scholarly journals Weakly Commensurable Zariski-Dense Subgroups of Algebraic Groups Defined Over Fields of Positive Characteristic

2015 ◽  
Author(s):  
Joshua Schwartz
Keyword(s):  
Positive Characteristic ◽  
Algebraic Groups ◽  
Dense Subgroups

RESIDUAL PROPERTIES OF FREE PRO-P GROUPS

2001 ◽  
Vol 33 (5) ◽  
pp. 578-582 ◽  
Author(s):  
YIFTACH BARNEA
Keyword(s):  
Positive Characteristic ◽  
Congruence Subgroup ◽  
Probabilistic Method ◽  
Algebraic Groups ◽  
Local Fields ◽  
Random Generation ◽  
Zero Characteristic ◽  
Residual Properties ◽  
Characteristic Case ◽  
Lie Methods

Recall that if S is a class of groups, then a group G is residually-S if, for any element 1 ≠ g ∈ G, there is a normal subgroup N of G such that g ∉ N and G/N ∈ S. Let Λ be a commutative Noetherian local pro-p ring, with a maximal ideal M. Recall that the first congruence subgroup of SLd(Λ) is: SL1d(Λ) = ker (SLd(Λ) → SLd(Λ/M)).Let K ⊆ ℕ. We define SΛ(K) = ∪d∈K{open subgroups of SL1d(Λ)}. We show that if K is infinite, then for Λ = [ ]p[[t]] and for Λ = ℤp a finitely generated non-abelian free pro-p group is residually-SΛ(K). We apply a probabilistic method, combined with Lie methods and a result on random generation in simple algebraic groups over local fields. It is surprising that the case of zero characteristic is deduced from the positive characteristic case.

scholarly journals Loewy series of parabolically induced -Verma modules

2014 ◽  
Vol 14 (1) ◽  
pp. 185-220 ◽  
Author(s):  
Abe Noriyuki ◽  
Kaneda Masaharu
Keyword(s):  
Algebraic Group ◽  
Positive Characteristic ◽  
Algebraic Groups ◽  
Minimal Length ◽  
Closed Field ◽  
Verma Modules ◽  
Parabolic Subgroups ◽  
Reductive Algebraic Group ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

AbstractWe show that the modules for the Frobenius kernel of a reductive algebraic group over an algebraically closed field of positive characteristic $p$ induced from the $p$-regular blocks of its parabolic subgroups can be $\mathbb{Z}$-graded. In particular, we obtain that the modules induced from the simple modules of $p$-regular highest weights are rigid and determine their Loewy series, assuming the Lusztig conjecture on the irreducible characters for the reductive algebraic groups, which is now a theorem for large $p$. We say that a module is rigid if and only if it admits a unique filtration of minimal length with each subquotient semisimple, in which case the filtration is called the Loewy series.

scholarly journals AX–SCHANUEL CONDITION IN ARBITRARY CHARACTERISTIC

2017 ◽  
Vol 18 (06) ◽  
pp. 1157-1213
Author(s):  
Piotr Kowalski
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Group ◽  
Algebraic Variety ◽  
Positive Characteristic ◽  
Algebraic Groups ◽  
General Context ◽  
Arbitrary Characteristic ◽  
Analytic Subgroup

We prove a positive characteristic version of Ax’s theorem on the intersection of an algebraic subvariety and an analytic subgroup of an algebraic group [Ax, Some topics in differential algebraic geometry. I. Analytic subgroups of algebraic groups, Amer. J. Math. 94 (1972), 1195–1204]. Our result is stated in a more general context of a formal map between an algebraic variety and an algebraic group. We derive transcendence results of Ax–Schanuel type.

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