In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite representation type. In order to investigate completions of bounded derived categories of algebras of finite global dimension, we define image and preimage metrics under a triangulated functor and use them to induce a triangulated equivalence between two completions. Furthermore, for a given metric on a triangulated category we construct a new, closely related good metric called the improvement and compare the respective completions.
{"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}
Let $W$ be a simply laced Weyl group of finite type and rank $n$. If $W$ has type $E_7$, $E_8$, or $D_n$ for $n$ even, then the root system of $W$ has subsystems of type $nA_1$. This gives rise to an irreducible Macdonald representation of $W$ spanned by $n$-roots, which are products of $n$ orthogonal roots in the symmetric algebra of the reflection representation. We prove that in these cases, the set of all maximal sets of orthogonal positive roots has the structure of a quasiparabolic set in the sense of Rains--Vazirani. The quasiparabolic structure can be described in terms of certain quadruples of orthogonal positive roots which we call crossings, nestings, and alignments. This leads to nonnesting and noncrossing bases for the Macdonald representation, as well as some highly structured partially ordered sets. We use the $8$-roots in type $E_8$ to give a concise description of a graph that is known to be non-isomorphic but quantum isomorphic to the orthogonality graph of the $E_8$ root system.
{"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}
The representation theory of the Nappi-Witten VOA was initiated in arXiv:1104.3921 and arXiv:2011.14453. In this paper we use the technique of inverse quantum hamiltonian reduction to investigate the representation theory of the Nappi-Witten VOA $ V^1(mathfrak h_4)$. We first prove that the quantum hamiltonian reduction of $ V^1(mathfrak h_4)$ is the Heisenberg-Virasoro VOA $L^{HVir}$ of level zero investigated in arXiv:math/0201314 and arXiv:1405.1707. We invert the quantum hamiltonian reduction in this case and prove that $ V^1(mathfrak h_4)$ is realized as a vertex subalgebra of $L^{HVir} otimes Pi$, where $Pi$ is a certain lattice-like vertex algebra. Using such an approach we shall realize all relaxed highest weight modules which were classified in arXiv:2011.14453. We show that every relaxed highest weight module, whose top components is neither highest nor lowest weight $mathfrak h_4$-module, has the form $M_1 otimes Pi_{1} (lambda)$ where $M_1$ is an irreducible, highest weight $L^{HVir}$-module and $Pi_{1} (lambda)$ is an irreducible weight $Pi$-module. Using the fusion rules for $L^{HVir}$-modules and the previously developed methods of constructing logarithmic modules we are able to construct a family of logarithmic $V^1(mathfrak h_4)$-modules. The Loewy diagrams of these logarithmic modules are completely analogous to the Loewy diagrams of projective modules of weight $L_k(mathfrak{sl}(2))$-modules, so we expect that our logarithmic modules are also projective in a certain category of weight $ V^1(mathfrak h_4)$-modules.
{"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}
In this paper, we consider the mixed tensor space of a $G$-graded vector space where $G$ is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the general linear group which we refer to as the general linear color group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the general linear color group, on color analogues of several copies of matrices. We show that in this special case, this is the set of trace monomials, which coincides with the set of generators obtained by Berele.
{"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}
We prove that the Eaton-Moreto conjecture is true for the principal blocks of the p-solvable groups
我们证明了伊顿-莫雷托猜想对于可解 p 群的主块是真实的
{"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}
We introduce the Macdonald piece polynomial $operatorname{I}_{mu,lambda,k}[X;q,t]$, which is a vast generalization of the Macdonald intersection polynomial in the science fiction conjecture by Bergeron and Garsia. We demonstrate a remarkable connection between $operatorname{I}_{mu,lambda,k}$, $nabla s_{lambda}$, and the Loehr--Warrington formula $operatorname{LW}_{lambda}$, thereby obtaining the Loehr--Warrington conjecture as a corollary. To connect $operatorname{I}_{mu,lambda,k}$ and $nabla s_{lambda}$, we employ the plethystic formula for the Macdonald polynomials of Garsia--Haiman--Tesler, and to connect $operatorname{I}_{mu,lambda,k}$ and $operatorname{LW}_{lambda}$, we use our new findings on the combinatorics of $P$-tableaux together with the column exchange rule. We also present an extension of the science fiction conjecture and the Macdonald positivity by exploiting $operatorname{I}_{mu,lambda,k}$.
{"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}
For given linear action of a finite group on a lattice and a positive integer q, we prove that the mod q permutation representation is a quasi-polynomial in q. Additionally, we establish several results that can be considered as mod q-analogues of results by Stapledon for equivariant Ehrhart quasi-polynomials. We also prove a reciprocity-type result for multiplicities of irreducible decompositions.
{"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}
Remarkable work of Kaledin, based on earlier joint work with Bezrukavnikov, has constructed a tilting generator of the category of coherent sheaves on a very general class of symplectic resolutions of singularities. In this paper, we give a concrete construction of this tilting generator on the cotangent bundle of $Gr(2,4)$, the Grassmannian of 2-planes in $mathbb{C}^4$. This construction builds on work of the second author describing these tilting bundles in terms of KLRW algebras, but in this low-dimensional case, we are able to describe our tilting generator as a sum of geometrically natural bundles on $T^*Gr(2,4)$: line bundles and their extensions, as well as the tautological bundle and its perpendicular.
{"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}
Generalizing the dihedral picture for G(M,M,2), we construct Hecke algebras (and categories) and asymptotic counterparts. We think of these as associated with the complex reflection group G(M,M,N).
{"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}
Nadia Lafrenière, Rosa Orellana, Anna Pun, Sheila Sundaram, Stephanie van Willigenburg, Tamsen Whitehead McGinley
The immaculate Hecke poset was introduced and investigated by Niese, Sundaram, van Willigenburg, Vega and Wang, who established the full poset structure, and determined modules for the 0-Hecke algebra action on immaculate and row-strict immaculate tableaux. In this paper, we extend their results by introducing the skew immaculate Hecke poset. We investigate the poset structure, and construct modules for the 0-Hecke algebra action on skew immaculate and skew row-strict immaculate tableaux, thus showing that the skew immaculate Hecke poset captures representation-theoretic information analogous to the immaculate Hecke poset. We also describe branching rules for the resulting skew modules.
Niese、Sundaram、van Willigenburg、Vega 和 Wang 介绍并研究了无暇赫克正集，他们建立了完整的正集结构，并确定了 0 赫克代数作用于无暇和行严格无暇表元的模块。在本文中，我们通过引入斜无暇赫克正集扩展了他们的成果。我们研究了正集结构，并构建了0-Hecke代数作用于斜无暇和斜行-严格无暇表元的模块，从而证明斜无暇Hecke正集捕获了类似于无暇Hecke正集的表述理论信息。
{"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}
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