scholarly journals Rational smoothness of varieties of representations for quivers of Dynkin type

2004 ◽  
Vol 54 (2) ◽  
pp. 295-315 ◽  
Author(s):  
Philippe Caldero ◽  
Ralf Schiffler
Keyword(s):  
Dynkin Type ◽  
Rational Smoothness

scholarly journals Quotients of cluster categories

2010 ◽  
Vol 140 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Peter Jørgensen
Keyword(s):  
The Other ◽  
Dynkin Type ◽  
Cluster Categories

Higher cluster categories were recently introduced as a generalization of cluster categories. This paper shows that in Dynkin types A and D, half of all higher cluster categories can be obtained as quotients of cluster categories. The other half are quotients of 2-cluster categories, the ‘lowest’ type of higher cluster categories. Hence, in Dynkin types A and D, all higher cluster phenomena are implicit in cluster categories and 2-cluster categories. In contrast, the same is not true in Dynkin type E.

A Graph Theoretical Framework for the Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸n

2022 ◽  
Vol 184 (1) ◽  
pp. 49-82
Author(s):  
Jesús Arturo Jiménez González
Keyword(s):  
Theoretical Framework ◽  
Line Graphs ◽  
Positive Unit ◽  
Dynkin Type ◽  
Principal Unit

In the context of signed line graphs, this article introduces a modified inflation technique to study strong Gram congruence of non-negative (integral quadratic) unit forms, and uses it to show that weak and strong Gram congruence coincide among positive unit forms of Dynkin type 𝔸n. The concept of inverse of a quiver is also introduced, and is used to obtain and analyze the Coxeter matrix of non-negative unit forms of Dynkin type 𝔸n. With these tools, connected principal unit forms of Dynkin type 𝔸n are also classified up to strong congruence.

Deformed mesh algebras of Dynkin type Cn

2012 ◽  
Vol 126 (2) ◽  
pp. 217-230 ◽  
Author(s):  
Jerzy Białkowski ◽  
Karin Erdmann ◽  
Andrzej Skowroński
Keyword(s):  
Dynkin Type

THE MINIMAL MODULAR FORM ON QUATERNIONIC

2020 ◽  
pp. 1-34
Author(s):  
Aaron Pollack
Keyword(s):  
Modular Form ◽  
Eisenstein Series ◽  
Explicit Form ◽  
Modular Forms ◽  
Fourier Expansion ◽  
Reductive Group ◽  
Automorphic Function ◽  
Unipotent Radical ◽  
Exceptional Groups ◽  
Dynkin Type

Suppose that $G$ is a simple reductive group over $\mathbf{Q}$ , with an exceptional Dynkin type and with $G(\mathbf{R})$ quaternionic (in the sense of Gross–Wallach). In a previous paper, we gave an explicit form of the Fourier expansion of modular forms on $G$ along the unipotent radical of the Heisenberg parabolic. In this paper, we give the Fourier expansion of the minimal modular form $\unicode[STIX]{x1D703}_{Gan}$ on quaternionic $E_{8}$ and some applications. The $Sym^{8}(V_{2})$ -valued automorphic function $\unicode[STIX]{x1D703}_{Gan}$ is a weight 4, level one modular form on $E_{8}$ , which has been studied by Gan. The applications we give are the construction of special modular forms on quaternionic $E_{7},E_{6}$ and $G_{2}$ . We also discuss a family of degenerate Heisenberg Eisenstein series on the groups $G$ , which may be thought of as an analogue to the quaternionic exceptional groups of the holomorphic Siegel Eisenstein series on the groups $\operatorname{GSp}_{2n}$ .

Gröbner-Shirshov Basis of Quantum Groups

2015 ◽  
Vol 22 (03) ◽  
pp. 495-516 ◽  
Author(s):  
Gulshadam Yunus ◽  
Zhenzhen Gao ◽  
Abdukadir Obul
Keyword(s):  
Quantum Groups ◽  
Hall Algebra ◽  
Dynkin Type ◽  
Algebra Method

In this paper, by using the Ringel-Hall algebra method, we prove that the set of the skew-commutator relations of quantum root vectors forms a minimal Gröbner-Shirshov basis for the quantum groups of Dynkin type. As an application, we give an explicit basis for the types E7 and Dn.

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