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Centralizers of irreducible subgroups in the projective special linear group Guerin, Clément in Journal of Group Theory (2018) In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PGL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n ... [more ▼] In this paper, we classify conjugacy classes of centralizers of irreducible subgroups in $PGL(n,\mathbb{C})$ using alternate modules a.k.a. finite abelian groups with an alternate bilinear form. When $n$ is squarefree, we prove that these conjugacy classes are classified by their isomorphism classes. More generally, we define a finite graph related to this classification whose combinatorial properties are expected to help us describe the stratification of the singular (orbifold) locus in some character varieties. [less ▲] Detailed reference viewed: 87 (5 UL)Bad irreducible subgroups and singular locus for character varieties in PSL(p,C) Guerin, Clément in Geometriae Dedicata (2017) We give the centralizers of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where p is a prime number. This leads to a description of the singular locus (te ... [more ▼] We give the centralizers of irreducible representations from a finitely generated group $\Gamma$ to $PSL(p,\mathbb{C})$ where p is a prime number. This leads to a description of the singular locus (te (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety $\chi^i(\Gamma,PSL(p,\mathbb{C}))$. When $\Gamma$ is a free group of rank $l\geq 2$ or the fundamental group of a closed Riemann surface of genus $g\geq 2$, we give a complete description of this locus and prove that this locus is exactly the set of algebraic singularities of the irreducible part of the character variety. [less ▲] Detailed reference viewed: 84 (5 UL)Singularités orbifoldes de la variété des caractères Guerin, Clément Doctoral thesis (2016) In this thesis, we want to understand some singularities in the character variety. In a first chapter, we justify that the characters of irreducible representations from a Fuchsian group to a complex semi ... [more ▼] In this thesis, we want to understand some singularities in the character variety. In a first chapter, we justify that the characters of irreducible representations from a Fuchsian group to a complex semi-simple Lie group is an orbifold. The orbifold locus is, then, the characters of bad representations. In the second chapter, we focus on the case where the Lie group is PSL(p,C) with p a prime number. In particular we give an explicit description of this locus. In the third and fourth chapter, we describe the isotropy groups (i.e. the centralizers of bad subgroups) arising in the cases when the Lie group is a quotient SL(n,C) (third chapter) and when the Lie group is a quotient of Spin(n,C) in the fourth chapter. [less ▲] Detailed reference viewed: 132 (7 UL)Alternate modules are subsymplectic Guerin, Clément E-print/Working paper (2016) In this paper, an alternate module $(A,\phi)$ is a finite abelian group $A$ with a $\mathbb{Z}$-bilinear application $\phi:A\times A\rightarrow \mathbb{Q}/\mathbb{Z}$ which is alternate (i.e. zero on the ... [more ▼] In this paper, an alternate module $(A,\phi)$ is a finite abelian group $A$ with a $\mathbb{Z}$-bilinear application $\phi:A\times A\rightarrow \mathbb{Q}/\mathbb{Z}$ which is alternate (i.e. zero on the diagonal). We shall prove that any alternate module is subsymplectic, i.e. if $(A,\phi)$ has a Lagrangian of cardinal $n$ then there exists an abelian group $B$ of order $n$ such that $(A,\phi)$ is a submodule of the standard symplectic module $B\times B^*$. [less ▲] Detailed reference viewed: 37 (1 UL) |
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