A computational criterion for the Kac conjecture

Mozgovoy, Sergey

doi:10.1016/j.jalgebra.2007.02.018

Cited by 25 publications

(33 citation statements)

References 10 publications

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“…. , x r ]] with a structure of a λ-ring (see [7,Appendix]) in terms of Adams operations by ψ n (f (q, x 1 , . .…”

Section: Combinatorial Prerequisitesmentioning

confidence: 99%

“…Lemma 2.1 ([7,Lemma 22]). Let f ∈ 1 + m and g ∈ R. Define the elements g d ∈ R, d ≥ 1 by the formula d|n d · g d = ψ n (g), n ≥ 1.…”

Section: Combinatorial Prerequisitesmentioning

confidence: 99%

“…The maps Exp, Log and Pow can be defined for an arbitrary complete λ-ring, see [7,Appendix]. Their basic properties can be also found there.…”

Section: Combinatorial Prerequisitesmentioning

confidence: 99%

“…The first named author already established the usefulness of λ-ring techniques for the study of counting polynomials in [7], thereby deriving a combinatorial reformulation of the Kac conjecture mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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On the number of stable quiver representations over finite fields

Mozgovoy

Reineke

2009

Journal of Pure and Applied Algebra

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“…. , x r ]] with a structure of a λ-ring (see [7,Appendix]) in terms of Adams operations by ψ n (f (q, x 1 , . .…”

Section: Combinatorial Prerequisitesmentioning

confidence: 99%

“…Lemma 2.1 ([7,Lemma 22]). Let f ∈ 1 + m and g ∈ R. Define the elements g d ∈ R, d ≥ 1 by the formula d|n d · g d = ψ n (g), n ≥ 1.…”

Section: Combinatorial Prerequisitesmentioning

confidence: 99%

“…The maps Exp, Log and Pow can be defined for an arbitrary complete λ-ring, see [7,Appendix]. Their basic properties can be also found there.…”

Section: Combinatorial Prerequisitesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the number of stable quiver representations over finite fields

Mozgovoy

Reineke

2009

Journal of Pure and Applied Algebra

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“…Let k be a finite field with q elements, put H D (r, d) = (−q 1 2 ) −lr 2 |Higgs ss D (r, d)(k)| , and define the DT-invariants Ω D (r, d) by the formulawhere Log is the plethystic logarithm (see below or e.g. [13]). It was conjectured in [15] that Ω D (r, d) is a polynomial in the Weil numbers of X which is independent of d, regardless of whether gcd(r, d) = 1 or not.…”

mentioning

confidence: 99%

Counting Higgs bundles and type quiver bundles

Mozgovoy¹,

Schiffmann²

2020

Compositio Math.

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We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve defined over a finite field of genus g, when the degree of twisting line bundle is at least 2g − 2 (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson-Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type A (finite or affine), obtaining in particular a Harder-Narasimhan-type formula counting semistable U (p, q)-Higgs bundles over a smooth projective curve defined over a finite field. SERGEY MOZGOVOY AND OLIVIER SCHIFFMANNThere is a natural notion of semistability for these pairs and one can construct the moduli stack Higgs ss D (r, d) of semistable D-twisted Higgs bundles over X of rank r and degree d. Despite its importance in algebraic geometry, in the theory of integrable systems and more recently in the theory of automorphic forms, the topology Higgs ss D (r, d) still remains somewhat mysterious. Observe that twisting by a line bundle of degree one yields an isomorphism Higgs ss D (r, d) ≃ Higgs ss D (r, d + r) so that only the value of d in Z/rZ matters. In [10] (see also [9, Conj.5.6]), Hausel and Rodriguez-Villegas formulated a precise conjecture for the Poincaré polynomial of Higgs ss K (r, d) when k = C and gcd(r, d) = 1. This conjecture was later refined by the first author in [15, Conj.3] (see also [3]) to a conjecture for the motive [Higgs ss D (r, d)] for any divisor D of degree l ≥ 2g − 2. In the case of D = K this conjecture was verified in low ranks [5,6] as well as for the y-genus specialization [4]. Some very interesting results for coprime r and d were also obtained in [1,2]. In [18] the second author gave an explicit formula 1 for the Poincaré polynomial of Higgs ss K (r, d) when k = C and gcd(r, d) = 1, by counting the number of points of Higgs ss K (r, d) over a finite field of high enough characteristic and using the Weil conjectures. This point count in turn relies on a geometric deformation argument to show that (in high enough characteristic) |Higgs ss K (r, d)(F q )| = q 1+(g−1)r 2 A X,r,d , where A X,r,d stands for the number of geometrically indecomposable vector bundles on X of rank r and degree d. A closed expression for A X,r,d is derived in [18].The main aim of this paper is to generalize the above results to arbitrary meromorphic Higgs bundles (i.e. to Higgs ss D (r, d) for any D of degree l ≥ 2g − 2) and to an arbitrary pair (r, d) (i.e. dropping the coprimality assumption on r and d). Our approach is in part related to that of [18], but it replaces the geometric deformation argument (only available in the symplectic case K = D and in high enough characteristic) by an argument involving the Hall algebra of the category of meromorphic Higgs bundles, which works for all D ≥ K, in all characteristics and which yields at the same time the motive of Higgs ss D (r, d) for all r and d. We also partly extend these results to the moduli sp...

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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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A computational criterion for the Kac conjecture

Cited by 25 publications

References 10 publications

On the number of stable quiver representations over finite fields

On the number of stable quiver representations over finite fields

Counting Higgs bundles and type quiver bundles

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

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