Related papers: The Decomposition of Permutation Module for Infini…

We describe the structure of Sylow {\ell}-subgroups of a finite reduc-tive group G(Fq) when q $\not\equiv$ 0 (mod {\ell}) that we find governed by a complex reflection group attached to G and {\ell}, which depends on {\ell} only through the…

This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture…

Let p, ell be distinct primes and let q be a power of p. Let G be a connected compact Lie group. We show that there exists an integer b such that the mod ell cohomology of the classifying space of a finite Chevalley group G(F_q) is…

Suppose $\ell$ is a prime number, ${\mathbf Q}_\ell$ is the field of $\ell$-adic numbers, ${\mathbf F}_\ell$ is the finite field of $\ell$ elements, and $d$ is a positive integer. Suppose $G$ is a finite subgroup of a symplectic group…

We investigate infinite dimensional modules for a linear algebraic group $\mathbb G$ over a field of positive characteristic $p$. For any subcoalgebra $C \subset \mathcal O(\mathbb G)$ of the coordinate algebra of $\mathbb G$, we consider…

Let $G$ be a finite group and $p$ be a prime. We study the kernel of the map, between the Burnside ring of $G$ and the Grothendieck ring of $\mathbb{F}_p[G]$-modules, taking a $G$-set to its associated permutation module. We are able, for…

The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(\Phi,R)$ be a Chevalley group with a reduced irreducible root system $\Phi$ over a commutative ring $R$. Fix $h\in G(\Phi,R)$. Call…

Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad…

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$…

We study the category of $\mathbf{P}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf{P}$ denotes the subgroup of the infinite general linear group $\mathbf{GL}(\mathbf{C}^\infty)$ consisting of elements…

Let $G$ be a simply-connected semisimple algebraic group scheme over an algebraically closed field of characteristic $p > 0$. Let $r \geq 1$ and set $q = p^r$. We show that if a rational $G$-module $M$ is projective over the $r$-th…

In a recent paper, Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we…

Let $\mathcal{S}_q$ denote the group of all square elements in the multiplicative group $\mathbb{F}_q^*$ of a finite field $\mathbb{F}_q$ of odd characteristic containing $q$ elements. Let $\mathcal{O}_q$ be the set of all odd order…

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 $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is…

If R is a commutative ring, we prove that every finitely generated module has a pure-composition series with indecomposable factors and any two such series are isomorphic if and only if R is a Bezout ring and a CF-ring.

We determine the decomposition numbers of the partition algebra when the characteristic of the ground field is zero or larger than the degree of the partition algebra. This will allow us to determine for which exact values of the parameter…

Let $G(\mathbb{Q})$ be a simply connected Chevalley group over $\mathbb{Q}$ corresponding to a simple Lie algebra $\mathfrak g$ over $\mathbb{C}$. Let $V$ be a finite dimensional faithful highest weight $\mathfrak g$-module and let…

Suppose that $Q$ is a finite quiver and $G\subseteq \Aut(Q)$ is a finite group, $k$ is an algebraic closed field whose characteristic does not divide the order of $G$. For any algebra $\Lambda=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an…

In Finite Group Modular Representation Theory, the basic objects are the indecomposable and simple modules. This paper offers a new classification of these objects that refines the Green Theory Classification of indecomposable and simple…