We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra $A^{(2)}_{2}$. We also give conjectural series for some level 2 modules of $A^{(2)}_{13}$.
{"title":"Andrews-Gordon type series for the level 5 and 7 standard modules of the affine Lie algebra $A^{(2)}_2$","authors":"Motoki Takigiku, Shunsuke Tsuchioka","doi":"10.1090/proc/15394","DOIUrl":"https://doi.org/10.1090/proc/15394","url":null,"abstract":"We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra $A^{(2)}_{2}$. We also give conjectural series for some level 2 modules of $A^{(2)}_{13}$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"246 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122521916","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}
Sp"ath showed that the Alperin-McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin-McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnaf'e-Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.
{"title":"Jordan Decomposition for the Alperin-McKay Conjecture","authors":"L. Ruhstorfer","doi":"10.25926/PXEY-HD44","DOIUrl":"https://doi.org/10.25926/PXEY-HD44","url":null,"abstract":"Sp\"ath showed that the Alperin-McKay conjecture in the representation theory of finite groups holds if the so-called inductive Alperin-McKay condition holds for all finite simple groups. In a previous article, we showed that the Bonnaf'e-Rouquier equivalence for blocks of finite groups of Lie type can be lifted to include automorphisms of groups of Lie type. We use our results to reduce the verification of the inductive condition for groups of Lie type to quasi-isolated blocks.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116896278","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 formulate the transfer factor of character lifting from orthogonal groups to symplectic groups by Adams in the framework of symplectic Dirac cohomology for the Lie superalgebras and the Rittenberg-Scheunert correspondence of representations of the Lie superalgebra $frofrsp(1|2n)$ and the Lie algebra $fro(2n+1)$. This leads to formulation of a direct lifting of characters from the linear symplectic group $Sp(2n,bbR)$ to its nonlinear covering metaplectic group $Mp(2n,bbR)$.
{"title":"Symplectic Dirac cohomology and lifting of\u0000 characters to metaplectic groups","authors":"Jing Huang","doi":"10.1090/CONM/768/15452","DOIUrl":"https://doi.org/10.1090/CONM/768/15452","url":null,"abstract":"We formulate the transfer factor of character lifting from orthogonal groups to symplectic groups by Adams in the framework of symplectic Dirac cohomology for the Lie superalgebras and the Rittenberg-Scheunert correspondence of representations of the Lie superalgebra $frofrsp(1|2n)$ and the Lie algebra $fro(2n+1)$. This leads to formulation of a direct lifting of characters from the linear symplectic group $Sp(2n,bbR)$ to its nonlinear covering metaplectic group $Mp(2n,bbR)$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"19 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125634852","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 : 2020-05-26DOI: 10.1142/S0219199721500231
T. Creutzig, Matt Rupert
We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $overline{U}_q^H(mfg)$ of a simple Lie algebra $mfg$ at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the $mfg=mathfrak{sl}_2$ case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra $W_Q(r)$ of Feigin and Tipunin cite{FT} and the $B_Q(r)$ algebras of cite{C1}.
{"title":"Uprolling unrolled quantum groups","authors":"T. Creutzig, Matt Rupert","doi":"10.1142/S0219199721500231","DOIUrl":"https://doi.org/10.1142/S0219199721500231","url":null,"abstract":"We construct families of commutative (super) algebra objects in the category of weight modules for the unrolled restricted quantum group $overline{U}_q^H(mfg)$ of a simple Lie algebra $mfg$ at roots of unity, and study their categories of local modules. We determine their simple modules and derive conditions for these categories being finite, non-degenerate, and ribbon. Motivated by numerous examples in the $mfg=mathfrak{sl}_2$ case, we expect some of these categories to compare nicely to categories of modules for vertex operator algebras. We focus in particular on examples expected to correspond to the higher rank triplet vertex algebra $W_Q(r)$ of Feigin and Tipunin cite{FT} and the $B_Q(r)$ algebras of cite{C1}.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132936779","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 study a semisimple extension of a Takiff superalgebra which turns out to have a remarkably rich representation theory. We determine the blocks in both the finite-dimensional and BGG module categories and also classify the Borel subalgebras. We further compute all extension groups between two finite-dimensional simple objects and prove that all non-principal blocks in the finite-dimensional module category are Koszul.
{"title":"Representation Theory of a Semisimple Extension of the Takiff Superalgebra","authors":"Shun-Jen Cheng, K. Coulembier","doi":"10.1093/IMRN/RNAB149","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB149","url":null,"abstract":"We study a semisimple extension of a Takiff superalgebra which turns out to have a remarkably rich representation theory. We determine the blocks in both the finite-dimensional and BGG module categories and also classify the Borel subalgebras. We further compute all extension groups between two finite-dimensional simple objects and prove that all non-principal blocks in the finite-dimensional module category are Koszul.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134029618","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 : 2020-05-15DOI: 10.1142/s0219498821501693
Hanpeng Gao
It is well known that the relation-extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove some extension of silted algebras are cluster-tilted algebras.
{"title":"Extending silted algebras to cluster-tilted algebras","authors":"Hanpeng Gao","doi":"10.1142/s0219498821501693","DOIUrl":"https://doi.org/10.1142/s0219498821501693","url":null,"abstract":"It is well known that the relation-extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove some extension of silted algebras are cluster-tilted algebras.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129761055","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 show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero mode of an appropriate combinatorial generating function. As an application, we obtain explicit formulas for all polynomial tau-functions of these hierarchies in terms of Schur polynomials and Q-Schur polynomials respectively. We also obtain formulas for polynomial tau-functions of the reductions of the s-component KP hierarchy associated to partitions in s parts.
{"title":"Polynomial tau-functions of the KP, BKP, and the s-component KP hierarchies","authors":"V. Kac, N. Rozhkovskaya, J. van de Leur","doi":"10.1063/5.0013017","DOIUrl":"https://doi.org/10.1063/5.0013017","url":null,"abstract":"We show that any polynomial tau-function of the s-component KP and the BKP hierarchies can be interpreted as a zero mode of an appropriate combinatorial generating function. As an application, we obtain explicit formulas for all polynomial tau-functions of these hierarchies in terms of Schur polynomials and Q-Schur polynomials respectively. We also obtain formulas for polynomial tau-functions of the reductions of the s-component KP hierarchy associated to partitions in s parts.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"167 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121138015","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 consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $Uto u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$.
通过有限全局维的Noetherian Hopf代数考虑具有光滑变形的有限维Hopf代数。这类Hopf代数的例子包括复上的小量子群、有限特征的受限包络代数和高度$1$群的Drinfeld双精度方案。通过在相应的参数化空间上与函数f$相关的超曲面$u /(f)$的奇异范畴，我们提供了一种分析这种$u$上同调支持的方法。对于有限群格式上的函数、一定高度1可解有限群格式的Drinfeld双元、玻色子化量子完全交和类型$ a $的小量子Borel，我们使用这种超曲面方法建立了上同调支持的张量积性质。
{"title":"Support for integrable Hopf algebras via noncommutative hypersurfaces","authors":"C. Negron, J. Pevtsova","doi":"10.1093/IMRN/RNAB264","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB264","url":null,"abstract":"We consider finite-dimensional Hopf algebras $u$ which admit a smooth deformation $Uto u$ by a Noetherian Hopf algebra $U$ of finite global dimension. Examples of such Hopf algebras include small quantum groups over the complex numbers, restricted enveloping algebras in finite characteristic, and Drinfeld doubles of height $1$ group schemes. We provide a means of analyzing (cohomological) support for representations over such $u$, via the singularity categories of the hypersurfaces $U/(f)$ associated to functions $f$ on the corresponding parametrization space. We use this hypersurface approach to establish the tensor product property for cohomological support, for the following examples: functions on a finite group scheme, Drinfeld doubles of certain height 1 solvable finite group schemes, bosonized quantum complete intersections, and the small quantum Borel in type $A$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"303 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129046266","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 $k$ be a field, $ngeq 3$ an integer and $mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $mathcal{T}^{fd}$ and a subcategory $mathcal{M}=text{add}(M)$ such that $(mathcal{T}, mathcal{T}^{fd}, mathcal{M})$ is an $n$-Calabi-Yau triple. For every integer $m$ and every object $X$ in $mathcal{T}$, there is a unique, up to isomorphism, truncation triangle of the form begin{align*} X^{leq m}rightarrow Xrightarrow X^{geq m+1}rightarrowSigma X^{leq m}, end{align*} with respect to the $t$-structure $((Sigma^{ -m}mathcal{M})^{perp_mathcal{T}})$. In this paper, we prove some properties of the triangulated categories $mathcal{T}$ and $mathcal{T}/mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $mathcal{T}$, showing when the truncation triangles split. Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $mathcal{T}/mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $mathcal{T}$ and over $mathcal{T}/mathcal{T}^{fd}$ are isomorphic. Note that these properties make $mathcal{T}/mathcal{T}^{fd}$ a generalisation of the cluster category.
{"title":"Properties of triangulated and quotient\u0000categories arising from n-Calabi–Yau triples","authors":"F. Fedele","doi":"10.2140/PJM.2021.310.1","DOIUrl":"https://doi.org/10.2140/PJM.2021.310.1","url":null,"abstract":"Let $k$ be a field, $ngeq 3$ an integer and $mathcal{T}$ a $k$-linear triangulated category with a triangulated subcategory $mathcal{T}^{fd}$ and a subcategory $mathcal{M}=text{add}(M)$ such that $(mathcal{T}, mathcal{T}^{fd}, mathcal{M})$ is an $n$-Calabi-Yau triple. For every integer $m$ and every object $X$ in $mathcal{T}$, there is a unique, up to isomorphism, truncation triangle of the form \u0000begin{align*} \u0000X^{leq m}rightarrow Xrightarrow X^{geq m+1}rightarrowSigma X^{leq m}, \u0000end{align*} with respect to the $t$-structure $((Sigma^{ -m}mathcal{M})^{perp_mathcal{T}})$. In this paper, we prove some properties of the triangulated categories $mathcal{T}$ and $mathcal{T}/mathcal{T}^{fd}$. Our first result gives a relation between the Hom-spaces in these categories, using limits and colimits. Our second result is a Gap Theorem in $mathcal{T}$, showing when the truncation triangles split. \u0000Moreover, we apply our two theorems to present an alternative proof to a result by Guo, originally stated in a more specific setup of dg $k$-algebras $A$ and subcategories of the derived category of dg $A$-modules. This proves that $mathcal{T}/mathcal{T}^{fd}$ is Hom-finite and $(n-1)$-Calabi-Yau, its object $M$ is $(n-1)$-cluster tilting and the endomorphism algebras of $M$ over $mathcal{T}$ and over $mathcal{T}/mathcal{T}^{fd}$ are isomorphic. Note that these properties make $mathcal{T}/mathcal{T}^{fd}$ a generalisation of the cluster category.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126335215","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 equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the $p$-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable.
{"title":"Categorifying Hecke algebras at prime roots of unity, part I","authors":"Ben Elias, You Qi","doi":"10.1090/tran/8908","DOIUrl":"https://doi.org/10.1090/tran/8908","url":null,"abstract":"We equip the type A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p it becomes a p-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. We conjecture that the $p$-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p-canonical basis. More precise conjectures will be found in the sequel. \u0000Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A. We also examine a particular Bott-Samelson bimodule in type A_7, which is indecomposable in characteristic 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p-dg setting of being indecomposable.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128285592","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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