Let $mathbb S$ be a Clifford module for the complexified Clifford algebra $Cell(mathbb R^n)$, $mathbb S'$ its dual, $rho$ and $rho'$ be the corresponding representations of the spin group $Spin(mathbb R^n)$. The group $G=Spin(mathbb R^{1,n+1})$ is the (twofold covering) of the conformal group of $mathbb R^n$. For $lambda, muin mathbb C$, let $pi_{rho, lambda}$ (resp. $pi_{rho',mu}$) be the spinorial representation of $G$ on $ mathbb S$-valued $lambda$-densities (resp. $mathbb S'$-valued $mu$-densities) on $mathbb R^n$. For $0leq kleq n$ and $min mathbb N$, we construct a symmetry breaking differential operator $B_{k;lambda,mu}^{(m)}$ from $C^infty(mathbb R^n times mathbb R^n, mathbb Sotimes mathbb S')$ into $C^infty(mathbb R^n, Lambda^*_k(mathbb R^n))$ which intertwines the representations $pi_{rho, lambda}otimes pi_{rho',mu} $ and $pi_{tau^*_k,lambda+mu+2m}$, where $tau^*_k$ is the representation of $Spin(mathbb R^n)$ on $Lambda^*_k(mathbb R^n)$.
{"title":"Symmetry breaking differential operators for tensor products of spinorial representations.","authors":"J. Clerc, K. Koufany","doi":"10.3842/SIGMA.2021.049","DOIUrl":"https://doi.org/10.3842/SIGMA.2021.049","url":null,"abstract":"Let $mathbb S$ be a Clifford module for the complexified Clifford algebra $Cell(mathbb R^n)$, $mathbb S'$ its dual, $rho$ and $rho'$ be the corresponding representations of the spin group $Spin(mathbb R^n)$. The group $G=Spin(mathbb R^{1,n+1})$ is the (twofold covering) of the conformal group of $mathbb R^n$. For $lambda, muin mathbb C$, let $pi_{rho, lambda}$ (resp. $pi_{rho',mu}$) be the spinorial representation of $G$ on $ mathbb S$-valued $lambda$-densities (resp. $mathbb S'$-valued $mu$-densities) on $mathbb R^n$. For $0leq kleq n$ and $min mathbb N$, we construct a symmetry breaking differential operator $B_{k;lambda,mu}^{(m)}$ from $C^infty(mathbb R^n times mathbb R^n, mathbb Sotimes mathbb S')$ into $C^infty(mathbb R^n, Lambda^*_k(mathbb R^n))$ which intertwines the representations $pi_{rho, lambda}otimes pi_{rho',mu} $ and $pi_{tau^*_k,lambda+mu+2m}$, where $tau^*_k$ is the representation of $Spin(mathbb R^n)$ on $Lambda^*_k(mathbb R^n)$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121049717","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 an explicit description of the irreducible components of two-row Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of two-row Springer fibers. Given a cup diagram, we explicitly write down all flags contained in the component associated to the cup diagram. This generalizes results by Stroppel--Webster and Fung to all classical types.
{"title":"Irreducible components of two-row Springer fibers for all classical types","authors":"Mee Seong Im, C. Lai, A. Wilbert","doi":"10.1090/proc/15965","DOIUrl":"https://doi.org/10.1090/proc/15965","url":null,"abstract":"We give an explicit description of the irreducible components of two-row Springer fibers for all classical types using cup diagrams. Cup diagrams can be used to label the irreducible components of two-row Springer fibers. Given a cup diagram, we explicitly write down all flags contained in the component associated to the cup diagram. This generalizes results by Stroppel--Webster and Fung to all classical types.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131241158","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 : 2020-11-22DOI: 10.1007/S00209-021-02754-2
A. Boysal
{"title":"PRV for the fusion product, the case $$uplambda gg mu $$","authors":"A. Boysal","doi":"10.1007/S00209-021-02754-2","DOIUrl":"https://doi.org/10.1007/S00209-021-02754-2","url":null,"abstract":"","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128365195","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 develop the general Theory of Cayley Hamilton algebras using norms and compare with the approach, valid only in characteristic 0, using traces and presented in a previous paper $T$-ideals of Cayley Hamilton algebras, 2020, arXiv:2008.02222
{"title":"Norms and Cayley–Hamilton algebras","authors":"C. Procesi","doi":"10.4171/rlm/925","DOIUrl":"https://doi.org/10.4171/rlm/925","url":null,"abstract":"We develop the general Theory of Cayley Hamilton algebras using norms and compare with the approach, valid only in characteristic 0, using traces and presented in a previous paper $T$-ideals of Cayley Hamilton algebras, 2020, arXiv:2008.02222","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116351066","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 finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products and affine Hopf algebras of low Gelfand-Kirillov dimension.
{"title":"Locally finite representations over Noetherian Hopf algebras","authors":"Can Hat.ipouglu, C. Lomp","doi":"10.1090/proc/15747","DOIUrl":"https://doi.org/10.1090/proc/15747","url":null,"abstract":"We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products and affine Hopf algebras of low Gelfand-Kirillov dimension.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123286004","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 ${mathscr{C}}$ be an $n$-cluster tilting subcategory of an exact category $({mathscr{A}}, {mathscr{E}})$, where $n geq 1$ is an integer. It is proved by Jasso that if $n> 1$, then ${mathscr{C}}$ although is no longer exact, but has a nice structure known as $n$-exact structure. In this new structure conflations are called admissible $n$-exact sequences and are ${mathscr{E}}$-acyclic complexes with $n+2$ terms in ${mathscr{C}}$. Since their introduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely to their links and applications to many research areas, many of them unexpected. On the other hand, ideal approximation theory, that is a gentle generalization of the classical approximation theory and deals with morphisms and ideals instead of objects and subcategories, is an active area that has been the subject of several researches. Our aim in this paper is to introduce the so-called `ideal approximation theory' into `higher homological algebra'. To this end, we introduce some important notions in approximation theory into the theory of $n$-exact categories and prove some results. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be introduced and studied. Our results motivate the definitions and show that $n$-exact categories are the appropriate context for the study of `higher ideal approximation theory'.
{"title":"Higher ideal approximation theory","authors":"J. Asadollahi, S. Sadeghi","doi":"10.1090/tran/8562","DOIUrl":"https://doi.org/10.1090/tran/8562","url":null,"abstract":"Let ${mathscr{C}}$ be an $n$-cluster tilting subcategory of an exact category $({mathscr{A}}, {mathscr{E}})$, where $n geq 1$ is an integer. It is proved by Jasso that if $n> 1$, then ${mathscr{C}}$ although is no longer exact, but has a nice structure known as $n$-exact structure. In this new structure conflations are called admissible $n$-exact sequences and are ${mathscr{E}}$-acyclic complexes with $n+2$ terms in ${mathscr{C}}$. Since their introduction by Iyama, cluster tilting subcategories has gained a lot of traction, due largely to their links and applications to many research areas, many of them unexpected. On the other hand, ideal approximation theory, that is a gentle generalization of the classical approximation theory and deals with morphisms and ideals instead of objects and subcategories, is an active area that has been the subject of several researches. Our aim in this paper is to introduce the so-called `ideal approximation theory' into `higher homological algebra'. To this end, we introduce some important notions in approximation theory into the theory of $n$-exact categories and prove some results. In particular, the higher version of the notions such as ideal cotorsion pairs, phantom ideals, Salce's Lemma and Wakamatsu's Lemma for ideals will be introduced and studied. Our results motivate the definitions and show that $n$-exact categories are the appropriate context for the study of `higher ideal approximation theory'.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121754338","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 : 2020-10-15DOI: 10.21538/0134-4889-2021-27-1-220-239
C. G. Ardito
We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.
{"title":"Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64","authors":"C. G. Ardito","doi":"10.21538/0134-4889-2021-27-1-220-239","DOIUrl":"https://doi.org/10.21538/0134-4889-2021-27-1-220-239","url":null,"abstract":"We classify the Morita equivalence classes of principal blocks with elementary abelian defect groups of order 64 with respect to a complete discrete valuation ring with an algebraically closed residue field of characteristic two.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"111 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116577092","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 : 2020-10-14DOI: 10.1142/s0219498822500414
Xiaojin Zhang
Let $Lambda$ be a radical square zero Nakayama algebra with $n$ simple modules and let $Gamma$ be the Auslander algebra of $Lambda$. Then every indecomposable direct summand of a tilting $Gamma$-module is either simple or projective. Moreover, if $Lambda$ is self-injective, then the number of tilting $Gamma$-modules is $2^n$; otherwise, the number of tilting $Gamma$-modules is $2^{n-1}$.
{"title":"Classifying tilting modules over the Auslander algebras of radical square zero Nakayama algebras","authors":"Xiaojin Zhang","doi":"10.1142/s0219498822500414","DOIUrl":"https://doi.org/10.1142/s0219498822500414","url":null,"abstract":"Let $Lambda$ be a radical square zero Nakayama algebra with $n$ simple modules and let $Gamma$ be the Auslander algebra of $Lambda$. Then every indecomposable direct summand of a tilting $Gamma$-module is either simple or projective. Moreover, if $Lambda$ is self-injective, then the number of tilting $Gamma$-modules is $2^n$; otherwise, the number of tilting $Gamma$-modules is $2^{n-1}$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115335793","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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