Semisimple Algebraic Group Research Articles

Twisted group ring isomorphism problem and infinite cohomology groups

We continue our investigation of a variation of the group ring isomorphism problem for twisted group algebras. Contrary to previous work, we include cohomology classes which do not contain any cocycle of finite order. This allows us to study the problem in particular over any field of characteristic 0. We prove that there are finite groups G and H which can not be distinguished by their rational twisted group algebras, while G and H can be identified by their semi-simple twisted group algebras over other fields. This is in contrast with the fact that the structural information on G which can be obtained from all the semi-simple group algebras of G is already encoded in its rational group algebra.We further show that for an odd prime p there are groups of order p4 which can not be distinguished by their twisted group algebras over F for any field F of characteristic different from p. On the other hand we prove that the groups constructed by E. Dade, which have isomorphic group algebras over any field, can be distinguished by their rational twisted group algebras. We also answer a question about sufficient conditions for the twisted group ring isomorphism problem to hold over the complex numbers.

Invariant Measures for Horospherical Actions and Anosov Groups

AbstractLet $\Gamma $ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group $G$. For a maximal horospherical subgroup $N$ of $G$, we show that the space of all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on $\Gamma \backslash G$, up to proportionality, is homeomorphic to ${\mathbb {R}}^{\text {rank}\,G-1}$, where $A$ is a maximal real split torus and $M$ is a maximal compact subgroup that normalizes $N$. One of the main ingredients is to establish the $NM$-ergodicity of all Burger–Roblin measures.

Unit group of some finite semisimple group algebras

We provide the structure of the unit group of {mathbb {F}}_{p^k}S_n, where p>n is a prime and S_n denotes the symmetric group on n letters. We also provide the complete characterization of the unit group of the group algebra {mathbb {F}}_{p^k}A_6 for pge 7, where A_6 is the alternating group on 6 letters.

A Note on the Wedderburn Decomposition and Unit Group of the Semisimple Group Algebra FpSn

Abstract: In this short note, we give an algorithm to compute the unit group of a semisimple group algebra for any . To show the practicality of the algorithm, we explicitly find the unit groups of the semisimple group algebras

Uniqueness of Conformal Measures and Local Mixing for Anosov Groups

In the late seventies, Sullivan showed that, for a convex cocompact subgroup Γ of SO∘(n,1) with critical exponent δ>0, any Γ-conformal measure on ∂Hn of dimension δ is necessarily supported on the limit set Λ and that the conformal measure of dimension δ exists uniquely. We prove an analogue of this theorem for any Zariski dense Anosov subgroup Γ of a connected semisimple real algebraic group G of rank at most 3. We also obtain the local mixing for generalized BMS measures on Γ∖G including Haar measures.

Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ from a subalgebra of factor group of $G$

In this paper, we derive a condition under which the Wedderburn decomposition of a semisimple group algebra $\mathbb{F}_qG$ can be deduced from the Wedderburn decomposition of $\mathbb{F}_q(G/H)$, where $H$ is a normal subgroup of $G$ having two elements and $q=p^k$ for some prime $p$ and $k\in \mathbb{Z}^+$. In order to complement the abstract theory of the paper, we deduce the Wedderburn decomposition and hence the unit group of semisimple group algebra $\mathbb{F}_q(A_5\rtimes C_4)$, where $A_5\rtimes C_4$ is a non-metabelian group and $C_4$ is a cyclic group of order $4$.

Hecke action on the principal block

In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical$p$-Kazhdan–Lusztig polynomials.

BOTT–SAMELSON–DEMAZURE–HANSEN VARIETIES FOR PROJECTIVE HOMOGENEOUS VARIETIES WITH NONREDUCED STABILIZERS

Over a field of positive characteristic, a semisimple algebraic group $G$ may have some nonreduced parabolic subgroup $P$. In this paper, we study the Schubert and Bott-Samelson-Demazure-Hansen (BSDH) varieties of $G/P$, with $P$ nonreduced, when the base field is perfect. It is shown that in general the Schubert and BSDH varieties of such a $G/P$ are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties are also studied. It is shown that the decomposition theorem holds for such morphisms, and the pushforward of intersection complexes by such morphisms are Frobenius semisimple.

Twisted conjugacy in linear algebraic groups II

Let G be a linear algebraic group over an algebraically closed field k and Autalg(G) the group of all algebraic group automorphisms of G. For every φ∈Autalg(G) let R(φ) denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by (g,x)↦gxφ(g−1), for all g,x∈G). We say that G has the algebraic R∞-property if R(φ) is infinite for every φ∈Autalg(G). In [1] we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group G has the algebraic R∞-property, then Gφ (the fixed-point subgroup of G under φ) is infinite for all φ∈Autalg(G). In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic R∞-property and identify certain classes of solvable algebraic groups for which the property fails.

On simple modules with singular highest weights for so2l+1(K)

In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.

On the normal complement problem in modular and semisimple group algebras

Let p and q be odd primes such that Let F be the field with p elements and be a group, where A is an abelian group of order In this article, we prove that if then G does not have a normal complement in Further, for any integer we prove that if F is a finite field such that then and do not have a normal complement in and respectively.

Torus orbit closures in flag varieties and retractions on Weyl groups

A finite Coxeter group W has a natural metric d and if [Formula: see text] is a subset of W, then for each [Formula: see text], there is [Formula: see text] such that [Formula: see text]. Such q is not unique in general but if [Formula: see text] is a Coxeter matroid, then it is unique, and we define a retraction [Formula: see text] so that [Formula: see text]. The T-fixed point set [Formula: see text] of a T-orbit closure Y in a flag variety G/B is a Coxeter matroid, where G is a semi-simple algebraic group, B is a Borel subgroup, and T is a maximal torus of G contained in B. We define a retraction [Formula: see text] geometrically, where W is the Weyl group of [Formula: see text], and show that [Formula: see text]. We introduce another retraction [Formula: see text] algebraically for an arbitrary subset [Formula: see text] of W when W is a Weyl group of classical Lie type, and show that [Formula: see text] when [Formula: see text] is a Coxeter matroid.

A Compactification of the Universal Moduli Space of Principal G-Bundles

Let G be a semisimple linear algebraic group, rho :Ghookrightarrow text {SL}(V) a finite-dimensional faithful representation, gge 2 a natural number, and delta a positive rational number. We prove the existence of a compactification of the universal moduli space of semistable principal G-bundles over overline{text {M}}_{g}, provided that delta is sufficiently large, having the following property: the fibers over singular curves are the moduli spaces of delta -semistable singular principal G-bundles.

Small toric resolutions of toric varieties of string polytopes with small indices

Let G be a semisimple algebraic group over [Formula: see text]. For a reduced word [Formula: see text] of the longest element in the Weyl group of G and a dominant integral weight [Formula: see text], one can construct the string polytope [Formula: see text], whose lattice points encode the character of the irreducible representation [Formula: see text]. The string polytope [Formula: see text] is singular in general and combinatorics of string polytopes heavily depends on the choice of [Formula: see text]. In this paper, we study combinatorics of string polytopes when [Formula: see text], and present a sufficient condition on [Formula: see text] such that the toric variety [Formula: see text] of the string polytope [Formula: see text] has a small toric resolution. Indeed, when [Formula: see text] has small indices and [Formula: see text] is regular, we explicitly construct a small toric resolution of the toric variety [Formula: see text] using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when [Formula: see text]. As a byproduct, we show that if [Formula: see text] has small indices then [Formula: see text] is integral for any dominant integral weight [Formula: see text], which in particular implies that the anticanonical limit toric variety [Formula: see text] of a partial flag variety [Formula: see text] is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold [Formula: see text] and obtain a formula of the disk potential of the Lagrangian torus fibration on [Formula: see text] obtained from a flat toric degeneration of [Formula: see text] to the toric variety [Formula: see text].

Deformations of symplectic singularities and orbit method for semisimple Lie algebras

We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of coadjoint orbits of a semisimple algebraic group to the set of primitive ideals in the universal enveloping algebra. We show that the map is injective for classical Lie algebras and conjecture that in that case the image consists of the primitive ideals corresponding to one-dimensional representations of W-algebras. Along the way, we get several new results on the Lusztig-Spaltenstein induction for coadjoint orbits.

Central and non central codes of dihedral 2-groups

In this paper, the central and non central codesof semisimple dihedral group algebra FqG, over a finite field Fq, are constructed. Further the distances of these central and non central codes are computed.

Existence of non-obvious divergent trajectories in homogeneous spaces

We prove a modified version for a conjecture of Weiss from 2004. Let G be a semisimple real algebraic group defined over ℚ, Γ be an arithmetic subgroup of G. A trajectory in G/Γ is divergent if eventually it leaves every compact subset, and is obvious divergent if there is a finite collection of algebraic data which cause the divergence. Let A be a diagonalizable subgroup of G of positive dimension. We show that if the projection of A to any ℚ-factor of G is of small enough dimension (relatively to the ℚ-rank of the ℚ-factor), then there are non-obvious divergent trajectories for the action of A on G/Γ.

Numerical flatness and principal bundles on Fujiki manifolds

Let M be a compact connected Fujiki manifold, G a semisimple affine algebraic group over C with one simple factor and P a fixed proper parabolic subgroup of G. For a holomorphic principal G–bundle EG over M, let EP be the holomorphic principal P–bundle EG⟶EG/P given by the quotient map. We prove that the following three statements are equivalent: (1) ad(EG) is numerically flat, (2) the holomorphic line bundle ⋀topad(EP)⁎ is nef, and (3) for every reduced irreducible compact complex analytic space Z with a Kähler form ω, holomorphic map γ:Z⟶M, and holomorphic reduction of structure group EP⊂γ⁎EG to P, the inequality degree(ad(EP))≤0 holds.

The Canonical Dimension of a Semisimple Group and the Unimodular Degree of a Root System

AbstractWe produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko [6–9] and Devyatov [5] as well as we produce new bounds (e.g., for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

Unit group of semisimple group algebras of some non-metabelian groups of order 120

In this paper, we give the characterization of the unit groups of semisimple group algebras of some non-metabelian groups of order [Formula: see text]. This study completes the study of unit groups of semisimple group algebras of all groups up to order [Formula: see text], except that of the symmetric group [Formula: see text] and groups of order [Formula: see text].