In this paper, we formulate a conjecture that describes the local theta correspondences in terms of the local Langland correspondences for rigid inner twists, which contain the correspondences for quaternionic dual pairs. Moreover, we verify the conjecture holds in some specific cases.
{"title":"Local theta correspondences and Langlands parameters for rigid inner twists","authors":"Hirotaka Kakuhama","doi":"arxiv-2409.00805","DOIUrl":"https://doi.org/arxiv-2409.00805","url":null,"abstract":"In this paper, we formulate a conjecture that describes the local theta\u0000correspondences in terms of the local Langland correspondences for rigid inner\u0000twists, which contain the correspondences for quaternionic dual pairs.\u0000Moreover, we verify the conjecture holds in some specific cases.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187157","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}
By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More explicitly, instead of algebra objects in finite tensor categories, we consider quasi-finite coalgebra objects in locally finite tensor categories. Moreover, we show that module categories over non-rigid monoidal categories can be reconstructed via lax module monads, which generalize algebra objects. For the category of finite-dimensional comodules over a (non-Hopf) bialgebra, we give this result a more concrete form, realizing module categories as categories of contramodules over Hopf trimodule algebras -- this specializes to our tensor-categorical results in the Hopf case. Using lax module functors we give a categorical proof of the variant of the fundamental theorem of Hopf modules which applies to Hopf trimodules. We also give a characterization of fusion operators for a Hopf monad as coherence cells for a module functor structure, using which we similarly reinterpret and reprove the Hopf-monadic fundamental theorem of Hopf modules due to Brugui`eres, Lack, and Virelizier.
{"title":"Reconstruction of module categories in the infinite and non-rigid settings","authors":"Mateusz Stroiński, Tony Zorman","doi":"arxiv-2409.00793","DOIUrl":"https://doi.org/arxiv-2409.00793","url":null,"abstract":"By building on the notions of internal projective and injective objects in a\u0000module category introduced by Douglas, Schommer-Pries, and Snyder, we extend\u0000the reconstruction theory for module categories of Etingof and Ostrik. More\u0000explicitly, instead of algebra objects in finite tensor categories, we consider\u0000quasi-finite coalgebra objects in locally finite tensor categories. Moreover,\u0000we show that module categories over non-rigid monoidal categories can be\u0000reconstructed via lax module monads, which generalize algebra objects. For the\u0000category of finite-dimensional comodules over a (non-Hopf) bialgebra, we give\u0000this result a more concrete form, realizing module categories as categories of\u0000contramodules over Hopf trimodule algebras -- this specializes to our\u0000tensor-categorical results in the Hopf case. Using lax module functors we give\u0000a categorical proof of the variant of the fundamental theorem of Hopf modules\u0000which applies to Hopf trimodules. We also give a characterization of fusion\u0000operators for a Hopf monad as coherence cells for a module functor structure,\u0000using which we similarly reinterpret and reprove the Hopf-monadic fundamental\u0000theorem of Hopf modules due to Brugui`eres, Lack, and Virelizier.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"393 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187166","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}
It is proven that if a finite group $G$ has a normal subgroup $H$ with $p'$-index (where $p$ is a prime) and $G/H$ is solvable, then for a $p$-subgroup $P$ of $H$, if the Scott $kH$-module with vertex $P$ is Brauer indecomposable, then so is the Scott $kG$-module with vertex $P$, where $k$ is a field of characteristic $p>0$. This has several applications.
{"title":"Lifting Brauer indecomposability of a Scott module","authors":"Shigeo Koshitani, İpek Tuvay","doi":"arxiv-2409.00403","DOIUrl":"https://doi.org/arxiv-2409.00403","url":null,"abstract":"It is proven that if a finite group $G$ has a normal subgroup $H$ with\u0000$p'$-index (where $p$ is a prime) and $G/H$ is solvable, then for a\u0000$p$-subgroup $P$ of $H$, if the Scott $kH$-module with vertex $P$ is Brauer\u0000indecomposable, then so is the Scott $kG$-module with vertex $P$, where $k$ is\u0000a field of characteristic $p>0$. This has several applications.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187160","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}
By inventing the notion of honeycombs, A. Knutson and T. Tao proved the saturation conjecture for Littlewood-Richardson coefficients. The Newell-Littlewood numbers are a generalization of the Littlewood-Richardson coefficients. By introducing honeycombs on a M"obius strip, we prove the saturation conjecture for Newell-Littlewood numbers posed by S. Gao, G. Orelowitz and A. Yong.
通过发明蜂窝概念，A. Knutson 和 T. Tao 证明了 Littlewood-Richardson 系数的饱和猜想。纽厄尔-利特尔伍德数是利特尔伍德-理查德森系数的广义化。通过在 M"obius 带上引入蜂窝，我们证明了由 S. Gao、G.Orelowitz 和 A. Yong 提出的纽厄尔-利特尔伍德数饱和猜想。
{"title":"Proof of the Newell-Littlewood saturation conjecture","authors":"Jaewon Min","doi":"arxiv-2409.00233","DOIUrl":"https://doi.org/arxiv-2409.00233","url":null,"abstract":"By inventing the notion of honeycombs, A. Knutson and T. Tao proved the\u0000saturation conjecture for Littlewood-Richardson coefficients. The\u0000Newell-Littlewood numbers are a generalization of the Littlewood-Richardson\u0000coefficients. By introducing honeycombs on a M\"obius strip, we prove the\u0000saturation conjecture for Newell-Littlewood numbers posed by S. Gao, G.\u0000Orelowitz and A. Yong.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187159","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 discuss the extension of the faithfulness question for the Burau representation of braid groups to the case of Artin--Tits groups. We prove that the Burau representation is not faithful in affine type $tilde{A_3}$, and not faithful over several finite rings in type $D_4$, using an algorithmic approach based on categorical methods that generalize Bigelow's curve strategy outside of type $A$.
{"title":"Some remarks about the faithfulness of the Burau representation of Artin--Tits groups","authors":"Asilata Bapat, Hoel Queffelec","doi":"arxiv-2409.00144","DOIUrl":"https://doi.org/arxiv-2409.00144","url":null,"abstract":"We discuss the extension of the faithfulness question for the Burau\u0000representation of braid groups to the case of Artin--Tits groups. We prove that\u0000the Burau representation is not faithful in affine type $tilde{A_3}$, and not\u0000faithful over several finite rings in type $D_4$, using an algorithmic approach\u0000based on categorical methods that generalize Bigelow's curve strategy outside\u0000of type $A$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142187169","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 paper studies how to compute irreducible characters of the generalized symmetric group $C_kwr{S}_n$ by iterative algorithms. After reproving the Murnaghan-Nakayama rule by vertex algebraic method, we formulate a new iterative formula for characters of the generalized symmetric group. As applications, we find a numerical relation between the character values of $C_kwr S_n$ and modular characters of $S_{kn}$.
{"title":"Irreducible characters of the generalized symmetric group","authors":"Huimin Gao, Naihuan Jing","doi":"arxiv-2408.04921","DOIUrl":"https://doi.org/arxiv-2408.04921","url":null,"abstract":"The paper studies how to compute irreducible characters of the generalized\u0000symmetric group $C_kwr{S}_n$ by iterative algorithms. After reproving the\u0000Murnaghan-Nakayama rule by vertex algebraic method, we formulate a new\u0000iterative formula for characters of the generalized symmetric group. As\u0000applications, we find a numerical relation between the character values of\u0000$C_kwr S_n$ and modular characters of $S_{kn}$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937769","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 prove a novel result on two categories that appear in the local Langlands correspondence, for generalized Steinberg representations. Let $G$ be a semisimple reductive group split over a $p$-adic field $F$. The main result of this paper shows that category of modules over the extension algebra of generalized Steinberg representations of $G(F)$ appears as a full subcategory of equivariant perverse sheaves on the variety of Langlands parameters for these representations.
{"title":"Koszul duality for generalized steinberg representations of $p$-adic groups","authors":"Clifton Cunningham, James Steele","doi":"arxiv-2408.05103","DOIUrl":"https://doi.org/arxiv-2408.05103","url":null,"abstract":"In this paper we prove a novel result on two categories that appear in the\u0000local Langlands correspondence, for generalized Steinberg representations. Let\u0000$G$ be a semisimple reductive group split over a $p$-adic field $F$. The main\u0000result of this paper shows that category of modules over the extension algebra\u0000of generalized Steinberg representations of $G(F)$ appears as a full\u0000subcategory of equivariant perverse sheaves on the variety of Langlands\u0000parameters for these representations.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937770","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, $R=K[x, y]$ the polynomial ring and $mathcal{M}(K)$ the set of all pairs of square matrices of the same size over $K.$ Pairs $P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $mathcal{M}(K)$ are called similar if $A_2=X^{-1}A_1X$ and $B_2=X^{-1}B_1X$ for some invertible matrix $X$ over $K$. Denote by $mathcal{N}(K)$ the subset of $mathcal{M}(K)$, consisting of all pairs of commuting nilpotent matrices. A pair $P$ will be called {it polynomially equivalent} to a pair $overline{P}=(overline{A}, overline{B})$ if $overline{A}=f(A,B), overline{B}=g(A ,B)$ for some polynomials $f, gin K[x,y]$ satisfying the next conditions: $f(0,0)=0, g(0,0)=0$ and $ {rm det} J(f, g)(0, 0)not =0,$ where $J(f, g)$ is the Jacobi matrix of polynomials $f(x, y)$ and $g(x, y).$ Further, pairs of matrices $P(A,B)$ and $widetilde{P}(widetilde{A}, widetilde{B})$ from $mathcal{N}(K)$ will be called {it polynomially similar} if there exists a pair $overline{P}(overline{A}, overline{B})$ from $mathcal{N}(K)$ such that $P$, $overline{P}$ are polynomially equivalent and $overline{P}$, $widetilde{P}$ are similar. The main result of the paper: it is proved that the problem of classifying pairs of matrices up to polynomial similarity is wild, i.e. it contains the classical unsolvable problem of classifying pairs of matrices up to similarity.
{"title":"Polynomial similarity of pairs of matrices","authors":"Vitaliy Bondarenko, Anatoliy Petravchuk, Maryna Styopochkina","doi":"arxiv-2408.04244","DOIUrl":"https://doi.org/arxiv-2408.04244","url":null,"abstract":"Let $K$ be a field, $R=K[x, y]$ the polynomial ring and $mathcal{M}(K)$ the\u0000set of all pairs of square matrices of the same size over $K.$ Pairs\u0000$P_1=(A_1,B_1)$ and $P_2=(A_2,B_2)$ from $mathcal{M}(K)$ are called similar if\u0000$A_2=X^{-1}A_1X$ and $B_2=X^{-1}B_1X$ for some invertible matrix $X$ over $K$.\u0000Denote by $mathcal{N}(K)$ the subset of $mathcal{M}(K)$, consisting of all\u0000pairs of commuting nilpotent matrices. A pair $P$ will be called {it\u0000polynomially equivalent} to a pair $overline{P}=(overline{A}, overline{B})$\u0000if $overline{A}=f(A,B), overline{B}=g(A ,B)$ for some polynomials $f, gin\u0000K[x,y]$ satisfying the next conditions: $f(0,0)=0, g(0,0)=0$ and $ {rm det}\u0000J(f, g)(0, 0)not =0,$ where $J(f, g)$ is the Jacobi matrix of polynomials\u0000$f(x, y)$ and $g(x, y).$ Further, pairs of matrices $P(A,B)$ and\u0000$widetilde{P}(widetilde{A}, widetilde{B})$ from $mathcal{N}(K)$ will be\u0000called {it polynomially similar} if there exists a pair\u0000$overline{P}(overline{A}, overline{B})$ from $mathcal{N}(K)$ such that $P$,\u0000$overline{P}$ are polynomially equivalent and $overline{P}$, $widetilde{P}$\u0000are similar. The main result of the paper: it is proved that the problem of\u0000classifying pairs of matrices up to polynomial similarity is wild, i.e. it\u0000contains the classical unsolvable problem of classifying pairs of matrices up\u0000to similarity.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937684","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}
Over fields of characteristic two, we construct an infinite ascending chain of GL-stable ideals in the coordinate ring of infinite (skew-)symmetric matrices. This construction provides the first known example of a non-noetherian GL-algebra, thereby resolving a long-standing open question in the area. Our results build on the work of Draisma, Krasilnikov, and Krone.
{"title":"Non-noetherian GL-algebras in characteristic two","authors":"Karthik Ganapathy","doi":"arxiv-2408.04630","DOIUrl":"https://doi.org/arxiv-2408.04630","url":null,"abstract":"Over fields of characteristic two, we construct an infinite ascending chain\u0000of GL-stable ideals in the coordinate ring of infinite (skew-)symmetric\u0000matrices. This construction provides the first known example of a\u0000non-noetherian GL-algebra, thereby resolving a long-standing open question in\u0000the area. Our results build on the work of Draisma, Krasilnikov, and Krone.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937687","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 the second adjointness in the setting of the categorical local Langlands correspondence. Moreover, we study the relation between Eisenstein series and cuspidal supports and present a conjectural characterization of irreducible smooth representations with supercuspidal $L$-parameters regarding geometric constant terms. The main technical ingredient is an induction principle for geometric Eisenstein series which allows us to reduce to the situations already treated in the literature.
{"title":"Second adjointness and cuspidal supports at the categorical level","authors":"Yuta Takaya","doi":"arxiv-2408.04582","DOIUrl":"https://doi.org/arxiv-2408.04582","url":null,"abstract":"We prove the second adjointness in the setting of the categorical local\u0000Langlands correspondence. Moreover, we study the relation between Eisenstein\u0000series and cuspidal supports and present a conjectural characterization of\u0000irreducible smooth representations with supercuspidal $L$-parameters regarding\u0000geometric constant terms. The main technical ingredient is an induction\u0000principle for geometric Eisenstein series which allows us to reduce to the\u0000situations already treated in the literature.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937772","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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