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Metric completions of triangulated categories from finite dimensional algebras 来自有限维代数的三角范畴的公设补全
arXiv - MATH - Representation Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01828
Cyril Matoušek
{"title":"Metric completions of triangulated categories from finite dimensional algebras","authors":"Cyril Matoušek","doi":"arxiv-2409.01828","DOIUrl":"https://doi.org/arxiv-2409.01828","url":null,"abstract":"In this paper, we study metric completions of triangulated categories in a\u0000representation-theoretic context. We provide a concrete description of\u0000completions of bounded derived categories of hereditary finite dimensional\u0000algebras of finite representation type. In order to investigate completions of\u0000bounded derived categories of algebras of finite global dimension, we define\u0000image and preimage metrics under a triangulated functor and use them to induce\u0000a triangulated equivalence between two completions. Furthermore, for a given\u0000metric on a triangulated category we construct a new, closely related good\u0000metric called the improvement and compare the respective completions.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187155","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}
引用次数: 0
Orthogonal roots, Macdonald representations, and quasiparabolic sets 正交根、麦克唐纳表示和准抛物集合
arXiv - MATH - Representation Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01948
R. M. Green, Tianyuan Xu
{"title":"Orthogonal roots, Macdonald representations, and quasiparabolic sets","authors":"R. M. Green, Tianyuan Xu","doi":"arxiv-2409.01948","DOIUrl":"https://doi.org/arxiv-2409.01948","url":null,"abstract":"Let $W$ be a simply laced Weyl group of finite type and rank $n$. If $W$ has\u0000type $E_7$, $E_8$, or $D_n$ for $n$ even, then the root system of $W$ has\u0000subsystems of type $nA_1$. This gives rise to an irreducible Macdonald\u0000representation of $W$ spanned by $n$-roots, which are products of $n$\u0000orthogonal roots in the symmetric algebra of the reflection representation. We\u0000prove that in these cases, the set of all maximal sets of orthogonal positive\u0000roots has the structure of a quasiparabolic set in the sense of\u0000Rains--Vazirani. The quasiparabolic structure can be described in terms of\u0000certain quadruples of orthogonal positive roots which we call crossings,\u0000nestings, and alignments. This leads to nonnesting and noncrossing bases for\u0000the Macdonald representation, as well as some highly structured partially\u0000ordered sets. We use the $8$-roots in type $E_8$ to give a concise description\u0000of a graph that is known to be non-isomorphic but quantum isomorphic to the\u0000orthogonality graph of the $E_8$ root system.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187161","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}
引用次数: 0
Nappi-Witten vertex operator algebra via inverse Quantum Hamiltonian Reduction 通过逆量子哈密顿还原的纳比-维滕顶点算子代数
arXiv - MATH - Representation Theory Pub Date : 2024-09-03 DOI: arxiv-2409.02093
Drazen Adamovic, Andrei Babichenko
{"title":"Nappi-Witten vertex operator algebra via inverse Quantum Hamiltonian Reduction","authors":"Drazen Adamovic, Andrei Babichenko","doi":"arxiv-2409.02093","DOIUrl":"https://doi.org/arxiv-2409.02093","url":null,"abstract":"The representation theory of the Nappi-Witten VOA was initiated in\u0000arXiv:1104.3921 and arXiv:2011.14453. In this paper we use the technique of\u0000inverse quantum hamiltonian reduction to investigate the representation theory\u0000of the Nappi-Witten VOA $ V^1(mathfrak h_4)$. We first prove that the quantum\u0000hamiltonian reduction of $ V^1(mathfrak h_4)$ is the Heisenberg-Virasoro VOA\u0000$L^{HVir}$ of level zero investigated in arXiv:math/0201314 and\u0000arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and\u0000prove that $ V^1(mathfrak h_4)$ is realized as a vertex subalgebra of\u0000$L^{HVir} otimes Pi$, where $Pi$ is a certain lattice-like vertex algebra.\u0000Using such an approach we shall realize all relaxed highest weight modules\u0000which were classified in arXiv:2011.14453. We show that every relaxed highest\u0000weight module, whose top components is neither highest nor lowest weight\u0000$mathfrak h_4$-module, has the form $M_1 otimes Pi_{1} (lambda)$ where\u0000$M_1$ is an irreducible, highest weight $L^{HVir}$-module and $Pi_{1}\u0000(lambda)$ is an irreducible weight $Pi$-module. Using the fusion rules for $L^{HVir}$-modules and the previously developed\u0000methods of constructing logarithmic modules we are able to construct a family\u0000of logarithmic $V^1(mathfrak h_4)$-modules. The Loewy diagrams of these\u0000logarithmic modules are completely analogous to the Loewy diagrams of\u0000projective modules of weight $L_k(mathfrak{sl}(2))$-modules, so we expect that\u0000our logarithmic modules are also projective in a certain category of weight $\u0000V^1(mathfrak h_4)$-modules.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187170","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}
引用次数: 0
Mixed tensor invariants of Lie color algebra 列色代数的混合张量不变式
arXiv - MATH - Representation Theory Pub Date : 2024-09-03 DOI: arxiv-2409.02068
Santosha Pattanayak, Preena Samuel
{"title":"Mixed tensor invariants of Lie color algebra","authors":"Santosha Pattanayak, Preena Samuel","doi":"arxiv-2409.02068","DOIUrl":"https://doi.org/arxiv-2409.02068","url":null,"abstract":"In this paper, we consider the mixed tensor space of a $G$-graded vector\u0000space where $G$ is a finite abelian group. We obtain a spanning set of\u0000invariants of the associated symmetric algebra under the action of a color\u0000analogue of the general linear group which we refer to as the general linear\u0000color group. As a consequence, we obtain a generating set for the polynomial\u0000invariants, under the simultaneous action of the general linear color group, on\u0000color analogues of several copies of matrices. We show that in this special\u0000case, this is the set of trace monomials, which coincides with the set of\u0000generators obtained by Berele.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187150","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}
引用次数: 0
The Eaton-Moreto Conjecture and p-Solvable Groups 埃顿-莫雷托猜想和可解 p 群
arXiv - MATH - Representation Theory Pub Date : 2024-09-03 DOI: arxiv-2409.01634
Gabriel Navarro
{"title":"The Eaton-Moreto Conjecture and p-Solvable Groups","authors":"Gabriel Navarro","doi":"arxiv-2409.01634","DOIUrl":"https://doi.org/arxiv-2409.01634","url":null,"abstract":"We prove that the Eaton-Moreto conjecture is true for the principal blocks of\u0000the p-solvable groups","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187164","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}
引用次数: 0
Extending the science fiction and the Loehr--Warrington formula 扩展科幻小说和罗尔--沃林顿公式
arXiv - MATH - Representation Theory Pub Date : 2024-09-02 DOI: arxiv-2409.01041
Donghyun Kim, Jaeseong Oh
{"title":"Extending the science fiction and the Loehr--Warrington formula","authors":"Donghyun Kim, Jaeseong Oh","doi":"arxiv-2409.01041","DOIUrl":"https://doi.org/arxiv-2409.01041","url":null,"abstract":"We introduce the Macdonald piece polynomial\u0000$operatorname{I}_{mu,lambda,k}[X;q,t]$, which is a vast generalization of\u0000the Macdonald intersection polynomial in the science fiction conjecture by\u0000Bergeron and Garsia. We demonstrate a remarkable connection between\u0000$operatorname{I}_{mu,lambda,k}$, $nabla s_{lambda}$, and the\u0000Loehr--Warrington formula $operatorname{LW}_{lambda}$, thereby obtaining the\u0000Loehr--Warrington conjecture as a corollary. To connect\u0000$operatorname{I}_{mu,lambda,k}$ and $nabla s_{lambda}$, we employ the\u0000plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and\u0000to connect $operatorname{I}_{mu,lambda,k}$ and\u0000$operatorname{LW}_{lambda}$, we use our new findings on the combinatorics of\u0000$P$-tableaux together with the column exchange rule. We also present an\u0000extension of the science fiction conjecture and the Macdonald positivity by\u0000exploiting $operatorname{I}_{mu,lambda,k}$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187163","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}
引用次数: 0
The quasi-polynomiality of mod q permutation representation for a linear finite group action on a lattice 网格上线性有限群作用的模q置换表示的准多项式性
arXiv - MATH - Representation Theory Pub Date : 2024-09-02 DOI: arxiv-2409.01084
Ryo Uchiumi, Masahiko Yoshinaga
{"title":"The quasi-polynomiality of mod q permutation representation for a linear finite group action on a lattice","authors":"Ryo Uchiumi, Masahiko Yoshinaga","doi":"arxiv-2409.01084","DOIUrl":"https://doi.org/arxiv-2409.01084","url":null,"abstract":"For given linear action of a finite group on a lattice and a positive integer\u0000q, we prove that the mod q permutation representation is a quasi-polynomial in\u0000q. Additionally, we establish several results that can be considered as mod\u0000q-analogues of results by Stapledon for equivariant Ehrhart quasi-polynomials.\u0000We also prove a reciprocity-type result for multiplicities of irreducible\u0000decompositions.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187167","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}
引用次数: 0
Tilting Generator for the $T^*Gr(2,4)$ Coulomb Branch T^*Gr(2,4)$库仑分支的倾斜发生器
arXiv - MATH - Representation Theory Pub Date : 2024-09-02 DOI: arxiv-2409.01379
Aiden Suter, Ben Webster
{"title":"Tilting Generator for the $T^*Gr(2,4)$ Coulomb Branch","authors":"Aiden Suter, Ben Webster","doi":"arxiv-2409.01379","DOIUrl":"https://doi.org/arxiv-2409.01379","url":null,"abstract":"Remarkable work of Kaledin, based on earlier joint work with Bezrukavnikov,\u0000has constructed a tilting generator of the category of coherent sheaves on a\u0000very general class of symplectic resolutions of singularities. In this paper, we give a concrete construction of this tilting generator on\u0000the cotangent bundle of $Gr(2,4)$, the Grassmannian of 2-planes in\u0000$mathbb{C}^4$. This construction builds on work of the second author\u0000describing these tilting bundles in terms of KLRW algebras, but in this\u0000low-dimensional case, we are able to describe our tilting generator as a sum of\u0000geometrically natural bundles on $T^*Gr(2,4)$: line bundles and their\u0000extensions, as well as the tautological bundle and its perpendicular.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187162","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}
引用次数: 0
On Hecke and asymptotic categories for complex reflection groups 论复杂反射群的赫克和渐近范畴
arXiv - MATH - Representation Theory Pub Date : 2024-09-02 DOI: arxiv-2409.01005
Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz
{"title":"On Hecke and asymptotic categories for complex reflection groups","authors":"Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz","doi":"arxiv-2409.01005","DOIUrl":"https://doi.org/arxiv-2409.01005","url":null,"abstract":"Generalizing the dihedral picture for G(M,M,2), we construct Hecke algebras\u0000(and categories) and asymptotic counterparts. We think of these as associated\u0000with the complex reflection group G(M,M,N).","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187156","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}
引用次数: 0
The skew immaculate Hecke poset and 0-Hecke modules 斜无暇赫克正集和 0 赫克模块
arXiv - MATH - Representation Theory Pub Date : 2024-09-01 DOI: arxiv-2409.00709
Nadia Lafrenière, Rosa Orellana, Anna Pun, Sheila Sundaram, Stephanie van Willigenburg, Tamsen Whitehead McGinley
{"title":"The skew immaculate Hecke poset and 0-Hecke modules","authors":"Nadia Lafrenière, Rosa Orellana, Anna Pun, Sheila Sundaram, Stephanie van Willigenburg, Tamsen Whitehead McGinley","doi":"arxiv-2409.00709","DOIUrl":"https://doi.org/arxiv-2409.00709","url":null,"abstract":"The immaculate Hecke poset was introduced and investigated by Niese,\u0000Sundaram, van Willigenburg, Vega and Wang, who established the full poset\u0000structure, and determined modules for the 0-Hecke algebra action on immaculate\u0000and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate\u0000Hecke poset. We investigate the poset structure, and construct modules for the\u00000-Hecke algebra action on skew immaculate and skew row-strict immaculate\u0000tableaux, thus showing that the skew immaculate Hecke poset captures\u0000representation-theoretic information analogous to the immaculate Hecke poset.\u0000We also describe branching rules for the resulting skew modules.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187158","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}
引用次数: 0
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