Finite Chevalley Groups Research Articles

A Schur ring approach to supercharacters of groups associated with finite radical rings

We consider the central Schur ring associated with the standard supercharacters of the adjoint group G ( A ) of a finite radical ring A , and define supercharacters of the subgroup C G ( A ) ( σ ) consisting of elements fixed by an involution of G ( A ) naturally defined by an (anti-)involution of A . In particular, we extend known results for unipotent subgroups of the classical finite Chevalley groups.

The character table of the finite Chevalley group F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} for q a power of 2

Let q be a prime power and F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} be the Chevalley group of type F4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4$$\\end{document} over a finite field with q elements. Marcelo and Shinoda (Tokyo J Math 18:303–340, 1995) determined the values of the unipotent characters of F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that q is a power of 2, we explain how to construct the complete character table of F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document}.

A multiplicity one theorem for groups of type 𝐴_{𝑛} over discrete valuation rings

Let G \mathbf {G} be the General Linear or Special Linear group with entries from the finite quotients of the ring of integers of a non-archimedean local field and U \mathbf {U} be the subgroup of G \mathbf {G} consisting of upper triangular unipotent matrices. We prove that the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) of G \mathbf {G} obtained from a non-degenerate character θ \theta of U \mathbf {U} is multiplicity free for all ℓ ≥ 2. \ell \geq 2. This is analogous to the multiplicity one theorem regarding Gelfand-Graev representation for the finite Chevalley groups. We prove that for many cases the regular representations of G \mathbf {G} are characterized by the property that these are the constituents of the induced representation Ind U G ⁡ ( θ ) \operatorname {Ind}^{\mathbf {G}}_{\mathbf {U}}(\theta ) for some non-degenerate character θ \theta of U \mathbf {U} . We use this to prove that the restriction of a regular representation of General Linear groups to the Special Linear groups is multiplicity free and also obtain the corresponding branching rules in many cases.

Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type V. Mixed classes in Chevalley and Steinberg groups

We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from mathbf {PSL}_n(q) collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is mathbf {PSp}_{2n}(q), mathbf {P}{varvec{Omega }}^+_{4n}(q), mathbf {P}{varvec{Omega }}^-_{4n}(q), ^3D_4(q), E_7(q), E_8(q), F_4(q), or G_2(q) with q even is the group algebra.

On the irreducible components of a Gelfand–Graev representation of a finite Chevalley group

On the irreducible components of a Gelfand–Graev representation of a finite Chevalley group

On the characters of Sylow $p$-subgroups of finite Chevalley groups $G(p^f)$ for arbitrary primes

We develop in this work a method to parametrize the set $\mathrm{Irr}(U)$ of irreducible characters of a Sylow $p$-subgroup $U$ of a finite Chevalley group $G(p^f)$ which is valid for arbitrary primes $p$, in particular when $p$ is a very bad prime for $G$. As an application, we parametrize $\mathrm{Irr}(U)$ when $G=\mathrm{F}_4(2^f)$.

THE THICKNESS OF SCHUBERT CELLS AS INCIDENCE STRUCTURES

This paper explores the possible use of Schubert cells and Schubert varieties in finite geometry, particularly in regard to the question of whether these objects might be a source of understanding of ovoids or provide new examples. The main result provides a characterization of those Schubert cells for finite Chevalley groups which have the first property (thinness) of ovoids. More importantly, perhaps this short paper can help to bridge the modern language barrier between finite geometry and representation theory. For this purpose, this paper includes very brief surveys of the powerful lattice theory point of view from finite geometry and the powerful method of indexing points of flag varieties by Chevalley generators from representation theory.

On the values of unipotent characters of finite Chevalley groups of type E6 in characteristic 3

Let G be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of G by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on G. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where G is one of the groups E6(q), E62(q), and q is a power of the bad prime p=3 for E6, by exploiting known facts about the representation theory of the Hecke algebra associated with G.

The Highest Dimension of Commutative Subalgebras in Chevalley Algebras

Let L (K) denotes a Chevalley algebra with the root system over a field K. In 1945 A. I. Mal’cev investigated the problem of describing abelian subgroups of highest dimension in complex simple Lie groups. He solved this problem by transition to complex Lie algebras and by reduction to the problem of describing commutative subalgebras of highest dimension in the niltriangular subalgebra. Later these methods were modified and applied for the problem of describing large abelian subgroups in finite Chevalley groups. The main result of this article allows to calculate the highest dimension of commutative subalgebras in a Chevalley algebra L (K) over an arbitrary field

The generic character table of a Sylow p-subgroup of a finite Chevalley group of type D4

ABSTRACTLet U be a Sylow p-subgroup of the finite Chevalley group of type D4 over the field of q elements, where q is a power of a prime p. We describe a construction of the generic character table of U.

Decomposition of Tensor Products Involving a Steinberg Module

We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G r or a finite Chevalley group $G(\mathbb {F}_q)$ . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G r or $G(\mathbb {F}_q)$ . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin’s tilting conjecture.

A tensor product of the Steinberg module and a certain simple kG(pr)-module

ABSTRACTLet G be a connected, semisimple, and simply connected algebraic group defined and split over the finite field of order p, and let G(q) be the corresponding finite Chevalley or twisted group, where q = pr. Recently, Anwar determines the direct sum decomposition of the tensor product of the rth Steinberg module and a simple G-module with a (p,r)-minuscule highest weight λ. In this paper, we determine that of the tensor product regarded as a module for G(q) under some weak assumptions for λ.

Constructing characters of Sylow p-subgroups of finite Chevalley groups

Let q be a power of a prime p, let G be a finite Chevalley group over Fq and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F4, where we observe that the parametrization is “uniform” over good primes p>3, but differs for the bad prime p=3. We also explain how it has been applied for all groups of rank 4 or less.

Notes on the structure constants of Hecke algebras of induced representations of finite Chevalley groups

Notes on the structure constants of Hecke algebras of induced representations of finite Chevalley groups

Cohomology of classifying spaces of loop groups and finite Chevalley groups associated with spin groups

Let q be a power of an odd prime p and Fq be the finite field with q elements. For n≥3, we prove the mod 2 cohomology of the finite Chevalley group Spinn(Fq) is isomorphic to that of the classifying space of the loop group of the spin group Spin(n).

A generalization of Pillen's theorem for principal series modules II

Let G be a connected, semisimple and simply connected algebraic group and G(q) the corresponding finite Chevalley group over the finite field of order q=pr. In a recent paper the author determined a direct sum decomposition of the kG(q)-submodule generated by a highest weight vector of a certain Weyl module when q is not too small, which is a generalization of Pillen's result in 1997. In this article, we claim that the result does not need the assumption on q.

Rational points and orbits on the variety of elementary subalgebras

For G a connected, reductive group over an algebraically closed field k of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of g:=Lie(G) and the unipotent variety of G to study the projective variety of elementary subalgebras of g of rank r, denoted E(r,g). In the case that G is defined over Fp, we define the category of Fq-expressible subalgebras of g for q=pd, and prove that this category is isomorphic to a subcategory of Quillen's category of elementary abelian subgroups of the finite Chevalley group G(Fq). This isomorphism of categories leads to a correspondence between G-orbits of E(r,g) defined over Fq and G-conjugacy classes of certain elementary abelian subgroups of rank rd in G(Fq) which satisfy a closure property characterized by the Springer isomorphism. We use Magma to compute examples for G=GLn, n≤5.

Extensions for finite Chevalley groups III: Rational and generic cohomology

Let G be a connected reductive algebraic group and B be a Borel subgroup defined over an algebraically closed field of characteristic p>0. In this paper, the authors study the existence of generic G-cohomology and its stability with rational G-cohomology groups via the use of methods from the authors' earlier work. New results on the vanishing of G and B-cohomology groups are presented. Furthermore, vanishing ranges for the associated finite group cohomology of G(Fq) are established which generalize earlier work of Hiller, in addition to stability ranges for generic cohomology which improve on seminal work of Cline, Parshall, Scott and van der Kallen.

Conjugacy Classes in Sylow p-Subgroups of Finite Chevalley Groups in Bad Characteristic

Let U = U(q) be a Sylow p-subgroup of a finite Chevalley group G = G(q). Röhrle and Goodwin in 2009 determined a parameterization of the conjugacy classes of U, for G of small rank when q is a power of a good prime for G. As a consequence they verified that the number k(U) of conjugacy classes of U is given by a polynomial in q with integer coefficients. In the present paper, we consider the case when p is a bad prime for G. Our motivation is to observe how the situation differs between good and bad characteristics. We obtain a parameterization of the conjugacy classes of U, when G has rank less than or equal to 4, and G is not of type F 4. In these cases we deduce that k(U) is given by a polynomial in q with integer coefficients; this polynomial is different from the polynomial for good primes.

Calculating conjugacy classes in Sylow -subgroups of finite Chevalley groups of rank six and seven

AbstractLet$G(q)$be a finite Chevalley group, where$q$is a power of a good prime$p$, and let$U(q)$be a Sylow$p$-subgroup of$G(q)$. Then a generalized version of a conjecture of Higman asserts that the number$k(U(q))$of conjugacy classes in$U(q)$is given by a polynomial in$q$with integer coefficients. In [S. M. Goodwin and G. Röhrle,J. Algebra321 (2009) 3321–3334], the first and the third authors of the present paper developed an algorithm to calculate the values of$k(U(q))$. By implementing it into a computer program using$\mathsf{GAP}$, they were able to calculate$k(U(q))$for$G$of rank at most five, thereby proving that for these cases$k(U(q))$is given by a polynomial in$q$. In this paper we present some refinements and improvements of the algorithm that allow us to calculate the values of$k(U(q))$for finite Chevalley groups of rank six and seven, except$E_7$. We observe that$k(U(q))$is a polynomial, so that the generalized Higman conjecture holds for these groups. Moreover, if we write$k(U(q))$as a polynomial in$q-1$, then the coefficients are non-negative.Under the assumption that$k(U(q))$is a polynomial in$q-1$, we also give an explicit formula for the coefficients of$k(U(q))$of degrees zero, one and two.