Given a finite group $G$ and its representation $rho$, the corresponding McKay graph is a graph $Gamma(G,rho)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $pi,tau$ of $Gamma(G,rho)$ is $dim Hom_G(pi otimes rho, tau) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) trees which are McKay graphs of finite groups and describe the corresponding pairs $(G,rho)$; this classification turns out to be very concise. Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests.
{"title":"McKay trees","authors":"Avraham Aizenbud, I. Entova-Aizenbud","doi":"10.5802/alco.270","DOIUrl":"https://doi.org/10.5802/alco.270","url":null,"abstract":"Given a finite group $G$ and its representation $rho$, the corresponding McKay graph is a graph $Gamma(G,rho)$ whose vertices are the irreducible representations of $G$; the number of edges between two vertices $pi,tau$ of $Gamma(G,rho)$ is $dim Hom_G(pi otimes rho, tau) $. The collection of all McKay graphs for a given group $G$ encodes, in a sense, its character table. Such graphs were also used by McKay to provide a bijection between the finite subgroups of $SU(2)$ and the affine Dynkin diagrams of types $A, D, E$, the bijection given by considering the appropriate McKay graphs. In this paper, we classify all (undirected) trees which are McKay graphs of finite groups and describe the corresponding pairs $(G,rho)$; this classification turns out to be very concise. Moreover, we give a partial classification of McKay graphs which are forests, and construct some non-trivial examples of such forests.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48678314","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 BSTRACT . We give closed product formulas for the irreducible characters of the symmetric groups related to rectangular ‘almost square’ Young diagrams p × ( p + δ ) for a fixed value of an integer δ and an arbitrary integer p .
{"title":"Symmetric group characters of almost square shape","authors":"Sho Matsumoto, Piotr Śniady","doi":"10.5802/alco.247","DOIUrl":"https://doi.org/10.5802/alco.247","url":null,"abstract":"A BSTRACT . We give closed product formulas for the irreducible characters of the symmetric groups related to rectangular ‘almost square’ Young diagrams p × ( p + δ ) for a fixed value of an integer δ and an arbitrary integer p .","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44086679","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 fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1x_{a+1}>...>x_b
栅栏是元素F = {x_1, x_2，…， x_n}，涵盖x_1x_{a+1}>…>x_b
{"title":"Rowmotion on fences","authors":"S. Elizalde, Matthew Plante, Tom Roby, B. Sagan","doi":"10.5802/alco.256","DOIUrl":"https://doi.org/10.5802/alco.256","url":null,"abstract":"A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1<x_2<...<x_a>x_{a+1}>...>x_b<x_{b+1}<... where a, b, ... are positive integers. We investigate rowmotion on antichains and ideals of F. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call orbomesy, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a homomesy result for all self-dual posets and show that any two Coxeter elements in certain toggle groups behave similarly with respect to homomesies which are linear combinations of ideal indicator functions. We end with some conjectures and avenues for future research.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41915588","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 Goulden–Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper, we lift the cluster method for permutations to the Malvenuto–Reutenauer algebra. Upon applying standard homomorphisms, our result specializes to both the cluster method for permutations as well as a q -analogue which keeps track of the inversion number statistic. We construct additional homomorphisms using the theory of shuffle-compatibility, leading to further specializations which keep track of various “inverse statistics”, including the inverse descent number, inverse peak number, and inverse left peak number. This approach is then used to derive formulas for counting permutations by occurrences of two families of consecutive patterns—monotone patterns and transpositional patterns—refined by these statistics.
{"title":"A lifting of the Goulden–Jackson cluster method to the Malvenuto–Reutenauer algebra","authors":"Zhuang Yan","doi":"10.5802/alco.255","DOIUrl":"https://doi.org/10.5802/alco.255","url":null,"abstract":"The Goulden–Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper, we lift the cluster method for permutations to the Malvenuto–Reutenauer algebra. Upon applying standard homomorphisms, our result specializes to both the cluster method for permutations as well as a q -analogue which keeps track of the inversion number statistic. We construct additional homomorphisms using the theory of shuffle-compatibility, leading to further specializations which keep track of various “inverse statistics”, including the inverse descent number, inverse peak number, and inverse left peak number. This approach is then used to derive formulas for counting permutations by occurrences of two families of consecutive patterns—monotone patterns and transpositional patterns—refined by these statistics.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":"76 S39","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41266629","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 result of Farahat and Higman shows that there is a ``universal'' algebra, $mathrm{FH}$, interpolating the centres of symmetric group algebras, $Z(mathbb{Z}S_n)$. We explain that this algebra is isomorphic to $mathcal{R} otimes Lambda$, where $mathcal{R}$ is the ring of integer-valued polynomials and $Lambda$ is the ring of symmetric functions. Moreover, the isomorphism is via ``evaluation at Jucys-Murphy elements'', which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products $Gamma wr S_n$ of a fixed finite group $Gamma$. This involves constructing wreath-product versions $mathcal{R}_Gamma$ and $Lambda(Gamma_*)$ of $mathcal{R}$ and $Lambda$, respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, $mathrm{FH}_Gamma$, is isomorphic to $mathcal{R}_Gamma otimes Lambda(Gamma_*)$ and use this to compute the $p$-blocks of wreath products.
{"title":"Stable centres of wreath products","authors":"Christopher Ryba","doi":"10.5802/alco.264","DOIUrl":"https://doi.org/10.5802/alco.264","url":null,"abstract":"A result of Farahat and Higman shows that there is a ``universal'' algebra, $mathrm{FH}$, interpolating the centres of symmetric group algebras, $Z(mathbb{Z}S_n)$. We explain that this algebra is isomorphic to $mathcal{R} otimes Lambda$, where $mathcal{R}$ is the ring of integer-valued polynomials and $Lambda$ is the ring of symmetric functions. Moreover, the isomorphism is via ``evaluation at Jucys-Murphy elements'', which leads to character formulae for symmetric groups. Then, we generalise this result to wreath products $Gamma wr S_n$ of a fixed finite group $Gamma$. This involves constructing wreath-product versions $mathcal{R}_Gamma$ and $Lambda(Gamma_*)$ of $mathcal{R}$ and $Lambda$, respectively, which are interesting in their own right (for example, both are Hopf algebras). We show that the universal algebra for wreath products, $mathrm{FH}_Gamma$, is isomorphic to $mathcal{R}_Gamma otimes Lambda(Gamma_*)$ and use this to compute the $p$-blocks of wreath products.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49647793","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}
Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Do{l}k{e}ga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^lambda_{mu^0,...,mu^k}(b)$ obtained by the expansion of these functions in the power-sum basis. These coefficients are indexed by $k+2$ integer partitions and the deformation parameter $b$, and can be considered as a generalization for $kgeq1$ of the connection coefficients introduced by Goulden and Jackson. We prove that when we take some marginal sums, these coefficients enumerate $b$-weighted $k$-tuples of matchings. This can be seen as an"unrooted"version of a recent result of Chapuy and Do{l}k{e}ga for constellations. For $k=1$, this gives a partial answer to Goulden and Jackson Matching-Jack conjecture. Lassale has formulated a positivity conjecture for the coefficients $theta^{(alpha)}_mu(lambda)$, defined as the coefficient of the Jack polynomial $J_lambda^{(alpha)}$ in the power-sum basis. We use the second main result of this paper to give a proof of this conjecture in the case of partitions $lambda$ with rectangular shape.
{"title":"Generating series of non-oriented constellations and marginal sums in the Matching-Jack conjecture","authors":"Houcine Ben Dali","doi":"10.5802/alco.207","DOIUrl":"https://doi.org/10.5802/alco.207","url":null,"abstract":"Using the description of hypermaps with matchings, Goulden and Jackson have given an expression of the generating series of rooted bipartite maps in terms of the zonal polynomials. We generalize this approach to the case of constellations on non-oriented surfaces that have recently been introduced by Chapuy and Do{l}k{e}ga. A key step in the proof is an encoding of constellations with tuples of matchings. We consider a one parameter deformation of the generating series of constellations using Jack polynomials and we introduce the coefficients $c^lambda_{mu^0,...,mu^k}(b)$ obtained by the expansion of these functions in the power-sum basis. These coefficients are indexed by $k+2$ integer partitions and the deformation parameter $b$, and can be considered as a generalization for $kgeq1$ of the connection coefficients introduced by Goulden and Jackson. We prove that when we take some marginal sums, these coefficients enumerate $b$-weighted $k$-tuples of matchings. This can be seen as an\"unrooted\"version of a recent result of Chapuy and Do{l}k{e}ga for constellations. For $k=1$, this gives a partial answer to Goulden and Jackson Matching-Jack conjecture. Lassale has formulated a positivity conjecture for the coefficients $theta^{(alpha)}_mu(lambda)$, defined as the coefficient of the Jack polynomial $J_lambda^{(alpha)}$ in the power-sum basis. We use the second main result of this paper to give a proof of this conjecture in the case of partitions $lambda$ with rectangular shape.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42564453","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 finite irreducible Coxeter group W with a fixed Coxeter element c , we define the Coxeter pop-tsack torsing operator Pop T : W → W by Pop T ( w ) = w · π T ( w ) − 1 , where π T ( w ) is the join in the noncrossing partition lattice NC( w,c ) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D , then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W . In each of these types, we obtain a natural lift from W to the dual braid monoid of W . We also prove that W is coincidental if and only if it has a unique forward orbit of size h . For arbitrary W , we show that the forward orbit of c − 1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.
给定具有固定Coxeter元素c的有限不可约Coxeter群W，我们定义了Coxeter pop-tack扭转算子pop T:W→ W by Pop T（W）=W·πT（W。这一定义是第一作者Coxeter pop堆栈排序算子概念的“Bessis对偶”版本，进而推广了对称群上的pop堆栈分类映射。我们证明，如果W是重合的或是D型的，那么W的单位元素是Pop T的唯一周期点，Pop T前向轨道的最大大小是W的Coxeter数h。在这些类型中的每一种中，我们都获得了从W到W的双辫半群的自然升力。我们还证明了W是巧合的，当且仅当它有一个大小为h的唯一前向轨道。对于任意W，我们证明了在Pop T下c−1的前向轨道的大小为h，并且是孤立的，因为轨道的非同一元素都没有位于轨道外的前像。
{"title":"Coxeter Pop-Tsack Torsing","authors":"Colin Defant, Nathan Williams","doi":"10.5802/alco.226","DOIUrl":"https://doi.org/10.5802/alco.226","url":null,"abstract":"Given a finite irreducible Coxeter group W with a fixed Coxeter element c , we define the Coxeter pop-tsack torsing operator Pop T : W → W by Pop T ( w ) = w · π T ( w ) − 1 , where π T ( w ) is the join in the noncrossing partition lattice NC( w,c ) of the set of reflections lying weakly below w in the absolute order. This definition serves as a “Bessis dual” version of the first author’s notion of a Coxeter pop-stack sorting operator, which, in turn, generalizes the pop-stack sorting map on symmetric groups. We show that if W is coincidental or of type D , then the identity element of W is the unique periodic point of Pop T and the maximum size of a forward orbit of Pop T is the Coxeter number h of W . In each of these types, we obtain a natural lift from W to the dual braid monoid of W . We also prove that W is coincidental if and only if it has a unique forward orbit of size h . For arbitrary W , we show that the forward orbit of c − 1 under Pop T has size h and is isolated in the sense that none of the non-identity elements of the orbit have preimages lying outside of the orbit.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46269043","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 transitive permutation group on a finite set Ω and recall that a base for G is a subset of Ω with trivial pointwise stabiliser. The base size of G , denoted b ( G ), is the minimal size of a base. If b ( G ) = 2 then we can study the Saxl graph Σ( G ) of G , which has vertex set Ω and two vertices are adjacent if and only if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most 2 when G is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of Σ( G ) and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of Σ( G ).
{"title":"On the Saxl graphs of primitive groups with soluble stabilisers","authors":"Timothy C. Burness, Hongdi Huang","doi":"10.5802/alco.238","DOIUrl":"https://doi.org/10.5802/alco.238","url":null,"abstract":"Let G be a transitive permutation group on a finite set Ω and recall that a base for G is a subset of Ω with trivial pointwise stabiliser. The base size of G , denoted b ( G ), is the minimal size of a base. If b ( G ) = 2 then we can study the Saxl graph Σ( G ) of G , which has vertex set Ω and two vertices are adjacent if and only if they form a base. This is a vertex-transitive graph, which is conjectured to be connected with diameter at most 2 when G is primitive. In this paper, we combine probabilistic and computational methods to prove a strong form of this conjecture for all almost simple primitive groups with soluble point stabilisers. In this setting, we also establish best possible lower bounds on the clique and independence numbers of Σ( G ) and we determine the groups with a unique regular suborbit, which can be interpreted in terms of the valency of Σ( G ).","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43607616","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 a case of a positivity conjecture of Mihalcea–Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassman- nian LG ( n, 2 n ). Combined with work of Aluffi–Mihalcea–Schürmann–Su, this further implies the positivity of the Mather classes for Schubert varieties in LG ( n, 2 n ), which Mihalcea–Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan–Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for LG ( n, 2 n ) the Euler obstructions e y,w may vanish for certain pairs ( y,w ) with y (cid:54) w in the Bruhat order. Our combinatorial description allows us to classify all the pairs ( y,w ) for which e y,w = 0. Restricting to the big opposite cell in LG ( n, 2 n ), which is naturally identified with the space of n × n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.
{"title":"Euler obstructions for the Lagrangian Grassmannian","authors":"P. LeVan, Claudiu Raicu","doi":"10.5802/alco.211","DOIUrl":"https://doi.org/10.5802/alco.211","url":null,"abstract":"We prove a case of a positivity conjecture of Mihalcea–Singh, concerned with the local Euler obstructions associated to the Schubert stratification of the Lagrangian Grassman- nian LG ( n, 2 n ). Combined with work of Aluffi–Mihalcea–Schürmann–Su, this further implies the positivity of the Mather classes for Schubert varieties in LG ( n, 2 n ), which Mihalcea–Singh had verified for the other cominuscule spaces of classical Lie type. Building on the work of Boe and Fu, we give a positive recursion for the local Euler obstructions, and use it to show that they provide a positive count of admissible labelings of certain trees, analogous to the ones describing Kazhdan–Lusztig polynomials. Unlike in the case of the Grassmannians in types A and D, for LG ( n, 2 n ) the Euler obstructions e y,w may vanish for certain pairs ( y,w ) with y (cid:54) w in the Bruhat order. Our combinatorial description allows us to classify all the pairs ( y,w ) for which e y,w = 0. Restricting to the big opposite cell in LG ( n, 2 n ), which is naturally identified with the space of n × n symmetric matrices, we recover the formulas for the local Euler obstructions associated with the matrix rank stratification.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42765881","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 semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application, we give new tableau models for the crystal bases of a level-zero fundamental representation and a level-zero extremal weight module over a quantum affine algebra of classical untwisted type, which we call quantum Kashiwara-Nakashima columns and semi-infinite Kashiwara-Nakashima tableaux. We give an explicit description of the crystal isomorphisms among three different realizations of the crystal basis of a level-zero fundamental representation by quantum Lakshmibai-Seshadri paths, quantum Kashiwara-Nakashima columns, and (ordinary) Kashiwara-Nakashima columns.
{"title":"Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras","authors":"Motohiro Ishii","doi":"10.5802/alco.242","DOIUrl":"https://doi.org/10.5802/alco.242","url":null,"abstract":"We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application, we give new tableau models for the crystal bases of a level-zero fundamental representation and a level-zero extremal weight module over a quantum affine algebra of classical untwisted type, which we call quantum Kashiwara-Nakashima columns and semi-infinite Kashiwara-Nakashima tableaux. We give an explicit description of the crystal isomorphisms among three different realizations of the crystal basis of a level-zero fundamental representation by quantum Lakshmibai-Seshadri paths, quantum Kashiwara-Nakashima columns, and (ordinary) Kashiwara-Nakashima columns.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47788702","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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