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Mckay Quivers

Author: FuErJuan
Tutor: XiaoJie
School: Tsinghua University
Course: Mathematics
Keywords: Dynkin diagram McKay quiver separated McKay quiver Gabriel quiver skew group algebra
CLC: O157.5
Type: Master's thesis
Year: 2013
Downloads: 5
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Abstract


In this thesis, we study the structure of McKay quivers and the relationship between McKay quivers and Dynkin diagrams coming from the classical McKay correspondence. Based on this, we deduce some properties of some skew group algebras, matrix algebras and their finitely generated modular categories.We consider a finite group G and its F-linear representation (ρ, V), where F is a algebraic closed field of characteristic not dividing the order of G. Denote the McKay quiver of (ρ, V) by Qρ, the McKay diagram by||Qρ||, the McKay matrix by Cρ, and the separated McKay quiver by Qρs.On the one hand, we describe the McKay diagram of a representation satisfying some conditions. First, we find that if p is a faithful representation and equals to its con-tragredient representation, then Cρ is an1-corank indecomposable positive semi-definite real symmetric matrix. By abstracting the properties of these McKay matrixes as the properties of incidence matrixes, we define the generalized McKay diagrams. Second, by the classification of generalized McKay diagrams, we obtain:the McKay diagram of a2-dimensional faithful representation with real character is the Dynkin diagram of type ADE or the generalized McKay diagram of type JK. In particular, we verify that the McKay diagram of the2-dimensional indecomposable representation of D3is the gener-alized McKay diagram of type K3. As an application of this result, we prove the classical McKay correspondence. Our method avoids analysing the eigenvalues of affine Coxeter elements by R. Steinberg in [1].On the other hand, we describe the separated McKay quiver for any representation. First, for a finite dimensional F-algebra A with Jacobson radical square zero, we prove QAs≌QΛA (QΛ denotes the Gabriel quiver of A and AΛ denotes the matrix algebra of A) and AAG≌AAG, and if A is indecomposable hereditary and G can F-linear act on it, we prove the type of connected components of|QAG|can be determined by the type of connected components of|QA|.Second, with the help of formal power series rings, we prove Qρs≌QARG (ΛR is an algebra defined by the formal power series ring). Third, by describing QΛR and proving AR is the algebra satisfying the above properties, we describe|QρS|\. Particularly, based on the result of Auslander and Reiten[2], we add the result of1-dimension:any connected component of|QρS|is the Dynkin diagram of type ADE. Furthermore, we prove|QρS|is the union of n Dynkin diagrams of type A2(n denotes the number of conjugate classes in G). Based on this result, for matrix algebras (FG is the normal group algebra, F*G denotes the skew group algebra determined by the action of G on F), we prove their finitely generated modular categories are equivalent, at the same time, they are F-algebraic isomorphism if and only if G is Abelian. At last, as an application of the description of|QρS|, we describe||Qρ||without the assumption F=C comparing with the above description of McKay diagrams.

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