The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category O is equivalent to knowledge of the Kazhdan-Lusztig character of the irreducible object. We then explore the extent to which Cherednik algebra techniques may be applied to ideals of linear subspace arrangements: we determine when the radical of the polynomial representation of the Cherednik algebra is a radical ideal, and, for the cyclotomic rational Cherednik algebra, determine the socle of the polynomial representation and characterize when it is a radical ideal. The subspace arrangements that arise include various generalizations of the k-equals arrangment. In the case of the socle, we give an explicit vector space basis in terms of certain specializations of non-symmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov. These results suggest several conjectures and questions about the submodule structure of the polynomial representation of the Cherednik algebra.
{"title":"Subspace Arrangements and Cherednik Algebras","authors":"Stephen Griffeth","doi":"10.1093/IMRN/RNAB016","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB016","url":null,"abstract":"The purpose of this article is to study the relationship between numerical invariants of certain subspace arrangements coming from reflection groups and numerical invariants arising in the representation theory of Cherednik algebras. For instance, we observe that knowledge of the equivariant graded Betti numbers (in the sense of commutative algebra) of any irreducible representation in category O is equivalent to knowledge of the Kazhdan-Lusztig character of the irreducible object. We then explore the extent to which Cherednik algebra techniques may be applied to ideals of linear subspace arrangements: we determine when the radical of the polynomial representation of the Cherednik algebra is a radical ideal, and, for the cyclotomic rational Cherednik algebra, determine the socle of the polynomial representation and characterize when it is a radical ideal. The subspace arrangements that arise include various generalizations of the k-equals arrangment. In the case of the socle, we give an explicit vector space basis in terms of certain specializations of non-symmetric Jack polynomials, which in particular determines its minimal generators and Hilbert series and answers a question posed by Feigin and Shramov. These results suggest several conjectures and questions about the submodule structure of the polynomial representation of the Cherednik algebra.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125624061","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}
Pub Date : 2019-05-03DOI: 10.1007/s00209-020-02561-1
Dor Mezer
{"title":"Multiplicity one theorem for $$(mathrm {GL}_{n+1},mathrm {GL}_n)$$ over a local field of positive characteristic","authors":"Dor Mezer","doi":"10.1007/s00209-020-02561-1","DOIUrl":"https://doi.org/10.1007/s00209-020-02561-1","url":null,"abstract":"","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130010803","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}
All finite-dimensional Leibniz algebra bimodules of a Lie algebra $mathfrak{sl}_2$ over a field of characteristic zero are described.
描述了李代数$mathfrak{sl}_2$在特征为零的域上的所有有限维莱布尼兹代数双模。
{"title":"Finite-dimensional Leibniz algebra representations of $mathfrak{sl}_2$","authors":"T. Kurbanbaev, R. Turdibaev","doi":"10.15672/hujms.788994","DOIUrl":"https://doi.org/10.15672/hujms.788994","url":null,"abstract":"All finite-dimensional Leibniz algebra bimodules of a Lie algebra $mathfrak{sl}_2$ over a field of characteristic zero are described.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124694561","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}
Pub Date : 2019-04-09DOI: 10.4007/annals.2020.192.1.4
M. Geck
The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.
{"title":"Green functions and Glauberman degree-divisibility","authors":"M. Geck","doi":"10.4007/annals.2020.192.1.4","DOIUrl":"https://doi.org/10.4007/annals.2020.192.1.4","url":null,"abstract":"The Glauberman correspondence is a fundamental bijection in the character theory of finite groups. In 1994, Hartley and Turull established a degree-divisibility property for characters related by that correspondence, subject to a congruence condition which should hold for the Green functions of finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present a general argument for completing the proof of that congruence condition. Consequently, the degree-divisibility property holds in complete generality.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"50 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129763203","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}
Pub Date : 2019-04-01DOI: 10.1142/S0219498821501310
Ayako Itaba, Kenta Ueyama
Let $A=A(alpha, beta)$ be a graded down-up algebra with $({rm deg},x, {rm deg},y)=(1,n)$ and $beta neq 0$, and let $nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $nabla A$ for the case $n geq 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether $left(begin{smallmatrix} 1&0 end{smallmatrix}right)left(begin{smallmatrix} alpha &1 beta &0 end{smallmatrix}right)^nleft(begin{smallmatrix} 1 0 end{smallmatrix}right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $ngeq 3$ in the context of Grothendieck groups.
{"title":"Hochschild cohomology related to graded down-up algebras with weights (1,n)","authors":"Ayako Itaba, Kenta Ueyama","doi":"10.1142/S0219498821501310","DOIUrl":"https://doi.org/10.1142/S0219498821501310","url":null,"abstract":"Let $A=A(alpha, beta)$ be a graded down-up algebra with $({rm deg},x, {rm deg},y)=(1,n)$ and $beta neq 0$, and let $nabla A$ be the Beilinson algebra of $A$. If $n=1$, then a description of the Hochschild cohomology group of $nabla A$ is known. In this paper, we calculate the Hochschild cohomology group of $nabla A$ for the case $n geq 2$. As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of $A$ is different depending on whether $left(begin{smallmatrix} 1&0 end{smallmatrix}right)left(begin{smallmatrix} alpha &1 beta &0 end{smallmatrix}right)^nleft(begin{smallmatrix} 1 0 end{smallmatrix}right)$ is zero or not. Moreover, it turns out that there is a difference between the cases $n=2$ and $ngeq 3$ in the context of Grothendieck groups.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115215057","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 give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.
给出了有限对称群不可约模表示维数的新的、显式的、渐近尖锐的下界。
{"title":"Lower bounds for dimensions of irreducible representations of symmetric groups","authors":"A. Kleshchev, Lucia Morotti, P. Tiep","doi":"10.1090/proc/14873","DOIUrl":"https://doi.org/10.1090/proc/14873","url":null,"abstract":"We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122564669","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}
For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.
{"title":"On solvability of the first Hochschild cohomology of a finite-dimensional algebra","authors":"F. Eisele, Theo Raedschelders","doi":"10.1090/tran/8064","DOIUrl":"https://doi.org/10.1090/tran/8064","url":null,"abstract":"For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116834541","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 establish a closed formula for a singular vector of weight $lambda-beta$ in the Verma module of highest weight $lambda$ for Lie superalgebra $mathfrak{gl}(m|n)$ when $lambda$ is atypical with respect to an odd positive root $beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $lambda$ and $lambda-beta$ are dominant integral.
{"title":"Odd Singular Vector Formula for General Linear Lie Superalgebras","authors":"Jie Liu, Lipeng Luo, Weiqiang Wang","doi":"10.21915/bimas.2019401","DOIUrl":"https://doi.org/10.21915/bimas.2019401","url":null,"abstract":"We establish a closed formula for a singular vector of weight $lambda-beta$ in the Verma module of highest weight $lambda$ for Lie superalgebra $mathfrak{gl}(m|n)$ when $lambda$ is atypical with respect to an odd positive root $beta$. It is further shown that this vector is unique up to a scalar multiple, and it descends to a singular vector, again unique up to a scalar multiple, in the corresponding Kac module when both $lambda$ and $lambda-beta$ are dominant integral.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123867863","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}
There are two aims in this paper. One is to give criteria on $tau$-tilting finiteness for two kinds of two-point algebras; another is to give criteria on $tau$-tilting finiteness for algebras from Table T and Table W introduced by Han [13].
{"title":"$tau$-tilting finiteness of two-point algebras I","authors":"Qi Wang","doi":"10.18926/mjou/62799","DOIUrl":"https://doi.org/10.18926/mjou/62799","url":null,"abstract":"There are two aims in this paper. One is to give criteria on $tau$-tilting finiteness for two kinds of two-point algebras; another is to give criteria on $tau$-tilting finiteness for algebras from Table T and Table W introduced by Han [13].","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130407723","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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