Character varieties

Sikora, Adam S.

doi:10.1090/s0002-9947-2012-05448-1

Cited by 101 publications

(163 citation statements)

References 131 publications

(146 reference statements)

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“…in [Ben02]). When ρ is scheme smooth, the link between this tangent space and H 1 (Γ, g Ad•ρ ) has been studied in [Sik12], Paragraph 13 by Sikora (see also Proposition 5.2 in [HP04]) using Luna'sÉtale Slice Theorem. Theorem 53 in [Sik12] states that if ρ is a scheme smooth completely reducible representation then :…”

Section: Orbifold Singularities and Algebraic Singularitiesmentioning

confidence: 99%

“…Sikora proved (see [Sik12], Corollary 17) that a completely reducible subgroup H of G is irreducible if and only if its centralizer Z G (H) in G is a finite extension of Z(G).…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, our goal in this paper is to study bad representations from a finitely generated group to P SL(p, C) when p is a prime number. We make a brief recall of some notions on representation/character varieties (see [Sik12] and also [LM85] for a complete exposition).…”

Section: Introductionmentioning

confidence: 99%

“…We denote Hom i (Γ, G) the set of irreducible representations from Γ to G. Since being non-irreducible is a closed condition, the set Hom i (Γ, G) is open in Hom(Γ, G) (see Proposition 27 in [Sik12]). …”

Section: Introductionmentioning

confidence: 99%

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Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

Guerin

2017

Geom Dedicata

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We give the centralizers of irreducible representations from a finitely generated group Γ to P SL(p, C) where p is a prime number. This leads to a description of the singular locus (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 χ i (Γ, P SL(p, C)). When Γ is a free group of rank l ≥ 2 or the fundamental group of a closed Riemann surface of genus g ≥ 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.

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Section: Orbifold Singularities and Algebraic Singularitiesmentioning

confidence: 99%

“…Sikora proved (see [Sik12], Corollary 17) that a completely reducible subgroup H of G is irreducible if and only if its centralizer Z G (H) in G is a finite extension of Z(G).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

Guerin

2017

Geom Dedicata

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“…For some other references on this subject, see [1,2,3,4,5,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,28,29,30,31,33,35,36,37].…”

Section: Theorem 12 ([7]mentioning

confidence: 99%

Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

Gamse¹,

Weitsman²

2016

J. Differential Geom.

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Abstract. We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.

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Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

Florentino

Nozad

Zamora

2022

Mathematische Nachrichten

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With 𝐺 = 𝐺𝐿(𝑛, ℂ), let  Γ 𝐺 be the 𝐺-character variety of a given finitely presented group Γ, and let  𝑖𝑟𝑟 Γ 𝐺 ⊂  Γ 𝐺 be the locus of irreducible representation conjugacy classes. We provide a concrete relation, in terms of plethystic functions, between the generating series for 𝐸-polynomials of  Γ 𝐺 and the one for  𝑖𝑟𝑟 Γ 𝐺, generalizing a formula of Mozgovoy-Reineke. The proof uses a natural stratification of  Γ 𝐺 coming from affine GIT, the combinatorics of partitions, and the formula of MacDonald-Cheah for symmetric products; we also adapt it to the so-called Cartan brane in the moduli space of Higgs bundles. Combining our methods with arithmetic ones yields explicit expressions for the 𝐸-polynomials, and Euler characteristics, of the irreducible stratum of 𝐺𝐿(𝑛, ℂ)-character varieties of some groups Γ, including surface groups, free groups, and torus knot groups, for low values of 𝑛. K E Y W O R D Scharacter varieties, E-polynomials, Hodge theory, representations of finitely presented groups INTRODUCTIONLet 𝐺 be a complex reductive algebraic group, Γ be a finitely presented group, such as the fundamental group of a compact manifold or a finite 𝐶𝑊-complex, and letThis is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Character varieties

Abstract: We study properties of irreducible and completely reducible representations of finitely generated groups Γ into reductive algebraic groups G.In particular, we study the geometric invariant theory of the G action on the space of G-representations of Γ by conjugation.

Cited by 101 publications

References 131 publications

Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

Geometry of the intersection ring and vanishing relations in the cohomology of the moduli space of parabolic bundles on a curve

Generating series for the E‐polynomials of GL(n,C)$GL(n,{\mathbb {C}})$‐character varieties

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