Tomoyuki Arakawa, Thomas Creutzig, Kazuya Kawasetsu
We discuss a possible generalization of a result by the third-named author on the rationality of non-admissible minimal W-algebras. We then apply this generalization to finding rational non-admissible principal W-algebras.
我们讨论了第三位作者关于不可容许的最小 W 结构的合理性的一个结果的可能推广。然后，我们将这一概括应用于寻找合理的不可容许主 W 轴。
{"title":"On lisse non-admissible minimal and principal W-algebras","authors":"Tomoyuki Arakawa, Thomas Creutzig, Kazuya Kawasetsu","doi":"arxiv-2408.04584","DOIUrl":"https://doi.org/arxiv-2408.04584","url":null,"abstract":"We discuss a possible generalization of a result by the third-named author on\u0000the rationality of non-admissible minimal W-algebras. We then apply this\u0000generalization to finding rational non-admissible principal W-algebras.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937686","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 compute (exact and asymptotic) formulas for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero, we obtain in addition a general exact formula for the growth rate and give a complete solution to the growth problems in terms of the character table. We also provide code used to compute our formulas.
{"title":"Growth Problems for Representations of Finite Groups","authors":"David He","doi":"arxiv-2408.04196","DOIUrl":"https://doi.org/arxiv-2408.04196","url":null,"abstract":"We compute (exact and asymptotic) formulas for the growth rate of the number\u0000of indecomposable summands in the tensor powers of representations of finite\u0000groups, over a field of arbitrary characteristic. In characteristic zero, we\u0000obtain in addition a general exact formula for the growth rate and give a\u0000complete solution to the growth problems in terms of the character table. We\u0000also provide code used to compute our formulas.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"22 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937774","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 $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over a field $F$ of characteristic $p>0$. This paper studies ring theoretic properties of the representation ring $T^Delta(B,B)$ of perfect $p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra $kotimes_mathbb{Z} T^Delta(B,B)$, for a field $k$. We show that if the Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not primitive in $T^Delta(B,B)$. If $B$ has cyclic defect groups we determine a primitive decomposition of $[B]$ in $T^Delta(B,B)$. Moreover, if $k$ is a field of characteristic different from $p$ and $B$ has cyclic defect groups of order $p^n$ we describe $kotimes_mathbb{Z} T^Delta(B,B)$ explicitly as a direct product of a matrix algebra and $n$ group algebras.
{"title":"The ring of perfect $p$-permutation bimodules for blocks with cyclic defect groups","authors":"Robert Boltje, Nariel Monteiro","doi":"arxiv-2408.04134","DOIUrl":"https://doi.org/arxiv-2408.04134","url":null,"abstract":"Let $B$ be a block algebra of a group algebra $FG$ of a finite group $G$ over\u0000a field $F$ of characteristic $p>0$. This paper studies ring theoretic\u0000properties of the representation ring $T^Delta(B,B)$ of perfect\u0000$p$-permutation $(B,B)$-bimodules and properties of the $k$-algebra\u0000$kotimes_mathbb{Z} T^Delta(B,B)$, for a field $k$. We show that if the\u0000Cartan matrix of $B$ has $1$ as an elementary divisor then $[B]$ is not\u0000primitive in $T^Delta(B,B)$. If $B$ has cyclic defect groups we determine a\u0000primitive decomposition of $[B]$ in $T^Delta(B,B)$. Moreover, if $k$ is a\u0000field of characteristic different from $p$ and $B$ has cyclic defect groups of\u0000order $p^n$ we describe $kotimes_mathbb{Z} T^Delta(B,B)$ explicitly as a\u0000direct product of a matrix algebra and $n$ group algebras.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937786","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 $D$ be the Auslander algebra of $mathbb{C}[t]/(t^n)$, which is quasi-hereditary, and $mathcal{F}_Delta$ the subcategory of good $D$-modules. For any $mathsf{J}subseteq[1, n-1]$, we construct a subcategory $mathcal{F}_Delta(mathsf{J})$ of $mathcal{F}_Delta$ with an exact structure $mathcal{E}$. We show that under $mathcal{E}$, $mathcal{F}_Delta(mathsf{J})$ is Frobenius stably 2-Calabi-Yau and admits a cluster structure consisting of cluster tilting objects. This then leads to an additive categorification of the cluster structure on the coordinate ring $mathbb{C}[operatorname{Fl}(mathsf{J})]$ of the (partial) flag variety $operatorname{Fl}(mathsf{J})$. We further apply $mathcal{F}_Delta(mathsf{J})$ to study flag combinatorics and the quantum cluster structure on the flag variety $operatorname{Fl}(mathsf{J})$. We show that weak and strong separation can be detected by the extension groups $operatorname{ext}^1(-, -)$ under $mathcal{E}$ and the extension groups $operatorname{Ext}^1(-,-)$, respectively. We give a interpretation of the quasi-commutation rules of quantum minors and identify when the product of two quantum minors is invariant under the bar involution. The combinatorial operations of flips and geometric exchanges correspond to certain mutations of cluster tilting objects in $mathcal{F}_Delta(mathsf{J})$. We then deduce that any (quantum) minor is reachable, when $mathsf{J}$ is an interval. Building on our result for the interval case, Geiss-Leclerc-Schr"{o}er's result on the quantum coordinate ring for the open cell of $operatorname{Fl}(mathsf{J})$ and Kang-Kashiwara-Kim-Oh's enhancement of that to the integral form, we prove that $mathbb{C}_q[operatorname{Fl}(mathsf{J})]$ is a quantum cluster algebra over $mathbb{C}[q,q^{-1}]$.
{"title":"Auslander algebras, flag combinatorics and quantum flag varieties","authors":"Bernt Tore Jensen, Xiuping Su","doi":"arxiv-2408.04753","DOIUrl":"https://doi.org/arxiv-2408.04753","url":null,"abstract":"Let $D$ be the Auslander algebra of $mathbb{C}[t]/(t^n)$, which is\u0000quasi-hereditary, and $mathcal{F}_Delta$ the subcategory of good $D$-modules.\u0000For any $mathsf{J}subseteq[1, n-1]$, we construct a subcategory\u0000$mathcal{F}_Delta(mathsf{J})$ of $mathcal{F}_Delta$ with an exact\u0000structure $mathcal{E}$. We show that under $mathcal{E}$,\u0000$mathcal{F}_Delta(mathsf{J})$ is Frobenius stably 2-Calabi-Yau and admits a\u0000cluster structure consisting of cluster tilting objects. This then leads to an\u0000additive categorification of the cluster structure on the coordinate ring\u0000$mathbb{C}[operatorname{Fl}(mathsf{J})]$ of the (partial) flag variety\u0000$operatorname{Fl}(mathsf{J})$. We further apply $mathcal{F}_Delta(mathsf{J})$ to study flag combinatorics\u0000and the quantum cluster structure on the flag variety\u0000$operatorname{Fl}(mathsf{J})$. We show that weak and strong separation can be\u0000detected by the extension groups $operatorname{ext}^1(-, -)$ under\u0000$mathcal{E}$ and the extension groups $operatorname{Ext}^1(-,-)$,\u0000respectively. We give a interpretation of the quasi-commutation rules of\u0000quantum minors and identify when the product of two quantum minors is invariant\u0000under the bar involution. The combinatorial operations of flips and geometric\u0000exchanges correspond to certain mutations of cluster tilting objects in\u0000$mathcal{F}_Delta(mathsf{J})$. We then deduce that any (quantum) minor is\u0000reachable, when $mathsf{J}$ is an interval. Building on our result for the interval case, Geiss-Leclerc-Schr\"{o}er's\u0000result on the quantum coordinate ring for the open cell of\u0000$operatorname{Fl}(mathsf{J})$ and Kang-Kashiwara-Kim-Oh's enhancement of that\u0000to the integral form, we prove that\u0000$mathbb{C}_q[operatorname{Fl}(mathsf{J})]$ is a quantum cluster algebra over\u0000$mathbb{C}[q,q^{-1}]$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"42 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937771","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}
Endosplit $p$-permutation resolutions play an instrumental role in verifying Brou'{e}'s abelian defect group conjecture in numerous cases. In this article, we give a complete classification of endosplit $p$-permutation resolutions and reduce the question of Galois descent of an endosplit $p$-permutation resolution to the Galois descent of the module it resolves. This is shown using techniques from the study of endotrivial complexes, the invertible objects of the bounded homotopy category of $p$-permutation modules. As an application, we show that a refinement of Brou'{e}'s conjecture proposed by Kessar and Linckelmann holds for all blocks of $p$-solvable groups.
{"title":"On endosplit $p$-permutation resolutions and Broué's conjecture for $p$-solvable groups","authors":"Sam K. Miller","doi":"arxiv-2408.04094","DOIUrl":"https://doi.org/arxiv-2408.04094","url":null,"abstract":"Endosplit $p$-permutation resolutions play an instrumental role in verifying\u0000Brou'{e}'s abelian defect group conjecture in numerous cases. In this article,\u0000we give a complete classification of endosplit $p$-permutation resolutions and\u0000reduce the question of Galois descent of an endosplit $p$-permutation\u0000resolution to the Galois descent of the module it resolves. This is shown using\u0000techniques from the study of endotrivial complexes, the invertible objects of\u0000the bounded homotopy category of $p$-permutation modules. As an application, we\u0000show that a refinement of Brou'{e}'s conjecture proposed by Kessar and\u0000Linckelmann holds for all blocks of $p$-solvable groups.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937685","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 the previous paper, we proposed a practical method of constructing explicitly representation groups $R(G)$ for finite groups $G$, and apply it to certain typical finite groups $G$ with Schur multiplier $M(G)$ containing prime number 3. In this paper, we construct a complete list of irreducible projective (or spin) representations of $G$ and compute their characters (called spin characters). It is a continuation of our study of spin representations in the cases where $M(G)$ contains prime number 2 to the cases where other prime $p$ appears, firstly $p=3$. We classify irreducible spin representations and calculate spin characters according to their spin types.
{"title":"Projective (or spin) representations of finite groups. II","authors":"Tatsuya Tsurii, Satoe Yamanaka, Itsumi Mikami, Takeshi Hirai","doi":"arxiv-2408.03486","DOIUrl":"https://doi.org/arxiv-2408.03486","url":null,"abstract":"In the previous paper, we proposed a practical method of constructing\u0000explicitly representation groups $R(G)$ for finite groups $G$, and apply it to\u0000certain typical finite groups $G$ with Schur multiplier $M(G)$ containing prime\u0000number 3. In this paper, we construct a complete list of irreducible projective\u0000(or spin) representations of $G$ and compute their characters (called spin\u0000characters). It is a continuation of our study of spin representations in the\u0000cases where $M(G)$ contains prime number 2 to the cases where other prime $p$\u0000appears, firstly $p=3$. We classify irreducible spin representations and\u0000calculate spin characters according to their spin types.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937689","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}
L"uroth's theorem describes the dominant maps from rational curves over a field. In this note we study the dominant maps from cartesian powers $X^{Psi}$ of absolutely irreducible varieties $X$ over a field $k$ for infinite sets $Psi$ that are equivariant with respect to all permutations of the factors $X$. At least some of such maps arise as compositions $h:X^{Psi}xrightarrow{f^{Psi}}Y^{Psi}to Hbackslash Y^{Psi}$, where $Xxrightarrow{f}Y$ is a dominant $k$-map and $H$ is an automorphism group $H$ of $Y|k$, acting diagonally on $Y^{Psi}$. In characteristic 0, we show that this construction, when properly modified, gives all dominant equivariant maps from $X^{Psi}$, if $dim X=1$. For arbitrary $X$, the results are only partial. In a subsequent paper, the `quasicoherent' equivariant sheaves on the targets of such $h$'s will be studied. Some preliminary results have already appeared in arXiv:math/2205.15144. A somewhat similar problem is to check, whether the irreducible invariant subvarieties of $X^{Psi}$ arise as pullbacks under $f^{Psi}$ (for appropriate $f$'s) of subvarieties of $Y$ diagonally embedded into $Y^{Psi}$. This would be a complement to the famous theorem of D.E.Cohen on the noetherian property of the symmetric ideals. We show that this is the case if $dim X=1$.
{"title":"Lüroth's theorem for fields of rational functions in infinitely many permuted variables","authors":"M. Rovinsky","doi":"arxiv-2408.04028","DOIUrl":"https://doi.org/arxiv-2408.04028","url":null,"abstract":"L\"uroth's theorem describes the dominant maps from rational curves over a\u0000field. In this note we study the dominant maps from cartesian powers $X^{Psi}$ of\u0000absolutely irreducible varieties $X$ over a field $k$ for infinite sets $Psi$\u0000that are equivariant with respect to all permutations of the factors $X$. At\u0000least some of such maps arise as compositions\u0000$h:X^{Psi}xrightarrow{f^{Psi}}Y^{Psi}to Hbackslash Y^{Psi}$, where\u0000$Xxrightarrow{f}Y$ is a dominant $k$-map and $H$ is an automorphism group $H$\u0000of $Y|k$, acting diagonally on $Y^{Psi}$. In characteristic 0, we show that this construction, when properly modified,\u0000gives all dominant equivariant maps from $X^{Psi}$, if $dim X=1$. For\u0000arbitrary $X$, the results are only partial. In a subsequent paper, the `quasicoherent' equivariant sheaves on the targets\u0000of such $h$'s will be studied. Some preliminary results have already appeared\u0000in arXiv:math/2205.15144. A somewhat similar problem is to check, whether the irreducible invariant\u0000subvarieties of $X^{Psi}$ arise as pullbacks under $f^{Psi}$ (for appropriate\u0000$f$'s) of subvarieties of $Y$ diagonally embedded into $Y^{Psi}$. This would\u0000be a complement to the famous theorem of D.E.Cohen on the noetherian property\u0000of the symmetric ideals. We show that this is the case if $dim X=1$.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"307 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937694","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 take a novel lattice-theoretic approach to the $tau$-cluster morphism category $mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the category via the lattice of torsion classes $mathrm{tors } A$. Using the lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I: mathfrak{T}(A) to mathfrak{T}(A/I)$ and if $mathrm{tors } A$ is finite an inclusion $mathcal{I}: mathfrak{T}(A/I) to mathfrak{T}(A)$. We characterise when these functors are full, faithful and adjoint. As a consequence we find a new family of algebras for which $mathfrak{T}(A)$ admits a faithful group functor.
{"title":"$τ$-cluster morphism categories of factor algebras","authors":"Maximilian Kaipel","doi":"arxiv-2408.03818","DOIUrl":"https://doi.org/arxiv-2408.03818","url":null,"abstract":"We take a novel lattice-theoretic approach to the $tau$-cluster morphism\u0000category $mathfrak{T}(A)$ of a finite-dimensional algebra $A$ and define the\u0000category via the lattice of torsion classes $mathrm{tors } A$. Using the\u0000lattice congruence induced by an ideal $I$ of $A$ we establish a functor $F_I:\u0000mathfrak{T}(A) to mathfrak{T}(A/I)$ and if $mathrm{tors } A$ is finite an\u0000inclusion $mathcal{I}: mathfrak{T}(A/I) to mathfrak{T}(A)$. We characterise\u0000when these functors are full, faithful and adjoint. As a consequence we find a\u0000new family of algebras for which $mathfrak{T}(A)$ admits a faithful group\u0000functor.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937773","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}
After Gross, Hacking, Keel, Kontsevich [GHKK18] introduced the theta basis which is shown to be indexed by its highest term exponent in cluster variables of any given seed, we are interested in all the non-vanishing exponents in these cluster variables. We prove that the coefficients of the exponents of any cluster variable of type $A_n$ are log-concave. We show that the cluster monomials of $A_2$ type are log-concave. As for larger generality, we conjecture that the log-concavity of cluster monomials is also true.
{"title":"Log-concavity of cluster algebras of type $A_n$","authors":"Zhichao Chen, Guanhua Huang, Zhe Sun","doi":"arxiv-2408.03792","DOIUrl":"https://doi.org/arxiv-2408.03792","url":null,"abstract":"After Gross, Hacking, Keel, Kontsevich [GHKK18] introduced the theta basis\u0000which is shown to be indexed by its highest term exponent in cluster variables\u0000of any given seed, we are interested in all the non-vanishing exponents in\u0000these cluster variables. We prove that the coefficients of the exponents of any\u0000cluster variable of type $A_n$ are log-concave. We show that the cluster\u0000monomials of $A_2$ type are log-concave. As for larger generality, we\u0000conjecture that the log-concavity of cluster monomials is also true.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937624","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 define and study quasi-biserial algebras, special quasi-biserial algebras and labeled Brauer graph algebras. These algebras naturally generalize biserial and special biserial algebras to algebras of wild representation type, in a different direction of multiserial and special multiserial algebras. We show that all special quasi-biserial algebras are quasi-biserial, all special quasi-biserial algebras are quotients of symmetric special quasi-biserial algebras and all symmetric special quasi-biserial algebras are labeled Brauer graph algebras. Moreover, we show that the labeled Brauer graph algebras are derived equivalent under Kauer moves.
{"title":"Quasi-biserial algebras, special quasi-biserial algebras and labeled Brauer graph algebras","authors":"Yuming Liu, Bohan Xing","doi":"arxiv-2408.03778","DOIUrl":"https://doi.org/arxiv-2408.03778","url":null,"abstract":"In this paper, we define and study quasi-biserial algebras, special\u0000quasi-biserial algebras and labeled Brauer graph algebras. These algebras\u0000naturally generalize biserial and special biserial algebras to algebras of wild\u0000representation type, in a different direction of multiserial and special\u0000multiserial algebras. We show that all special quasi-biserial algebras are\u0000quasi-biserial, all special quasi-biserial algebras are quotients of symmetric\u0000special quasi-biserial algebras and all symmetric special quasi-biserial\u0000algebras are labeled Brauer graph algebras. Moreover, we show that the labeled\u0000Brauer graph algebras are derived equivalent under Kauer moves.","PeriodicalId":501038,"journal":{"name":"arXiv - MATH - Representation Theory","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937688","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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