Related papers: On Endo-trivial Modules for p-Solvable Groups

Let $G$ be a finite group such that $\text{SL}(n,q)\subseteq G \subseteq \text{GL}(n,q)$ and $Z$ be a central subgroup of $G$. In this paper we determine the group $T(G/Z)$ consisting of the equivalence classes of endotrivial…

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial finite dimensional irreducible rational $KG$-module.…

Let k be a field, let G be an affine algebraic k-group and V a finite-dimensional G-module. We say V is rigid if the socle series and radical series coincide for the action of G on each indecomposable summand of V; say V is geometrically…

In this paper, I show that if $p$ is an odd prime, and if $P$ is a finite $p$-group, then there exists an exact sequence of abelian groups $$0\to T(P)\to D(P)\to\lproj{P}\to H^1\big(\apdeux(P),\Z\big)^{(P)},$$ where $D(P)$ is the Dade group…

Let $G$ be a finite group and $k$ a field of prime characteristic $p$. We examine the Lefschetz homomorphism $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$ from the group of endotrivial complexes, i.e. the Picard group of the bounded homotopy…

Let $p$ be a prime number and $\Bbbk=\bar{\mathbb{F}}_p$, the algebraic closure of the finite field $\mathbb{F}_p$ of $p$ elements. Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_p$ and ${\bf B}$ be a Borel subgroup…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

Let $k$ be a field, and $G$ be a $k$-group scheme of finite type. Let $G_{\mathrm{ad}}$ be the $k$-scheme $G$ with the adjoint action of $G$. We call $\lambda_{G,G}=H^0(\mathop{\mathrm{Spec}} k,e^*(\omega_{G_{\mathrm{ad}}}))$ the Knop…

J.F. Carlson conjectured in 1995 that if G is a finite group and k is a field whose characteristic p divides the order of G that the depth of H*(G,k) equals the minimum of the dimensions of associated primes of H*(G,k). This is obviously…

Let $G$ be a finite group and $k$ a field of characteristic $p$. We conjecture that if $M$ is a $kG$-module with $H^*(G,M)$ finitely generated as a module over $H^*(G,k)$ then as an element of the stable module category…

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that…

We finish off the classification of the endo-trivial modules of finite groups with Sylow $2$-subgroups isomorphic to a semi-dihedral $2$-group started by Carlson, Mazza and Th\'evenaz in their article "Endotrivial modules over groups with…

In this paper the authors provide a complete answer to Donkin's Tilting Module Conjecture for all rank $2$ semisimple algebraic groups and $\text{SL}_{4}(k)$ where $k$ is an algebraically closed field of characteristic $p>0$. In the…

We give several criteria to decide whether a given tensor category is the abelian envelope of a fixed symmetric monoidal category. As a main result we prove that the category of finite-dimensional representations of a semisimple simply…

If $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$ is a Lie superalgebra over an algebraically closed field $k$ of characteristic 0, the notion of an endotrivial module has recently been extended to…

We prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V_p(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of…

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently…

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p>0. Here we classify all irreducible kG-modules for which the principal A1 has no repeated composition factors, extending the work of…

Let k be a perfect field and A a finite dimensional k-algebra of finite global dimension (e.g. the path algebra of a finite quiver without oriented cycles). Making use of the recent theory of noncommutative motives, we prove that the value…

Let $G$ be a finitely generated abelian-by-finite group and $k$ a field of characteristic $p\ge 0$. The Euler class $[k_G]$ of $G$ over $k$ is the class of the trivial $kG$-module in the Grothendieck group $G_0(kG)$. We show that $[k_G]$…