In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan–Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan–Lusztig, the inverse Kazhdan–Lusztig and the Z -polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan–Lusztig, inverse Kazhdan–Lusztig and Z -polynomial of all sparse paving matroids.
{"title":"Matroid relaxations and Kazhdan–Lusztig non-degeneracy","authors":"L. Ferroni, Lorenzo Vecchi","doi":"10.5802/alco.244","DOIUrl":"https://doi.org/10.5802/alco.244","url":null,"abstract":"In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan–Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan–Lusztig, the inverse Kazhdan–Lusztig and the Z -polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan–Lusztig, inverse Kazhdan–Lusztig and Z -polynomial of all sparse paving matroids.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47441182","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 G be a 2-generated finite group. The generating graph Γ( G ) is the graph whose vertices are the elements of G and where two vertices g 1 and g 2 are adjacent if G = h g 1 ,g 2 i . This graph encodes the combinatorial structure of the distribution of generating pairs across G. In this paper we study some graph theoretic properties of Γ( G ), with particular emphasis on those properties that can be formulated in terms of forbidden induced subgraphs. In particular we investigate when the generating graph Γ( G ) is a cograph (giving a complete description when G is soluble) and when it is perfect (giving a complete description when G is nilpotent and proving, among other things, that Γ( S n ) and Γ( A n ) are perfect if and only if n (cid:54) 4). Finally we prove that for a finite group G , the properties that Γ( G ) is split, chordal or C 4 -free are equivalent.
{"title":"Forbidden subgraphs in generating graphs of finite groups","authors":"A. Lucchini, Daniele Nemmi","doi":"10.5802/alco.229","DOIUrl":"https://doi.org/10.5802/alco.229","url":null,"abstract":"Let G be a 2-generated finite group. The generating graph Γ( G ) is the graph whose vertices are the elements of G and where two vertices g 1 and g 2 are adjacent if G = h g 1 ,g 2 i . This graph encodes the combinatorial structure of the distribution of generating pairs across G. In this paper we study some graph theoretic properties of Γ( G ), with particular emphasis on those properties that can be formulated in terms of forbidden induced subgraphs. In particular we investigate when the generating graph Γ( G ) is a cograph (giving a complete description when G is soluble) and when it is perfect (giving a complete description when G is nilpotent and proving, among other things, that Γ( S n ) and Γ( A n ) are perfect if and only if n (cid:54) 4). Finally we prove that for a finite group G , the properties that Γ( G ) is split, chordal or C 4 -free are equivalent.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43889023","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}
Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_lambda$ are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials and can be viewed as partition functions in $mathfrak{sl}(2)$ higher spin six vertex models. We obtain a refined Littlewood identity expressing a weighted sum of $F_lambda$'s over all partitions $lambda$ with even multiplicities as a certain Pfaffian. This Pfaffian can be derived as a partition function of the six vertex model in a triangle with suitably decorated domain wall boundary conditions. The proof is based on the Yang-Baxter equation.
{"title":"Refined Littlewood identity for spin Hall–Littlewood symmetric rational functions","authors":"S. Gavrilova","doi":"10.5802/alco.251","DOIUrl":"https://doi.org/10.5802/alco.251","url":null,"abstract":"Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_lambda$ are multiparameter deformations of the classical Hall-Littlewood symmetric polynomials and can be viewed as partition functions in $mathfrak{sl}(2)$ higher spin six vertex models. We obtain a refined Littlewood identity expressing a weighted sum of $F_lambda$'s over all partitions $lambda$ with even multiplicities as a certain Pfaffian. This Pfaffian can be derived as a partition function of the six vertex model in a triangle with suitably decorated domain wall boundary conditions. The proof is based on the Yang-Baxter equation.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48990129","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}
Two elements g and h of a permutation group G acting on a set V are said to be intersecting if vg = vh for some v ∈ V . More generally, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(G) of a transitive permutation group G is the maximum value of the quotient |F|/|Gv | where F runs over all intersecting sets in G and Gv is a stabilizer of v ∈ V . In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be either 1 or 2. In addition, it is proved that the intersection density of transitive groups of prime power degree is 1. 1. Introductory remarks For a finite set V let Sym(V ) and Alt(V ) denote the corresponding symmetric group and alternating group on V . (Of course, if |V | = n the standard notations Sn, An apply.) Let G 6 Sym(V ) be a permutation group acting on a set V . Two elements g, h ∈ G are said to be intersecting if v = v for some v ∈ V . Furthermore, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(F) of the intersecting set F is defined to be the quotient ρ(F) = |F| maxv∈V |Gv| , and the intersection density ρ(G) of a group G, first defined by Li, Song and Pantangi in [8], is the maximum value of ρ(F) where F runs over all intersecting sets in G, that is, ρ(G) = max{ρ(F) : F ⊆ G,F is intersecting} = max{|F| : F ⊂ G is intersecting} maxv∈V |Gv| . Manuscript received 26th September 2021, accepted 18th November 2021.
{"title":"Intersection density of transitive groups of certain degrees","authors":"Ademir Hujdurovi'c, Dragan Maruvsivc, vStefko Miklavivc, Klavdija Kutnar","doi":"10.5802/alco.209","DOIUrl":"https://doi.org/10.5802/alco.209","url":null,"abstract":"Two elements g and h of a permutation group G acting on a set V are said to be intersecting if vg = vh for some v ∈ V . More generally, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(G) of a transitive permutation group G is the maximum value of the quotient |F|/|Gv | where F runs over all intersecting sets in G and Gv is a stabilizer of v ∈ V . In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be either 1 or 2. In addition, it is proved that the intersection density of transitive groups of prime power degree is 1. 1. Introductory remarks For a finite set V let Sym(V ) and Alt(V ) denote the corresponding symmetric group and alternating group on V . (Of course, if |V | = n the standard notations Sn, An apply.) Let G 6 Sym(V ) be a permutation group acting on a set V . Two elements g, h ∈ G are said to be intersecting if v = v for some v ∈ V . Furthermore, a subset F of G is an intersecting set if every pair of elements of F is intersecting. The intersection density ρ(F) of the intersecting set F is defined to be the quotient ρ(F) = |F| maxv∈V |Gv| , and the intersection density ρ(G) of a group G, first defined by Li, Song and Pantangi in [8], is the maximum value of ρ(F) where F runs over all intersecting sets in G, that is, ρ(G) = max{ρ(F) : F ⊆ G,F is intersecting} = max{|F| : F ⊂ G is intersecting} maxv∈V |Gv| . Manuscript received 26th September 2021, accepted 18th November 2021.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44159808","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}
The toughness t(G) of a graph G = (V, E) is defined as t(G) = min { |S| c(G−S) } , in which the minimum is taken over all S ⊂ V such that G − S is disconnected, where c(G − S) denotes the number of components of G − S. We present two tight lower bounds for t(G) in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalues are obtained, which leads to a conjecture about Hamiltonicity and Laplacian eigenvalues.
韧性t (G)的图G = (V, E)的定义是t (G) =分钟{| S | c (G−)},最低的接管所有的S⊂V G−年代是断开连接,其中c (G−S)表示数量的组件G−S .我们现在两个紧下界t (G)的拉普拉斯算子特征值和提供强有力支持猜想到一个更好的约束,如果情况属实,意味着这两个范围,改进和推广了已知边界的阿龙,这和第一作者。作为应用，得到了关于拉普拉斯特征值的完美匹配、因子和行走的几个新结果，从而引出了关于拉普拉斯特征值和哈密顿性的猜想。
{"title":"Graph toughness from Laplacian eigenvalues","authors":"Xiaofeng Gu, W. Haemers","doi":"10.5802/alco.197","DOIUrl":"https://doi.org/10.5802/alco.197","url":null,"abstract":"The toughness t(G) of a graph G = (V, E) is defined as t(G) = min { |S| c(G−S) } , in which the minimum is taken over all S ⊂ V such that G − S is disconnected, where c(G − S) denotes the number of components of G − S. We present two tight lower bounds for t(G) in terms of the Laplacian eigenvalues and provide strong support for a conjecture for a better bound which, if true, implies both bounds, and improves and generalizes known bounds by Alon, Brouwer, and the first author. As applications, several new results on perfect matchings, factors and walks from Laplacian eigenvalues are obtained, which leads to a conjecture about Hamiltonicity and Laplacian eigenvalues.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48702617","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 this article, we consider involutions, called togglings, on the set of independent sets of the Dynkin diagram of type A, or a path graph. We are interested in the action of the subgroup of the symmetric group of the set of independent sets generated by togglings. We show that the subgroup coincides with the symmetric group.
{"title":"On the action of the toggle group of the Dynkin diagram of type A","authors":"Yasuhide Numata, Yuiko Yamanouchi","doi":"10.5802/alco.204","DOIUrl":"https://doi.org/10.5802/alco.204","url":null,"abstract":"In this article, we consider involutions, called togglings, on the set of independent sets of the Dynkin diagram of type A, or a path graph. We are interested in the action of the subgroup of the symmetric group of the set of independent sets generated by togglings. We show that the subgroup coincides with the symmetric group.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46009984","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}
Zachary Hamaker, A. Morales, I. Pak, Luis G. Serrano, N. Williams
We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge's symmetry of LR-coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of K-theory, leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020).
{"title":"Bijecting hidden symmetries for skew staircase shapes","authors":"Zachary Hamaker, A. Morales, I. Pak, Luis G. Serrano, N. Williams","doi":"10.5802/alco.285","DOIUrl":"https://doi.org/10.5802/alco.285","url":null,"abstract":"We present a bijection between the set of standard Young tableaux of staircase minus rectangle shape, and the set of marked shifted standard Young tableaux of a certain shifted shape. Numerically, this result is due to DeWitt (2012). Combined with other known bijections this gives a bijective proof of the product formula for the number of standard Young tableaux of staircase minus rectangle shape. This resolves an open problem by Morales, Pak and Panova (2019), and allows for efficient random sampling. Other applications include a bijection for semistandard Young tableaux, and a bijective proof of Stembridge's symmetry of LR-coefficients of the staircase shape. We also extend these results to set-valued standard Young tableaux in the combinatorics of K-theory, leading to new proofs of results by Lewis and Marberg (2019) and Abney-McPeek, An and Ng (2020).","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43251441","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}
Given a symmetric pair $(G,K)=(mathrm{GL}_{p+q}(mathbb{C}),mathrm{GL}_{p}(mathbb{C})times mathrm{GL}_{q}(mathbb{C}))$ of type AIII, we consider the diagonal action of $K$ on the double flag variety $mathfrak{X}=mathrm{Grass}(mathbb{C}^{p+q},r)times K/B_K$ whose first factor is a Grassmann variety for $G$ and whose second factor is a full flag variety of $K$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in $mathfrak{X}$ gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed $p=q=r$ and considered orbits of special forms.
{"title":"On generalized Steinberg theory for type AIII","authors":"Lucas Fresse, Kyo Nishiyama","doi":"10.5802/alco.245","DOIUrl":"https://doi.org/10.5802/alco.245","url":null,"abstract":"Given a symmetric pair $(G,K)=(mathrm{GL}_{p+q}(mathbb{C}),mathrm{GL}_{p}(mathbb{C})times mathrm{GL}_{q}(mathbb{C}))$ of type AIII, we consider the diagonal action of $K$ on the double flag variety $mathfrak{X}=mathrm{Grass}(mathbb{C}^{p+q},r)times K/B_K$ whose first factor is a Grassmann variety for $G$ and whose second factor is a full flag variety of $K$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in $mathfrak{X}$ gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed $p=q=r$ and considered orbits of special forms.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44578704","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 derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial techniques related to the intervals of quasi-Coxeter elements. In a subsequent work [4], we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations. Alongside the proof of the main results, we establish important properties related to the dual approach to Coxeter and Artin groups.
{"title":"Interval groups related to finite Coxeter groups I","authors":"B. Baumeister, Georges Neaime, Sarah Rees","doi":"10.5802/alco.266","DOIUrl":"https://doi.org/10.5802/alco.266","url":null,"abstract":"We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial techniques related to the intervals of quasi-Coxeter elements. In a subsequent work [4], we complete our analysis to cover all the exceptional cases of finite Coxeter groups, and establish that almost all the interval groups related to proper quasi-Coxeter elements are not isomorphic to the related Artin groups, hence establishing a new family of interval groups with nice presentations. Alongside the proof of the main results, we establish important properties related to the dual approach to Coxeter and Artin groups.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47475831","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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