Uniform cell decomposition with applications to Chevalley groups

Berman, Mark; Derakhshan, Jamshid; Onn, Uri; Paajanen, Pirita

doi:10.1112/jlms/jds056

Cited by 21 publications

(60 citation statements)

References 16 publications

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“…We note that

L_{3, 2} \approx {frakturn}_{3} false(K false)

and

L_{6, 19} false(- 1 false) \approx {frakturn}_{4} false(K false)

. A formula for

{sans-serifZ}_{{prefixU}_{3} false(frakturO false)}^{sans-serifcc} false(T false)

was previously given in [, § 8.2]. This formula is incorrect due to a sign mistake.…”

Section: Further Examplesmentioning

confidence: 94%

“…The following questions are inspired by Theorems and and [, Theorem C]. Question Let

k

be a number field with ring of integers

frakturo

.…”

Section: Orbits and Conjugacy Classes Of Linear Groupsmentioning

confidence: 99%

“…This formula is incorrect due to a sign mistake. More substantially, the computation in [, § 8.2] relies on [, Proposition 6.2] and what seems to be a variation of the integral formalism developed in for unipotent groups; this is however not explained. Said integral formalism in appears to be essentially different from the methods developed and applied here.…”

Section: Further Examplesmentioning

confidence: 99%

“…Berman et al . investigated

{sans-serifZ}_{G (O)}^{sans-serifcc} false(T false)

for Chevalley groups

sans-serifG

. Lins recently determined

{sans-serifZ}_{G (O)}^{sans-serifcc} false(T false)

for certain families of unipotent group schemes

sans-serifG

.…”

Section: Introductionmentioning

confidence: 99%

“…10 , L6,25 ; L 6,26 ≈ f 2,3 (K)( 1 − qT )/((1 − q 4 T )(1 − q 3 T )) qT )(1 − T )/((1 − q 2 T ) 2 (1 − q 3 T )) L 6,17 1−T −qT +q 2 T +q 3 T 2 −q 4 T 2 −q 5 T 2 +q 5 T 3 (1−q 6 T 2 )(1−q 3 T )(1−q 2 T ) L 6,18 (1 − T )/((1 − q 2 T )(1 − q 4 T )) L 6,19 (0) 1+T −3qT −q 2 T +q 3 T 2 +3q 4 T 2 −q 5 T 2 −q 5 T 3 (1−q 3 T ) 3 (1−q 2 T ) L 6,19 (−1) ≈ n 4 (K), L 6,21 (0) (1 − qT ) 2 /((1 − q 3 T ) 2 (1 − q 2 T )) L 6,21 (1) 1−T −qT +q 2 T +q 2 T 2 −q 3 T 2 −q 4 T 2 +q 4 T 3 (1−q 5 T 2 )(1−q 3 T )(1−q 2 T ) L 6,22 (0) 1−qT −q 2 T +q 3 T +q 4 T 2 −q 5 T 2 −q 6 T 2 +q 7 T 3 (1−q 7 T 2 )(1−q 4 T )(1−q 2 T ) L 6,23 , L 6,24 (0) 1−2qT +q 3 T +q 3 T 2 −2q 5 T 2 +q 6 T 3 (1−q 7 T 2 )(1−q 3 T )(1−q 2 T )…”

mentioning

confidence: 99%

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The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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“…We note that

L_{3, 2} \approx {frakturn}_{3} false(K false)

and

L_{6, 19} false(- 1 false) \approx {frakturn}_{4} false(K false)

. A formula for

{sans-serifZ}_{{prefixU}_{3} false(frakturO false)}^{sans-serifcc} false(T false)

was previously given in [, § 8.2]. This formula is incorrect due to a sign mistake.…”

Section: Further Examplesmentioning

confidence: 94%

“…The following questions are inspired by Theorems and and [, Theorem C]. Question Let

k

be a number field with ring of integers

frakturo

.…”

Section: Orbits and Conjugacy Classes Of Linear Groupsmentioning

confidence: 99%

Section: Further Examplesmentioning

confidence: 99%

“…Berman et al . investigated

{sans-serifZ}_{G (O)}^{sans-serifcc} false(T false)

for Chevalley groups

sans-serifG

. Lins recently determined

{sans-serifZ}_{G (O)}^{sans-serifcc} false(T false)

for certain families of unipotent group schemes

sans-serifG

.…”

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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On singularity properties of convolutions of algebraic morphisms

Glazer

Hendel

2019

Sel. Math. New Ser.

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Let K be a field of characteristic zero, X and Y be smooth K-varieties, and let G be a algebraic K-group. Given two algebraic morphisms ϕ : X → G and ψ : Y → G, we define their convolution ϕ * ψ : X × Y → G by ϕ * ψ(x, y) = ϕ(x) · ψ(y). We then show that this operation yields morphisms with improved smoothness properties. More precisely, we show that for any morphism ϕ : X → G which is dominant when restricted to each absolutely irreducible component of X, by convolving it with itself finitely many times, one can obtain a flat morphism with reduced fibers of rational singularities, generalizing the main result of our previous paper [GH]. Uniform bounds on families of morphisms are given as well. Moreover, as a key analytic step, we also prove the following result in motivic integration; if {f Qp : Q n p → C}p∈primes is a collection of functions which is motivic in the sense of Denef-Pas, and f Qp is L 1 for any p large enough, then in fact there exists ǫ > 0 such that f Qp is L 1+ǫ for any p large enough.

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