Pub Date : 2023-03-19DOI: 10.2140/pjm.2023.325.127
N. Jing, Ning Liu
An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{ximu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K^-_{ximu}(t)$ for $|xi|leq6$ are listed.
{"title":"Spin Kostka polynomials and vertex operators","authors":"N. Jing, Ning Liu","doi":"10.2140/pjm.2023.325.127","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.127","url":null,"abstract":"An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{ximu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K^-_{ximu}(t)$ for $|xi|leq6$ are listed.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41386687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-03DOI: 10.2140/pjm.2023.324.227
John M. Campbell
We introduce a lifting of West's stack-sorting map $s$ to partition diagrams, which are combinatorial objects indexing bases of partition algebras. Our lifting $mathscr{S}$ of $s$ is such that $mathscr{S}$ behaves in the same way as $s$ when restricted to diagram basis elements in the order-$n$ symmetric group algebra as a diagram subalgebra of the partition algebra $mathscr{P}_{n}^{xi}$. We then introduce a lifting of the notion of $1$-stack-sortability, using our lifting of $s$. By direct analogy with Knuth's famous result that a permutation is $1$-stack-sortable if and only if it avoids the pattern $231$, we prove a related pattern-avoidance property for partition diagrams, as opposed to permutations, according to what we refer to as stretch-stack-sortability.
我们将West的堆栈排序映射$s$提升到分区图，分区图是分区代数的组合对象索引基。我们将$ $ S $的$mathscr{S}$提升，使得$ $mathscr{S}$的行为与$ $ S $在作为分区代数$ $mathscr{P}_{n}^{xi}$的图子代数的序-$n$对称群代数中的图基元素的行为相同。然后我们引入$1$-堆栈可排序性的提升概念，使用我们的$s$提升。通过直接类比Knuth的著名结果，即排列是$1$-堆栈可排序的，当且仅当它避免了模式$231$，我们证明了与排列相反的分区图的相关模式避免性质，根据我们所说的拉伸堆栈可排序性。
{"title":"A lift of West’s stack-sorting map to partition\u0000diagrams","authors":"John M. Campbell","doi":"10.2140/pjm.2023.324.227","DOIUrl":"https://doi.org/10.2140/pjm.2023.324.227","url":null,"abstract":"We introduce a lifting of West's stack-sorting map $s$ to partition diagrams, which are combinatorial objects indexing bases of partition algebras. Our lifting $mathscr{S}$ of $s$ is such that $mathscr{S}$ behaves in the same way as $s$ when restricted to diagram basis elements in the order-$n$ symmetric group algebra as a diagram subalgebra of the partition algebra $mathscr{P}_{n}^{xi}$. We then introduce a lifting of the notion of $1$-stack-sortability, using our lifting of $s$. By direct analogy with Knuth's famous result that a permutation is $1$-stack-sortable if and only if it avoids the pattern $231$, we prove a related pattern-avoidance property for partition diagrams, as opposed to permutations, according to what we refer to as stretch-stack-sortability.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44609646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-31DOI: 10.2140/pjm.2022.321.309
B. Kalmár
{"title":"Fold maps on small dimensional manifolds with prescribed singular set","authors":"B. Kalmár","doi":"10.2140/pjm.2022.321.309","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.309","url":null,"abstract":"","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49392468","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-22DOI: 10.2140/pjm.2023.323.223
Swee Hong Chan, I. Pak
Motivated by the Lam--Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn's correlation inequality for the number of linear extensions of posets. We then give a multivariate generalization of the Daykin--Daykin--Paterson inequality proving log-concavity of the order polynomial of a poset. We also prove a multivariate $P$-partition version of the cross-product inequality by Brightwell--Felsner--Trotter. The proofs are based on a multivariate generalization of the Ahlswede--Daykin inequality.
{"title":"Multivariate correlation inequalities for\u0000P-partitions","authors":"Swee Hong Chan, I. Pak","doi":"10.2140/pjm.2023.323.223","DOIUrl":"https://doi.org/10.2140/pjm.2023.323.223","url":null,"abstract":"Motivated by the Lam--Pylyavskyy inequalities for Schur functions, we give a far reaching multivariate generalization of Fishburn's correlation inequality for the number of linear extensions of posets. We then give a multivariate generalization of the Daykin--Daykin--Paterson inequality proving log-concavity of the order polynomial of a poset. We also prove a multivariate $P$-partition version of the cross-product inequality by Brightwell--Felsner--Trotter. The proofs are based on a multivariate generalization of the Ahlswede--Daykin inequality.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43228902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-12-15DOI: 10.2140/pjm.2023.322.281
Nick Gill, M. Liebeck
We prove that the maximum length of an irredundant base for a primitive action of a finite simple group of Lie type is bounded above by a function which is a polynomial in the rank of the group. We give examples to show that this type of upper bound is best possible.
{"title":"Irredundant bases for finite groups of Lie type","authors":"Nick Gill, M. Liebeck","doi":"10.2140/pjm.2023.322.281","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.281","url":null,"abstract":"We prove that the maximum length of an irredundant base for a primitive action of a finite simple group of Lie type is bounded above by a function which is a polynomial in the rank of the group. We give examples to show that this type of upper bound is best possible.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43223410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-14DOI: 10.2140/pjm.2023.325.147
B. Klopsch, M. Quick
Just infinite groups play a significant role in profinite group theory. For each $c geq 0$, we consider more generally JNN$_c$F profinite (or, in places, discrete) groups that are Fitting-free; these are the groups $G$ such that every proper quotient of $G$ is virtually class-$c$ nilpotent whereas $G$ itself is not, and additionally $G$ does not have any non-trivial abelian normal subgroup. When $c = 1$, we obtain the just non-(virtually abelian) groups without non-trivial abelian normal subgroups. Our first result is that a finitely generated profinite group is virtually classnbd$c$ nilpotent if and only if there are only finitely many subgroups arising as the lower central series terms $gamma_{c+1}(K)$ of open normal subgroups $K$ of $G$. Based on this we prove several structure theorems. For instance, we characterize the JNN$_c$F profinite groups in terms of subgroups of the above form $gamma_{c+1}(K)$. We also give a description of JNN$_c$F profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNN$_c$F groups and, for instance, we show that a Fitting-free JNN$_c$F profinite (or discrete) group is hereditarily JNN$_cF$ if and only if every maximal subgroup of finite index is JNN$_c$F. Finally, we give a construction of hereditarily JNN$_c$F groups, which uses as an input known families of hereditarily just infinite groups.
{"title":"The structure of groups with all proper quotients virtually nilpotent","authors":"B. Klopsch, M. Quick","doi":"10.2140/pjm.2023.325.147","DOIUrl":"https://doi.org/10.2140/pjm.2023.325.147","url":null,"abstract":"Just infinite groups play a significant role in profinite group theory. For each $c geq 0$, we consider more generally JNN$_c$F profinite (or, in places, discrete) groups that are Fitting-free; these are the groups $G$ such that every proper quotient of $G$ is virtually class-$c$ nilpotent whereas $G$ itself is not, and additionally $G$ does not have any non-trivial abelian normal subgroup. When $c = 1$, we obtain the just non-(virtually abelian) groups without non-trivial abelian normal subgroups. Our first result is that a finitely generated profinite group is virtually classnbd$c$ nilpotent if and only if there are only finitely many subgroups arising as the lower central series terms $gamma_{c+1}(K)$ of open normal subgroups $K$ of $G$. Based on this we prove several structure theorems. For instance, we characterize the JNN$_c$F profinite groups in terms of subgroups of the above form $gamma_{c+1}(K)$. We also give a description of JNN$_c$F profinite groups as suitable inverse limits of virtually nilpotent profinite groups. Analogous results are established for the family of hereditarily JNN$_c$F groups and, for instance, we show that a Fitting-free JNN$_c$F profinite (or discrete) group is hereditarily JNN$_cF$ if and only if every maximal subgroup of finite index is JNN$_c$F. Finally, we give a construction of hereditarily JNN$_c$F groups, which uses as an input known families of hereditarily just infinite groups.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41444593","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-13DOI: 10.2140/pjm.2022.321.431
Mihir Sheth
Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called emph{cyclic diagrams}, and use them to show that the universal supersingular modules of $mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible admissible quotients.
{"title":"On irreducible supersingular representations of\u0000GL2(F)","authors":"Mihir Sheth","doi":"10.2140/pjm.2022.321.431","DOIUrl":"https://doi.org/10.2140/pjm.2022.321.431","url":null,"abstract":"Let $F$ be a non-archimedean local field of residual characteristic $p>3$ and residue degree $f>1$. We study a certain type of diagram, called emph{cyclic diagrams}, and use them to show that the universal supersingular modules of $mathrm{GL}_{2}(F)$ admit infinitely many non-isomorphic irreducible admissible quotients.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41913998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-03DOI: 10.2140/pjm.2023.322.369
Kang Li, Eduardo Scarparo
Given a commensurated subgroup $Lambda$ of a group $Gamma$, we completely characterize when the inclusion $Lambdaleq Gamma$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that $rm{PSL}(n,mathbb{Z})leqrm{PGL}(n,mathbb{Q})$ is $C^*$-irreducible for any $nin mathbb{N}$, and that the inclusion of a $C^*$-simple group into its abstract commensurator is $C^*$-irreducible. The main ingredient that we use is the fact that the action of a commensurated subgroup $LambdaleqGamma$ on its Furstenberg boundary $partial_FLambda$ can be extended in a unique way to an action of $Gamma$ on $partial_FLambda$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.
{"title":"C∗-irreducibility of commensurated\u0000subgroups","authors":"Kang Li, Eduardo Scarparo","doi":"10.2140/pjm.2023.322.369","DOIUrl":"https://doi.org/10.2140/pjm.2023.322.369","url":null,"abstract":"Given a commensurated subgroup $Lambda$ of a group $Gamma$, we completely characterize when the inclusion $Lambdaleq Gamma$ is $C^*$-irreducible and provide new examples of such inclusions. In particular, we obtain that $rm{PSL}(n,mathbb{Z})leqrm{PGL}(n,mathbb{Q})$ is $C^*$-irreducible for any $nin mathbb{N}$, and that the inclusion of a $C^*$-simple group into its abstract commensurator is $C^*$-irreducible. The main ingredient that we use is the fact that the action of a commensurated subgroup $LambdaleqGamma$ on its Furstenberg boundary $partial_FLambda$ can be extended in a unique way to an action of $Gamma$ on $partial_FLambda$. Finally, we also investigate the counterpart of this extension result for the universal minimal proximal space of a group.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49349654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-09-21DOI: 10.2140/pjm.2023.324.207
Cihan Bahran
We identify two recursively defined polynomial conditions for FI-modules in the literature. We characterize these conditions using homological invariants of FI-modules (namely the local degree and regularity, together with the stable degree) and clarify their relationship. For one of these conditions, we give improved twisted homological stability ranges for the symmetric groups. As another application, we improve the representation stability ranges for congruence subgroups with respect to the action of an appropriate linear group by a factor of 2 in its slope.
{"title":"Polynomial conditions and homology of FI-modules","authors":"Cihan Bahran","doi":"10.2140/pjm.2023.324.207","DOIUrl":"https://doi.org/10.2140/pjm.2023.324.207","url":null,"abstract":"We identify two recursively defined polynomial conditions for FI-modules in the literature. We characterize these conditions using homological invariants of FI-modules (namely the local degree and regularity, together with the stable degree) and clarify their relationship. For one of these conditions, we give improved twisted homological stability ranges for the symmetric groups. As another application, we improve the representation stability ranges for congruence subgroups with respect to the action of an appropriate linear group by a factor of 2 in its slope.","PeriodicalId":54651,"journal":{"name":"Pacific Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43970236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：