A basis of a certain module for the hyperalgebra of (SL2)r and some applications

2021 ◽  
pp. 2250184
Author(s):  
Yutaka Yoshii
Keyword(s):  
Algebraic Group ◽  
Frobenius Kernel

In the hyperalgebra [Formula: see text] of the [Formula: see text]th Frobenius kernel [Formula: see text] of the algebraic group [Formula: see text], we construct a basis of the [Formula: see text]-module generated by a certain element which was given by the author before. As its applications, we also prove some results on the [Formula: see text]-modules and the algebra [Formula: see text].

scholarly journals Counterexamples to the tilting and (p,r)-filtration conjectures

2020 ◽  
Vol 2020 (767) ◽  
pp. 193-202
Author(s):  
Christopher P. Bendel ◽  
Daniel K. Nakano ◽  
Cornelius Pillen ◽  
Paul Sobaje
Keyword(s):  
Algebraic Group ◽  
Indecomposable Module ◽  
Weyl Module ◽  
Tilting Module ◽  
Simple Algebraic Group ◽  
Frobenius Kernel ◽  
Projective Indecomposable Module

AbstractIn this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the {(p,r)}-Filtration Conjecture.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

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.

Additive Groups in a Non-Symbolic ‘Artificial Algebra’

2020 ◽  
Author(s):  
Randolph C Grace ◽  
Nicola J. Morton ◽  
Matt Grice ◽  
Kate Stuart ◽  
Simon Kemp
Keyword(s):  
Algebraic Group ◽  
Explicit Instruction ◽  
Strong Support ◽  
Line Length

Grace et al. (2018) developed an ‘artificial algebra’ task in which participants learn to make an analogue response based on a combination of non-symbolic magnitudes by feedback and without explicit instruction. Here we tested if participants could learn to add stimulus magnitudes in this task in accord with the properties of an algebraic group. Three pairs of experiments tested the group properties of commutativity (Experiments 1a-b), identity and inverse existence (Experiments 2a-b) and associativity (Experiments 3a-b), with both line length and brightness modalities. Transfer designs were used in which participants responded on trials with feedback based on sums of magnitudes and later were tested with novel stimulus configurations. In all experiments, correlations of average responses with magnitude sums were high on trials with feedback, r = .97 and .96 for Experiments 1a-b, r = .97 and .96 for Experiments 2a-b, and ranged between r = .97 and .99 for Experiment 3a and between r = .82 and .95 for Experiment 3b. Responding on transfer trials was accurate and provided strong support for commutativity, identity and inverse existence, and associativity with line length, and for commutativity and identity and inverse existence with brightness. Deviations from associativity in Experiment 3b suggested that participants were averaging rather than adding brightness magnitudes. Our results confirm that the artificial algebra task can be used to study implicit computation and suggest that representations of magnitudes may have a structure similar to an algebraic group.

scholarly journals MULTIPLE FLAG IND-VARIETIES WITH FINITELY MANY ORBITS

2021 ◽  
Author(s):  
LUCAS FRESSE ◽  
IVAN PENKOV
Keyword(s):  
Algebraic Group ◽  
Linear Algebraic Group ◽  
Interesting Case ◽  
Analogous Problem ◽  
Restrictive Condition ◽  
Dimensional Case ◽  
Essential Difference ◽  
Parabolic Subgroups ◽  
Finite Dimensional ◽  
Finite Dimensional Case

AbstractLet G be one of the ind-groups GL(∞), O(∞), Sp(∞), and let P1, ..., Pℓ be an arbitrary set of ℓ splitting parabolic subgroups of G. We determine all such sets with the property that G acts with finitely many orbits on the ind-variety X1 × × Xℓ where Xi = G/Pi. In the case of a finite-dimensional classical linear algebraic group G, the analogous problem has been solved in a sequence of papers of Littelmann, Magyar–Weyman–Zelevinsky and Matsuki. An essential difference from the finite-dimensional case is that already for ℓ = 2, the condition that G acts on X1 × X2 with finitely many orbits is a rather restrictive condition on the pair P1, P2. We describe this condition explicitly. Using the description we tackle the most interesting case where ℓ = 3, and present the answer in the form of a table. For ℓ ≥ 4 there always are infinitely many G-orbits on X1 × × Xℓ.

scholarly journals Complete reducibility: variations on a theme of Serre

2021 ◽  
Author(s):  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Spherical Building ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
General Properties ◽  
Complete Reducibility ◽  
Special Cases ◽  
Variations On A Theme

AbstractIn this note, we unify and extend various concepts in the area of G-complete reducibility, where G is a reductive algebraic group. By results of Serre and Bate–Martin–Röhrle, the usual notion of G-complete reducibility can be re-framed as a property of an action of a group on the spherical building of the identity component of G. We show that other variations of this notion, such as relative complete reducibility and $$\sigma $$ σ -complete reducibility, can also be viewed as special cases of this building-theoretic definition, and hence a number of results from these areas are special cases of more general properties.

scholarly journals Frobenius groups of low rank

2021 ◽  
Author(s):  
Wolfgang Knapp ◽  
Peter Schmid
Keyword(s):  
Frobenius Group ◽  
Low Rank ◽  
Special Situation ◽  
Frobenius Groups ◽  
Frobenius Kernel

AbstractLet G be a finite Frobenius group of degree n. We show, by elementary means, that n is a power of some prime p provided the rank $${\mathrm{rk}}(G)\le 3+\sqrt{n+1}$$ rk ( G ) ≤ 3 + n + 1 . Then the Frobenius kernel of G agrees with the (unique) Sylow p-subgroup of G. So our result implies the celebrated theorems of Frobenius and Thompson in a special situation.

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

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