. We consider a tower of generalized rook monoid algebras over the field C of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field k of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0 ≤ i ≤ p − 1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum G C of the Grothendieck groups of module categories over rook monoid algebras over k , these functors induce an action of the tensor product of the universal enveloping algebra U ( b sl p ( C )) and the monoid algebra C [ B ] of the bicyclic monoid B . Furthermore, we prove that G C is isomorphic to the tensor product of the basic representation of U ( b sl p ( C )) and the unique infinite-dimensional simple module over C [ B ], and also exhibit that G C is a bialgebra. Under some natural restrictions on the characteristic of k , we outline the corresponding result for generalized rook monoids.
{"title":"Jucys–Murphy elements and Grothendieck groups for generalized rook monoids","authors":"V. Mazorchuk, S. Srivastava","doi":"10.4171/JCA/65","DOIUrl":"https://doi.org/10.4171/JCA/65","url":null,"abstract":". We consider a tower of generalized rook monoid algebras over the field C of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys–Murphy elements for generalized rook monoid algebras. Over an algebraically closed field k of positive characteristic p , utilizing Jucys–Murphy elements of rook monoid algebras, for 0 ≤ i ≤ p − 1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum G C of the Grothendieck groups of module categories over rook monoid algebras over k , these functors induce an action of the tensor product of the universal enveloping algebra U ( b sl p ( C )) and the monoid algebra C [ B ] of the bicyclic monoid B . Furthermore, we prove that G C is isomorphic to the tensor product of the basic representation of U ( b sl p ( C )) and the unique infinite-dimensional simple module over C [ B ], and also exhibit that G C is a bialgebra. Under some natural restrictions on the characteristic of k , we outline the corresponding result for generalized rook monoids.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42904951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Developing an idea of M. Gromov in [5] 9.A, we study the intersection formula for random subsets with density. The density of a subset A in a finite set E is defined by densA := log|E|(|A|). The aim of this article is to give a precise meaning of Gromov’s intersection formula: "Random subsets" A and B of a finite set E satisfy dens(A∩B) = densA+densB−1. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density λ/2 for any 0 < λ < 1, characterizing the C(λ)-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol’shanskii in [2] from density 0 to density 0 ≤ d < 1 120m2 ln(2m) .
{"title":"Density of random subsets and applications to group theory","authors":"Tsung-Hsuan Tsai","doi":"10.4171/jca/63","DOIUrl":"https://doi.org/10.4171/jca/63","url":null,"abstract":"Developing an idea of M. Gromov in [5] 9.A, we study the intersection formula for random subsets with density. The density of a subset A in a finite set E is defined by densA := log|E|(|A|). The aim of this article is to give a precise meaning of Gromov’s intersection formula: \"Random subsets\" A and B of a finite set E satisfy dens(A∩B) = densA+densB−1. As an application, we exhibit a phase transition phenomenon for random presentations of groups at density λ/2 for any 0 < λ < 1, characterizing the C(λ)-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol’shanskii in [2] from density 0 to density 0 ≤ d < 1 120m2 ln(2m) .","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42280681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A regular left-order on a finitely generated group G is a total, left-multiplication invariant order on G whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and we give a classification of the groups whose left-orders are all regular left-orders. In addition, we prove that a solvable Baumslag-Solitar group B(1, n) admits a regular left-order if and only if n ≥ −1. Finally, Hermiller and S̆unić showed that no free product admits a regular left-order. We show that if A and B are groups with regular left-orders, then (A ∗B)× Z admits a regular left-order. MSC 2020 classification: 06F15, 20F60, 68Q45
{"title":"Regular left-orders on groups","authors":"Y. Antol'in, C. Rivas, H. Su","doi":"10.4171/jca/64","DOIUrl":"https://doi.org/10.4171/jca/64","url":null,"abstract":"A regular left-order on a finitely generated group G is a total, left-multiplication invariant order on G whose corresponding positive cone is the image of a regular language over the generating set of the group under the evaluation map. We show that admitting regular left-orders is stable under extensions and wreath products and we give a classification of the groups whose left-orders are all regular left-orders. In addition, we prove that a solvable Baumslag-Solitar group B(1, n) admits a regular left-order if and only if n ≥ −1. Finally, Hermiller and S̆unić showed that no free product admits a regular left-order. We show that if A and B are groups with regular left-orders, then (A ∗B)× Z admits a regular left-order. MSC 2020 classification: 06F15, 20F60, 68Q45","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47326007","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $mathcal O$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster. Applying our construction to nonlinear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.
{"title":"A category $mathcal{O}$ for oriented matroids","authors":"Ethan Kowalenko, C. Mautner","doi":"10.4171/jca/71","DOIUrl":"https://doi.org/10.4171/jca/71","url":null,"abstract":"We associate to a sufficiently generic oriented matroid program and choice of linear system of parameters a finite dimensional algebra, whose representation theory is analogous to blocks of Bernstein--Gelfand--Gelfand category $mathcal O$. When the data above comes from a generic linear program for a hyperplane arrangement, we recover the algebra defined by Braden--Licata--Proudfoot--Webster. Applying our construction to nonlinear oriented matroid programs provides a large new class of algebras. For Euclidean oriented matroid programs, the resulting algebras are quasi-hereditary and Koszul, as in the linear setting. In the non-Euclidean case, we obtain algebras that are not quasi-hereditary and not known to be Koszul, but still have a natural class of standard modules and satisfy numerical analogues of quasi-heredity and Koszulity on the level of graded Grothendieck groups.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44897016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant, generalizing the classical Sch"afli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.
{"title":"Iterated and mixed discriminants","authors":"A. Dickenstein, S. Rocco, Ralph Morrison","doi":"10.4171/jca/68","DOIUrl":"https://doi.org/10.4171/jca/68","url":null,"abstract":"We consider systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant, generalizing the classical Sch\"afli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre-Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42249498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighbourhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighbourhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory.
{"title":"Conical $SL(3)$ foams","authors":"M. Khovanov, Louis-Hadrien Robert","doi":"10.4171/jca/61","DOIUrl":"https://doi.org/10.4171/jca/61","url":null,"abstract":"In the unoriented SL(3) foam theory, singular vertices are generic singularities of two-dimensional complexes. Singular vertices have neighbourhoods homeomorphic to cones over the one-skeleton of the tetrahedron, viewed as a trivalent graph on the two-sphere. In this paper we consider foams with singular vertices with neighbourhoods homeomorphic to cones over more general planar trivalent graphs. These graphs are subject to suitable conditions on their Kempe equivalence Tait coloring classes and include the dodecahedron graph. In this modification of the original homology theory it is straightforward to show that modules associated to the dodecahedron graph are free of rank 60, which is still an open problem for the original unoriented SL(3) foam theory.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47384039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. The two fundamental questions about maximal green sequences are whether a given algebra admits such sequences and, if so, does it admit only finitely many. We study maximal green sequences in the case of string algebras and give sufficient conditions on the algebra that ensure an affirmative answer to these questions.
{"title":"Maximal green sequences for string algebras","authors":"Alexander Garver, K. Serhiyenko","doi":"10.4171/jca/60","DOIUrl":"https://doi.org/10.4171/jca/60","url":null,"abstract":"Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. The two fundamental questions about maximal green sequences are whether a given algebra admits such sequences and, if so, does it admit only finitely many. We study maximal green sequences in the case of string algebras and give sufficient conditions on the algebra that ensure an affirmative answer to these questions.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42085064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
To every group $G$ we associate a linear monoidal category $mathcal{P}mathit{ar}(G)$ that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of $mathcal{P}mathit{ar}(G)$ into the group Heisenberg category associated to $G$. This embedding intertwines the natural actions of both categories on modules for wreath products of $G$. Finally, we prove that the additive Karoubi envelope of $mathcal{P}mathit{ar}(G)$ is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.
{"title":"Group partition categories","authors":"Samuel Nyobe Likeng, Alistair Savage","doi":"10.4171/JCA/55","DOIUrl":"https://doi.org/10.4171/JCA/55","url":null,"abstract":"To every group $G$ we associate a linear monoidal category $mathcal{P}mathit{ar}(G)$ that we call a group partition category. We give explicit bases for the morphism spaces and also an efficient presentation of the category in terms of generators and relations. We then define an embedding of $mathcal{P}mathit{ar}(G)$ into the group Heisenberg category associated to $G$. This embedding intertwines the natural actions of both categories on modules for wreath products of $G$. Finally, we prove that the additive Karoubi envelope of $mathcal{P}mathit{ar}(G)$ is equivalent to a wreath product interpolating category introduced by Knop, thereby giving a simple concrete description of that category.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43778036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the C-enriched pre-Lie operad defined by Calaque and Willwacher for any Hopf cooperad C to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding enriched pre-Lie operads; we prove criteria for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré–Birkhoff–Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.
{"title":"Enriched pre-Lie operads and freeness theorems","authors":"V. Dotsenko, L. Foissy","doi":"10.4171/jca/58","DOIUrl":"https://doi.org/10.4171/jca/58","url":null,"abstract":"In this paper, we study the C-enriched pre-Lie operad defined by Calaque and Willwacher for any Hopf cooperad C to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding enriched pre-Lie operads; we prove criteria for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré–Birkhoff–Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45188908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups. We show that, for every such group $T$, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $mathbf{N}$. Also, using invariant means, we address a classical problem, initiated by ErdH{o}s and Graham and then generalized by Nash and Nathanson both in the case of $mathbf{N}$, of estimating the maximal order $X_T(h,k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that $X_T(h,k)=O(h^{2k+1})$ for every integer $k ge 1$. This result is new even in the case where $k=1$. Besides the maximal order $X_T(h,k)$, the typical order $S_T(h,k)$ is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in $mathbf{N}$ and in abelian groups.
在Lambert, Plagne和第三作者先前工作的基础上，我们研究了无限阿贝尔群中加性基的行为的各个方面。我们证明了对于每一个这样的群$T$，任何加性基的本质子集的数目是有限的，并且在阶不超过$h$的加性基中包含的基数$k$的本质子集的数目可以仅由$h$和$k$有界。这些结果扩展了两个定理的范围，一个是由德尚和法希提出的，另一个是由赫加蒂提出的，它们与$mathbf{N}$有关。此外，使用不变方法，我们解决了一个经典问题，该问题由ErdH{o}和Graham提出，然后由Nash和Nathanson在$mathbf{N}$的情况下推广，即估计最大阶$X_T(H,k)$，该阶$k$包含在最多$ H $阶的加法基中。在其他结果中，我们证明了对于每个整数$k ge 1$， $X_T(h,k)=O(h^{2k+1})$。即使在$k=1$的情况下，这个结果也是新的。除了最大阶$X_T(h,k)$外，还研究了典型阶$S_T(h,k)$。我们的方法实际上适用于更广泛的无限阿贝尔半群，从而将$mathbf{N}$中的加性基理论和阿贝尔群中的加性基理论统一在一个公理框架中。
{"title":"On additive bases in infinite abelian semigroups","authors":"Pierre-Yves Bienvenu, B. Girard, T. H. Lê","doi":"10.4171/jca/67","DOIUrl":"https://doi.org/10.4171/jca/67","url":null,"abstract":"Building on previous work by Lambert, Plagne and the third author, we study various aspects of the behavior of additive bases in infinite abelian groups. We show that, for every such group $T$, the number of essential subsets of any additive basis is finite, and also that the number of essential subsets of cardinality $k$ contained in an additive basis of order at most $h$ can be bounded in terms of $h$ and $k$ alone. These results extend the reach of two theorems, one due to Deschamps and Farhi and the other to Hegarty, bearing upon $mathbf{N}$. Also, using invariant means, we address a classical problem, initiated by ErdH{o}s and Graham and then generalized by Nash and Nathanson both in the case of $mathbf{N}$, of estimating the maximal order $X_T(h,k)$ that a basis of cocardinality $k$ contained in an additive basis of order at most $h$ can have. Among other results, we prove that $X_T(h,k)=O(h^{2k+1})$ for every integer $k ge 1$. This result is new even in the case where $k=1$. Besides the maximal order $X_T(h,k)$, the typical order $S_T(h,k)$ is also studied. Our methods actually apply to a wider class of infinite abelian semigroups, thus unifying in a single axiomatic frame the theory of additive bases in $mathbf{N}$ and in abelian groups.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43578896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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