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

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].

Frobenius groups of low rank

Let 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.

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

In 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.

HIGHER DEFORMATIONS OF LIE ALGEBRA REPRESENTATIONS II

Steinberg’s tensor product theorem shows that for semisimple algebraic groups, the study of irreducible representations of higher Frobenius kernels reduces to the study of irreducible representations of the first Frobenius kernel. In the preceding paper in this series, deforming the distribution algebra of a higher Frobenius kernel yielded a family of deformations called higher reduced enveloping algebras. In this paper, we prove that the Steinberg decomposition can be similarly deformed, allowing us to reduce representation theoretic questions about these algebras to questions about reduced enveloping algebras. We use this to derive structural results about modules over these algebras. Separately, we also show that many of the results in the preceding paper hold without an assumption of reductivity.

A note on Frobenius–Wielandt groups

Let G be a finite group. G is called a Frobenius–Wielandt group if there exists {H<G} such that {U=\langle H\cap H^{g}\mid g\in G-H\rangle} is a proper subgroup of H. The Wielandt theorem [H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen, Math. Nachr. 18 1958, 274–280; Mathematische Werke Vol. 1, 769–775] on the structure of G generalizes the celebrated Frobenius theorem. From a permutation group point of view, considering the action of G on the coset space {G/H}, it states in particular that the subgroup {D=D_{G}(H)} generated by all derangements (fixed-point-free elements) is a proper subgroup of G. Let {W=U^{G}}, the normal closure of U in G. Then W is the subgroup generated by all elements fixing at least two points. We present the proof of the Wielandt theorem in a new way (Theorem 1.6, Corollary 1.7, Theorem 1.8) such that the unique component whose proof is not elementary or by the Frobenius theorem is the equality {W\cap H=U}. This presentation shows what can be achieved by elementary arguments and how Frobenius groups are involved in one case of Frobenius–Wielandt groups. To be more precise, Theorem 1.6 shows that there are two possible cases for a Frobenius–Wielandt group G with {H<G}: (a) {W=D} and {G=HW}, or (b) {W<D} and {HW<G}. In the latter case, {G/W} is a Frobenius group with a Frobenius complement {HW/W} and Frobenius kernel {D/W}.

Saturation rank for finite group schemes: Finite groups and infinitesimal group schemes

We investigate the saturation rank of a finite group scheme defined over an algebraically closed field{\Bbbk}of positive characteristicp. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group{\operatorname{SL}_{n}}.

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

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.

Finite Groups with Four Relative Commutativity Degrees

It is shown that a finite group G has four relative commutativity degrees if and only if G/Z(G) is a p-group of order p3 and G has no abelian maximal subgroups, or G/Z(G) is a Frobenius group with Frobenius kernel and complement isomorphic to ℤp × ℤp and ℤq, respectively, and the Sylow p-subgroup of G is abelian, where p and q are distinct primes.

Loewy series of parabolically induced -Verma modules

We 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.

ON THE SUPPORT VARIETIES FOR DEMAZURE MODULES

The support varieties for the induced modules or Weyl modules for a reductive algebraic group G were computed over the first Frobenius kernel G1 by Nakano, Parshall and Vella. A natural generalization of this computation is the calculation of the support varieties of Demazure modules over the first Frobenius kernel, B1, of the Borel subgroup B. In this paper we initiate the study of such computations. We complete the entire picture for reductive groups with underlying root systems A1 and A2. Moreover, we give complete answers for Demazure modules corresponding to a particular (standard) element in the Weyl group, and provide results relating support varieties between different Demazure modules which depend on the Bruhat order.