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

Florentino, Carlos; Nozad, Azizeh; Zamora, Alfonso

doi:10.48550/arxiv.1902.06837

Cited by 3 publications

(13 citation statements)

References 8 publications

(13 reference statements)

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“…Proof. The proof for U n,r is analogous to the proof in [FNZ,Proposition 4.3] for R Γ GL n , observing that dealing with the usual Euclidean topology on U n,r works ipsis verbis as with the Zariski topology on R Γ GL n . The cases SU r,n and P U r,n follow as in Proposition 4.5.…”

Section: Now Definementioning

confidence: 88%

“…The character variety of good representations is a smooth algebraic variety, by [Sik]. For further details, we refer the reader to [FNZ,Section 3].…”

Section: Denote By R Irrmentioning

confidence: 99%

“…Any character variety admits a stratification by the dimension of the stabilizer of a given representation. When dealing with the general linear group GL n as well as the related groups SL n and P GL n , there is a more refined stratification which gives a lot more of information on the corresponding character varieties X Γ G. In this subsection, we recall this stratification, following [FNZ,Section 4.1], and describe its analogous versions for SL n and…”

Section: The Strictly Polystable Casementioning

confidence: 99%

“…Central action on representation spaces. We now define an stratification by polystable type of U r,n , SU r,n and P U r,n in complete analogy with the stratification of GL n -character varieties given in [FNZ,Section 4.1] and recalled in Section 4 for P GL n and SL n .…”

Section: Now Definementioning

confidence: 99%

See 3 more Smart Citations

Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups

Florentino,

Nozad,

Zamora

2019

Preprint

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Let G be a complex reductive group and XrG denote the G-character variety of the free group of rank r. Using geometric methods, we prove that E(XrSLn) = E(XrP GLn), for any n, r ∈ N, where E(X) denotes the Serre (also known as E-) polynomial of the complex quasi-projective variety X, settling a conjecture of Lawton-Muñoz in [LM]. The proof involves the stratification by polystable type introduced in [FNZ], and shows moreover that the equality of E-polynomials holds for every stratum and, in particular, for the irreducible stratum of XrSLn and XrP GLn. We also present explicit computations of these polynomials, and of the corresponding Euler characteristics, based on our previous results and on formulas of Mozgovoy-Reineke for GLn-character varieties over finite fields.

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Section: Now Definementioning

confidence: 88%

“…The character variety of good representations is a smooth algebraic variety, by [Sik]. For further details, we refer the reader to [FNZ,Section 3].…”

Section: Denote By R Irrmentioning

confidence: 99%

Section: The Strictly Polystable Casementioning

confidence: 99%

Section: Now Definementioning

confidence: 99%

See 2 more Smart Citations

Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups

Florentino,

Nozad,

Zamora

2019

Preprint

Self Cite

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“…y n ∈ x, y , follows step by step from the one in [FHH,FNZ1], written for s = 1. Since it is an easy computation, we include it here, for the reader's convenience.…”

Section: Plethystic Exponentials and Partitionsmentioning

confidence: 99%

Plethystic exponential calculus and characteristic polynomials of permutations

Florentino

2021

Preprint

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We prove a family of identities, expressing generating functions of powers of permutations polynomials, as finite or infinite products. These generalize formulae first obtained in a study of the topology/mixed Hodge structures of symmetric products of real/algebraic tori. The present proof comes from power series expansions of plethystic exponentials in rings of formal power series motivated by some recent applications of these combinatorial tools in supersymmetric gauge theories. Since the proof is elementary, we aimed at being self-contained and introduced all needed tools from plethystic calculus.

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On Hodge polynomials of Singular Character Varieties

Florentino¹,

Nozad²,

Silva³

et al. 2020

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Let X Γ G := Hom(Γ , G)//G be the G-character variety of Γ , where G is a complex reductive group and Γ a finitely presented group. We introduce new techniques for computing Hodge-Deligne and Serre polynomials of X Γ G, and present some applications, focusing on the cases when Γ is a free or free abelian group. Detailed constructions and proofs of the main results will appear elsewhere. IntroductionLet G be a connected reductive complex algebraic group, and Γ be a finitely presented group. The G-character variety of Γ is defined to be the (affine) geometric invariant theory (GIT) quotientThe most well studied families of character varieties include the cases when the group Γ is the fundamental group of a Riemann surface Σ , and its "twisted" variants. In these cases, the non-abelian Hodge correspondence (see, for example [Si]) shows that (components of) X Γ G are homeomorphic to certain moduli spaces of G-Higgs bundles which appear in connection to important problems in Mathematical-Physics: for example, these spaces play an important role in the quantum field theory interpretation of the geometric Langlands correspondence, in the context of mirror symmetry ([KW]).The study of geometric and topological properties of character varieties is an active topic and there are many recent advances in the computation of their Poincaré polynomials and other invariants. For the surface group case (Γ = π 1 (Σ ) and related groups) the calculations of Poincaré polynomials started with Hitchin and Gothen, and have been pursued more recently by Hausel, Lettelier, Mellit, Rodriguez-Villegas, Schiffmann and others, who also considered the parabolic version of these character varieties (see [HRV, Me, Sc]). Those recent results use arithmetic methods: it is shown that the number of points of the corresponding moduli space over finite fields is given by a polynomial,

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

Cited by 3 publications

References 8 publications

Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups

Serre polynomials of $SL_n$- and $PGL_n$-character varieties of free groups

Plethystic exponential calculus and characteristic polynomials of permutations

On Hodge polynomials of Singular Character Varieties

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