{"title":"Cellular subalgebras of the partition algebra","authors":"Travis Scrimshaw","doi":"10.4171/jca/84","DOIUrl":"https://doi.org/10.4171/jca/84","url":null,"abstract":"","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"136 6","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138953348","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}
The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart–Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant $K$-group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula.
与主导权相关的量子凹形模型在许多数学分支中起着重要作用，如组合表示理论、麦克唐纳多项式理论和舒伯特微积分。对于一个优势权值，Lenart-Lubovsky证明了量子凹形模型不依赖于约化凹形路径的选择，约化凹形路径是从基本凹形到给定优势权值平移的最短凹形路径。这是通过量子Yang-Baxter移动建立的，它将模型的对象与两个这样的凹形路径相关联，并且可以被视为对任意根系统的jeu de taquin滑动的概括。本文的目的是将量子Yang-Baxter运动推广到任意权值对应的量子凹形模型，并利用该模型来表示半无限标志流形的等变$K$群的一般Chevalley公式。广义量子Yang-Baxter移动产生了一个“双射”(符号集之间的双射)，并被证明可以保持某些重要的统计量，包括权重和高度。作为一个应用，我们证明了这些统计量的生成函数不依赖于缩凹路径的选择。此外，我们还得到了量子仿射代数上零级极值权模的Demazure子模的梯度特征的恒等式，它可以被认为是上述Chevalley公式的表示理论类比。
{"title":"New structure on the quantum alcove model with applications to representation theory and Schubert calculus","authors":"Takafumi Kouno, Cristian Lenart, Satoshi Naito","doi":"10.4171/jca/77","DOIUrl":"https://doi.org/10.4171/jca/77","url":null,"abstract":"The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant weight, it is proved by Lenart–Lubovsky that the quantum alcove model does not depend on the choice of a reduced alcove path, which is a shortest path of alcoves from the fundamental one to its translation by the given dominant weight. This is established through quantum Yang–Baxter moves, which biject the objects of the models associated to two such alcove paths, and can be viewed as a generalization of jeu de taquin slides to arbitrary root systems. The purpose of this paper is to give a generalization of quantum Yang–Baxter moves to the quantum alcove model corresponding to an arbitrary weight, which was used to express a general Chevalley formula for the equivariant $K$-group of semi-infinite flag manifolds. The generalized quantum Yang–Baxter moves give rise to a “sijection” (bijection between signed sets), and are shown to preserve certain important statistics, including weights and heights. As an application, we prove that the generating function of these statistics does not depend on the choice of a reduced alcove path. Also, we obtain an identity for the graded characters of Demazure submodules of level-zero extremal weight modules over a quantum affine algebra, which can be thought of as a representation-theoretic analogue of the mentioned Chevalley formula.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"22 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135765834","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}
For any Levi subalgebra of the form $mathfrak{l}=mathfrak{gl}{l{1}}opluscdotsoplusmathfrak{gl}{l{d}}subseteqmathfrak{gl}{n}$ we construct a quotient of the category of annular quantum $mathfrak{gl}{n}$ webs that is equivalent to the category of finite-dimensional representations of quantum $mathfrak{l}$ generated by exterior powers of the vector representation. This can be interpreted as an annular version of skew Howe duality, gives a description of the representation category of $mathfrak{l}$ by additive idempotent completion, and a web version of the generalized blob algebra.
{"title":"Annular webs and Levi subalgebras","authors":"Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz","doi":"10.4171/jca/76","DOIUrl":"https://doi.org/10.4171/jca/76","url":null,"abstract":"For any Levi subalgebra of the form $mathfrak{l}=mathfrak{gl}{l{1}}opluscdotsoplusmathfrak{gl}{l{d}}subseteqmathfrak{gl}{n}$ we construct a quotient of the category of annular quantum $mathfrak{gl}{n}$ webs that is equivalent to the category of finite-dimensional representations of quantum $mathfrak{l}$ generated by exterior powers of the vector representation. This can be interpreted as an annular version of skew Howe duality, gives a description of the representation category of $mathfrak{l}$ by additive idempotent completion, and a web version of the generalized blob algebra.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219719","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 give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra $mathrm{End}_{mathbf{U}_q(mathfrak{gl}_2)}(V^{otimes k})$, where ${V = V(0) oplus V(1)}$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $mathbf{U}_q(mathfrak{gl}_2)$. It is a proper subalgebra of the Motzkin algebra (the $mathbf{U}_q(fraksl_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.
{"title":"The partial Temperley–Lieb algebra and its representations","authors":"Stephen Doty, Anthony Giaquinto","doi":"10.4171/jca/74","DOIUrl":"https://doi.org/10.4171/jca/74","url":null,"abstract":"In this paper, we give a combinatorial description of a new diagram algebra, the partial Temperley–Lieb algebra, arising as the generic centralizer algebra $mathrm{End}_{mathbf{U}_q(mathfrak{gl}_2)}(V^{otimes k})$, where ${V = V(0) oplus V(1)}$ is the direct sum of the trivial and natural module for the quantized enveloping algebra $mathbf{U}_q(mathfrak{gl}_2)$. It is a proper subalgebra of the Motzkin algebra (the $mathbf{U}_q(fraksl_2)$-centralizer) of Benkart and Halverson. We prove a version of Schur–Weyl duality for the new algebras, and describe their generic representation theory.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136079072","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}
Anneleen De Schepper, Jeroen Schillewaert, Hendrik van Maldeghem
We characterise the projective varieties related to the second row of the Freudenthal–Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the $27$-dimensional $mathrm{E_6}$ module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.
{"title":"A uniform characterisation of the varieties of the second row of the Freudenthal–Tits magic square over arbitrary fields","authors":"Anneleen De Schepper, Jeroen Schillewaert, Hendrik van Maldeghem","doi":"10.4171/jca/75","DOIUrl":"https://doi.org/10.4171/jca/75","url":null,"abstract":"We characterise the projective varieties related to the second row of the Freudenthal–Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the $27$-dimensional $mathrm{E_6}$ module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136294076","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 find a combinatorial formula which computes the first cotangent cohomology module of Stanley–Reisner rings associated to matroids. For arbitrary simplicial complexes we provide upper bounds for the dimensions of the multigraded components of $T^1$. For specific degrees we prove that these bounds are reached if and only if the simplicial complex is a matroid, obtaining thus a new characterization for matroids. Furthermore, the graded first cotangent cohomology turns out to be a complete invariant for nondiscrete matroids.
{"title":"The first cotangent cohomology module for matroids","authors":"William Brehm, Alexandru Constantinescu","doi":"10.4171/jca/73","DOIUrl":"https://doi.org/10.4171/jca/73","url":null,"abstract":"We find a combinatorial formula which computes the first cotangent cohomology module of Stanley–Reisner rings associated to matroids. For arbitrary simplicial complexes we provide upper bounds for the dimensions of the multigraded components of $T^1$. For specific degrees we prove that these bounds are reached if and only if the simplicial complex is a matroid, obtaining thus a new characterization for matroids. Furthermore, the graded first cotangent cohomology turns out to be a complete invariant for nondiscrete matroids.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135346116","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 investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis and the question whether there is an anlogue of the well known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis and Reiner as well as Baumeister, Gobet, Roberts and the first author on the Hurwitz in finite Coxeter groups. Further we show that in an arbitrary Coxeter group all reduced reflection factorizations of the same element appear in the same Hurwitz orbit after a suitable extension by simple reflections. As parabolic quasi-Coxeter elements play an outstanding role in the study of the Hurwitz action, we aim to characterize these elements. We give characterizations of maximal parabolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups.
{"title":"Reflection factorizations and quasi-Coxeter elements","authors":"Patrick Wegener, S. Yahiatene","doi":"10.4171/jca/70","DOIUrl":"https://doi.org/10.4171/jca/70","url":null,"abstract":"We investigate the so-called dual Matsumoto property or Hurwitz action in finite, affine and arbitrary Coxeter groups. In particular, we want to investigate how to reduce reflection factorizations and how two reflection factorizations of the same element are related to each other. We are motivated by the dual approach to Coxeter groups proposed by Bessis and the question whether there is an anlogue of the well known Matsumoto property for reflection factorizations. Our aim is a substantial understanding of the Hurwitz action. We therefore reprove uniformly results of Lewis and Reiner as well as Baumeister, Gobet, Roberts and the first author on the Hurwitz in finite Coxeter groups. Further we show that in an arbitrary Coxeter group all reduced reflection factorizations of the same element appear in the same Hurwitz orbit after a suitable extension by simple reflections. As parabolic quasi-Coxeter elements play an outstanding role in the study of the Hurwitz action, we aim to characterize these elements. We give characterizations of maximal parabolic quasi-Coxeter elements in arbitrary Coxeter groups as well as a characterization of all parabolic quasi-Coxeter elements in affine Coxeter groups.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47626843","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}
The fusion of two Mirkovic-Vilonen cycles is a degeneration of their product, defined using the Beilinson-Drinfeld Grassmannian. In this paper, we put in place a conceptually elementary approach to computing this product in type $A$. We do so by transferring the problem to a fusion of generalized orbital varieties using the Mirkovic-Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of $GL_4$, confirming that all the cluster variables are contained in the Mirkovic-Vilonen basis.
两个Mirkovic-Vilonen循环的融合是它们乘积的退化，使用Beilinson-Drinfeld Grassmannian来定义。在本文中，我们提出了一种概念上基本的方法来计算类型为$ a $的这个乘积。我们通过使用Mirkovic-Vybornov同构将问题转化为广义轨道变体的融合来做到这一点。作为应用，我们显式计算了$GL_4$上三角子群的坐标环上的所有簇交换关系，确认了所有簇变量都包含在Mirkovic-Vilonen基中。
{"title":"Computing fusion products of MV cycles using the Mirkovi'c--Vybornov isomorphism","authors":"R. Bai, Anne Dranowski, J. Kamnitzer","doi":"10.4171/JCA/69","DOIUrl":"https://doi.org/10.4171/JCA/69","url":null,"abstract":"The fusion of two Mirkovic-Vilonen cycles is a degeneration of their product, defined using the Beilinson-Drinfeld Grassmannian. In this paper, we put in place a conceptually elementary approach to computing this product in type $A$. We do so by transferring the problem to a fusion of generalized orbital varieties using the Mirkovic-Vybornov isomorphism. As an application, we explicitly compute all cluster exchange relations in the coordinate ring of the upper-triangular subgroup of $GL_4$, confirming that all the cluster variables are contained in the Mirkovic-Vilonen basis.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45148290","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}
The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic p with the one-dimensional sign representation. It can also be interpreted as an isomorphism between crystal graphs for ŝlp. We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two applications of our new realisation of the Mullineux map, by providing purely combinatorial proofs of a conjecture of Lyle relating the Mullineux map with regularisation, and a theorem of Paget describing the Mullineux map in RoCK blocks of symmetric groups.
{"title":"Crystals, regularisation and the Mullineux map","authors":"M. Fayers","doi":"10.4171/jca/59","DOIUrl":"https://doi.org/10.4171/jca/59","url":null,"abstract":"The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic p with the one-dimensional sign representation. It can also be interpreted as an isomorphism between crystal graphs for ŝlp. We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two applications of our new realisation of the Mullineux map, by providing purely combinatorial proofs of a conjecture of Lyle relating the Mullineux map with regularisation, and a theorem of Paget describing the Mullineux map in RoCK blocks of symmetric groups.","PeriodicalId":48483,"journal":{"name":"Journal of Combinatorial Algebra","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2021-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42911735","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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