Bounding the Dimensions of Rational Cohomology Groups

Let G be a simple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Let σ : G → G be a strict endomorphism (i. e., the subgroup G(σ) of σ-fixed points is finite). Also, let Gσ be the scheme-theoretic kernel of σ, an infinitesimal subgroup of G. This paper shows that the degree m cohomology H m (G(σ), L) of any irreducible kG(σ)-module L is bounded by a constant depending on the root system Φ of G and the integer m. A similar result holds for the degree m cohomology of Gσ. These bounds are actually established for the degree m extension groups Ext m G(σ) (L, L ′ ) between irreducible kG(σ)-modules L, L ′ , with again a similar result holding for Gσ. In these Ext m results, of interest in their own right, the bounds depend also on L, or, more precisely, on length of the p-adic expansion of the highest weight associated to L. All bounds are, nevertheless, independent of the characteristic p. These results extend earlier work of Parshall and Scott for rational representations of algebraic groups G.We also show that one can find bounds independent of the prime for the Cartan invariants of G(σ) and Gσ, and even for the lengths of the underlying PIMs. These bounds, which depend only on the root system of G and the "height" of σ, provide in a strong way an affirmative answer to a question of Hiss, for the special case of finite groups G(σ) of Lie type in the defining characteristic.

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Bounding Cohomology for Finite Groups and Frobenius Kernels

Bendel

Nakano

Parshall

et al. 2015

Algebr Represent Theor

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Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

Lübeck¹

2020

Trans. Amer. Math. Soc.

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We discuss a practical algorithm to compute parabolic Kazhdan-Lusztig polynomials. As an application we compute Kazhdan-Lusztig polynomials which are needed to evaluate a character formula for reductive groups due to Lusztig.Some coefficients of these polynomials have interesting interpretations for certain finite groups. We find examples of finite dimensional modules for finite groups with much higher dimensional first cohomology group than in all previously known cases.Some of these examples lead to the construction of finite groups with many maximal subgroups, contradicting an old conjecture by G. E. Wall.1991 Mathematics Subject Classification. 20F55 20C08 20G05 20G10 20E28.

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Restricted Partitions and $Sl_2$-Cohomology

Benzel,

Conner,

Ngo

et al. 2023

Albanian J. Math.

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Bounding the Dimensions of Rational Cohomology Groups

Cited by 3 publications

References 29 publications

Bounding Cohomology for Finite Groups and Frobenius Kernels

Bounding Cohomology for Finite Groups and Frobenius Kernels

Computation of Kazhdan-Lusztig polynomials and some applications to finite groups

Restricted Partitions and $Sl_2$-Cohomology

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