prove analogues of these facts for Lagrangian orthogonal matroids in the sense of Gelfand–Serganova, which are equivalent to even Delta-matroids in the sense of Bouchet.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135475566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In January 1969, Peter M. Neumann wrote a paper entitled “Primitive permutation groups of degree 3p”. The main theorem placed restrictions on the parameters of a primitive but not 2-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree.
1969年1月，彼得·m·诺伊曼(Peter M. Neumann)写了一篇题为《3p度的原始置换群》的论文。主要定理对三次素数的原始而非2-传递置换群的参数进行了限制。这篇论文从未发表过，其结果已经被更强的定理所取代，这些定理依赖于有限单群的分类，例如奇数次原始群的分类。
{"title":"Association schemes with given stratum dimensions: on a paper of Peter M. Neumann","authors":"Marina Anagnostopoulou-Merkouri, Peter J. Cameron","doi":"10.5802/alco.307","DOIUrl":"https://doi.org/10.5802/alco.307","url":null,"abstract":"In January 1969, Peter M. Neumann wrote a paper entitled “Primitive permutation groups of degree 3p”. The main theorem placed restrictions on the parameters of a primitive but not 2-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135432350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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