Related papers: Root graded groups of type $ H_3 $ and $ H_4 $

In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we established commutator formulas for relative elementary subgroups in $GL(n,R)$, $n\ge 3$, and other similar groups, such as Bak's unitary groups, or Chevalley groups.…

Fix a flat and projective morphism $X\rightarrow\Sigma$ of schemes. We show, first, that any set of $\mathbb{P}^1$-fibrations on $X$ defines a set of simple roots, a set of simple coroots and a Cartan matrix $C$. Second, $X$ is an \'etale…

The computation of the cohomology for finite groups of Lie type in the describing characteristic is a challenging and difficult problem. In earlier work, the authors constructed an induction functor which takes modules over the finite group…

We compute the cohomology of the complement of toric arrangements associated to root systems as representations of the corresponding Weyl groups. Specifically, we develop an algorithm for computing the cohomology of the complement of toric…

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{<=2}(X), to the second nilpotent quotient, G/G_3. We define invariants of G/G_3 by counting normal subgroups of a fixed prime index p,…

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…

A class of groups C is root in a sense of K. W. Gruenberg if it is closed under taking subgroups and satisfies the Gruenberg condition: for any group X and for any subnormal sequence Z \leqslant Y \leqslant X with factors in C, there exists…

Attributed to J F Adams is the conjecture that, at odd primes, the mod-p cohomology ring of the classifying space of a connected compact Lie group is detected by its elementary abelian p-subgroups. In this note we rely on Toda's calculation…

We develop (quantum) cluster algebra structures over arbitrary commutative unital rings $\Bbbk$ and prove that the (quantized) coordinate rings of connected simply-connected complex simple algebraic groups $G$ over $\Bbbk$ admit such…

We prove explicit and elementary formulas for the group homology and cohomology of a finite group with coefficients in any module. We describe in elementary terms the cohomology algebra $H^*(G,k)$ as a graded algebra for a finite group $G$…

Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$ is a power of a prime $p$. We describe a construction of all complex irreducible characters of $U(q)$ and obtain a classification of these irreducible characters…

We show that every finite-dimensional pointed Hopf algebra over a finite simple Chevalley group, different from $PSL_2(q)$ with q= 3 mod 4 (and from $PSL_3(2)\simeq PSL_2(7)$), is isomorphic to the corresponding group algebra. To do this,…

Let $B$ denote the upper triangular subgroup of $SL_2(C)$, $T$ its diagonal torus and $U$ its unipotent radical. A complex projective variety $Y$ endowed with an algebraic action of $B$ such that the fixed point set $Y^U$ is a single point,…

Let e>1 be an integer. Among the finite 3-groups G with bicyclic commutator quotient G/G' ~ C(3^e) * C(3), having one non-elementary component with logarithmic exponent e, there exists a unique pair of coclass trees with distinguished rank…

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be…

Let $R$ be a reduced irreducible root system, $h$ its Coxeter number and $m$ a positive integer smaller than $h$. Choose of base of $R$, whence a corresponding height function, and let $R(m)$ be the set of roots whose height is a multiple…

A commutative order in a quaternion algebra is called selective if it is embeds into some, but not all, the maximal orders in the algebra. It is known that a given quadratic order over a number field can be selective in at most one…

We consider all connected and simply connected 7-dimensional Lie groups whose Lie algebras have nilradical $\g_{5,2} = \s \{X_1, X_2, X_3, X_4, X_5 \colon [X_1, X_2] = X_4, [X_1, X_3] = X_5\}$ of Dixmier. First, we give a geometric…

Let $\mathbb{K}$ be a number field with ring of integers $\mathfrak{O}$ and let $\mathcal{G}$ be a Chevalley group scheme not of type $\mathtt{E}_8$, $\mathtt{F}_4$ or $\mathtt{G}_2$. We use the theory of Tits buildings and a result of…

This paper examines a systematic method to construct a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of non-standard geometric representations. This method can be employed to construct generalizations of…