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See detailBad representations and homotopy groups of Character Varieties
Guerin, Clément UL; Lawton, Sean; Ramras, Daniel

E-print/Working paper (2019)

Let G be a connected, reductive, complex affine algebraic group, and let Xr denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in Xr ... [more ▼]

Let G be a connected, reductive, complex affine algebraic group, and let Xr denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in Xr resolving conjectures of Florentino-Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel-de Siebenthal groups. We then use this codimension to calculate some higher homotopy groups of the smooth locus of Xr, proving conjectures of Florentino-Lawton-Ramras. Lastly, using the earlier analysis of Borel-de Siebenthal groups, we prove a conjecture of Sikora about CI Lie groups. [less ▲]

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See detailDual pairs in the Pin-group and duality for the corresponding spinorial representation
Guerin, Clément UL; Liu, Gang; Merino, Allan

E-print/Working paper (2019)

In this paper, we give a complete picture of Howe correspondence for the setting (O(E,b),Pin(E,b),Π), where O(E,b) is an orthogonal group (real or complex), Pin(E,b) is the two-fold Pin-covering of O(E,b ... [more ▼]

In this paper, we give a complete picture of Howe correspondence for the setting (O(E,b),Pin(E,b),Π), where O(E,b) is an orthogonal group (real or complex), Pin(E,b) is the two-fold Pin-covering of O(E,b), and Π is the spinorial representation of Pin(E,b). More precisely, for a dual pair (G,G′) in O(E,b), we determine explicitly the nature of its preimages (G̃,G′~) in Pin(E,b), and prove that apart from some exceptions, (G̃,G′~) is always a dual pair in Pin(E,b); then we establish the Howe correspondence for Π with respect to (G̃,G′~). [less ▲]

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See detailCentralizers of irreducible subgroups in the projective special linear group
Guerin, Clément UL

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 ▲]

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See detailBad irreducible subgroups and singular locus for character varieties in PSL(p,C)
Guerin, Clément UL

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 ▲]

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See detailSingularités orbifoldes de la variété des caractères
Guerin, Clément UL

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 ▲]

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See detailAlternate modules are subsymplectic
Guerin, Clément UL

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 ▲]

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