Chevalley plocal finite groups carles broto and jesper m. Generation of relative commutator subgroups in chevalley. Property t for kacmoody groups over rings introduction. Theorem 1 for the root systems b2 and g2 is obtained in the. Diameters of chevalley groups over local rings springerlink advertisement. The goal of this series is to systematically develop methods of calculations in chevalley groups over rings, based on the use of their minimal modules.
Pdf nonabelian ktheory for chevalley groups over rings. Roughly speaking, an iwahori subgroup of an algebraic group gk, for a local field k with integers o and residue field k, is the inverse image in go of a borel subgroup of gk. The isomorphisms of linear groups over local rings, acta math. Apr 01, 2003 classical chevalley groups fall into this category and it follows from the results of 8 that k 1 are nilpotent for chevalley groups of types c l and d l. Thus, this branch of the theory of chevalley groups over rings suslins localglobal principle, the normality of elementary. Chevalley groups of polynomial rings over dedekind domains. Relative subgroups in chevalley groups journal of ktheory. Many papers have been devoted to the description of automorphisms of chevalley groups over di. The approaches are complementary, or, to put it plainly, antagonistic. Gauss decomposition for chevalley groups indian statistical institute. Let g be the universal chevalley demazure group scheme corresponding to a reduced irreducible root system of rank at least 2, and let r be a commutative ring. Steinberg characters for chevalley groups over finite local rings. On normal subgroups of chevalley groups over commutative rings vaserstein, leonid n.
Introduction given a simple lie algebra, one can study its automorphisms. Nov 17, 2012 moreover, we give an efficient algorithm for computing a short path between any two vertices in any cayley graph of the groups g n. Chevalley groups over commutative rings stein, michael r. Generators, relations and coverings of chevalley groups over commutative rings. We prove a uniform polylogarithmic diameter bound of the cayley graphs of gn with respect to arbitrary sets of generators. Relations and coverings of chevalley groups over commutative rings. In his many papers, vavilov works with the points of a chevalleydemazure group scheme but also with the usually smaller elementary chevalley group. A commutative ring with a unit that has a unique maximal ideal.
A new look at the decomposition of unipotents and the normal. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. This is the first in a series of papers dedicated to the structure of chevalley groups over commutative rings. Nonabelian ktheory for chevalley groups over rings.
If is a local ring with maximal ideal, then the quotient ring is a field, called the residue field of examples of local rings. Faithful representations of chevalley groups over quotient rings of nonarchimedean local fields mohammad bardestani, camelia karimianpour, keivan mallahikarai, and hadi salmasian abstract. Chevalley groups, elementary chevalley groups, triangular factorisations, rings of stable rank 1, parabolic subgroups, gauss decomposition, commutator width. Armand borel, properties and linear representations of chevalley groups. Chevalley groups we begin by outlining the construction of the chevalley groups over a finite local ring. Concerning notation, generally lkwill be an extension of number elds, with rings of integers o l and o k, and efwill be an extension of local elds. Structure of chevalley groups over commutative rings. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The set of primes in k that split completely in lis spllk. The whole theory of chevalley groups over semilocal rings rests upon the following analogue of. K1 of chevalley groups are nilpotent request pdf researchgate. Minimal permutation representations of finite groups.
Moreover, we give an efficient algorithm for computing a short path between any two vertices in any cayley graph of the groups g n. Property t for kacmoody groups over rings to e m zelmanov. Calculations in exceptional groups over rings mathnet. Follow ing 3 they are defined as groups of automorphisms of lie rings corresponding to nonabelian semisimple lie algebras over the complex numbers. Automorphismsofchevalleygroups oftypes a,d,e overlocal. Nakano complexity for modules over finite chevalley groups and classical lie algebras, invent. Chevalleys theorem asserts that every algebraic group over a perfect eld is built up from a linear algebraic group and an abelian variety in a way we will make precise shortly. Minimal dimension of faithful representations for pgroups, with keivan mallahikarai and hadi salmasian. Every chevalley group g is a zmodel of its split connected reductive. Abe, e chevalley groups over local rings, tohoku math. Let f be a nonarchimedean local eld with the ring of integers o and the prime ideal p and let g g adopn be the adjoint chevalley group. On abstract homomorphisms of chevalley groups over the. Some twisted variants of chevalley groups steinbach, anja, bulletin of the belgian mathematical society simon stevin, 2005.
Relative subgroups in chevalley groups journal of k. Automorphisms of chevalley groups of type f4 over local. Better still, over fields and semilocal rings there are canonical forms such as bruhat. On linear representations of chevalley groups over. Tang the fundamental theorem of projective geometry over some commutative rings, j. Abe, chevalley groups over local rings, tohoku mathematical journal 21 1969, no. Chevalley groups over local rings abe, eiichi, tohoku mathematical journal, 1969. In this paper, we prove that every automorphism of a chevalley group of type b l, l. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal. On parabolic subgroups of chevalley groups over local rings suzuki, kazuo, tohoku mathematical journal, 1976 the duality operation in the character ring of a finite chevalley group alvis, dean, bulletin new series of the american mathematical society, 1979. General theory of saturation and of saturated local rings ii. The concept of local rings was introduced by wolfgang krull in 1938 under the name stellenringe.
Apr 15, 2010 over local rings universal chevalley groups coincide with their elementary subgroups, consequently the chevalley group gr is also an elementary chevalley group. The main result is that if a is a semilocal ring generated by its units, the groups l, a. Groups play an important role nearly in every part of mathematics and can be used to study the symmetries of a mathematical object. Elementary equivalence of chevalley groups over local rings.
Plotkin developed a geometric approach towards calculations in exceptional groups at the level of k1, decomposition of unipotents. Eudml on some bruhat decomposition and the structure of the. Hijikata, maximal invariant orders of an involutive algebra over a local field, to appear. Research papers by zongzhu lin kansas state university. Faithful representations of chevalley groups over quotient rings of nonarchimedean local fields, with camelia karimianpour, keivan mallahikarai and hadi salmasian. Chevalley groups elias weber january 07, 2010 supervised by dr claude marion and prof donna testerman abstract this project deals with the construction of the chevalley group and the studies of its structure and properties through the analysis of its relevant subgroups. Iwahori, on the structure of hecke rings associated to general linear groups over padic fields, to appear. Abstract it is proved that elementary chevalley groups over local rings with invertible 2 are elementarily equivalent if and only if their types and weight lattices coincide and the initial rings are elementarily equivalent. For example, the proof of the n eronoggshafarevich criterion for good reduction of. Li canonical basis for algebras arising from quivers with loops. Theorem 1 for the root systems a l, d l, and e l was obtained by the author in 5, in 7 all auto morphisms of chevalley groups of given types over local rings with 12 were. Property t for kacmoody groups over rings 3 another interesting result on property t for kacmoody groups was obtained by hartnick and k ohl hk who proved that for any local eld f and any indecomposable 2spherical d dgeneralized cartan matrix a, with d 2, the group g af has property.
We also deduce some corollaries of the above result for regular rings r of higher dimension. Research papers postscript or pdf files joint with y. We also deduce some corollaries of the above result for regular rings r of higher. In the given paper we prove that every automorphism of a chevalley group of type f4 over a commutative local ring with 12 is standard, i. Over local rings universal chevalley groups coincide with their elementary subgroups, consequently the chevalley group gr is also an elementary chevalley group. Automorphisms and isomorphisms of chevalley groups of type g2. This is the case, for example, when the ground ring is a eld, or, more generally, a semilocal ring m. Automorphisms of chevalley groups of type b l over local. Namely, we describe in all details two new approaches in the study of exceptional chevalley groups over rings, sketched in 146 and 7. We describe automorphisms of chevalley groups of type bl over local rings with 12.
On the adjoint quotient of chevalley groups over arbitrary base. Steinberg characters for chevalley groups over finite. This paper continues the investigation of generators and relations for chevalley groups over commutative rings initiated in ll4j. Let r be a dedekind domain, and let g be a split reductive group, i. In fact 8 establishes much more general results, namely that certain slightly larger k 1 functors are nilpotentbyabelian for a huge class of unitary groups over form rings. Thus, by minkowskis theorem that every number eld k 6 q has a rami ed prime, specz has no nontrivial connected nite etale covers. Chevalley groups we begin by outlining the construction of the chevalley groups over a.
On the other hand, the polynomial ring with is not local. Publication date 1946 topics rmsc collection digitallibraryindia. The duality operation in the character ring of a finite chevalley group alvis, dean, bulletin new series of the american mathematical society, 1979. The ring of formal power series over a field or over any local ring is local. Aug 20, 2010 in this paper, we prove that every automorphism of a chevalley group of type b l, l. Eudml on some bruhat decomposition and the structure of. Sp department of mathematics, university of toronto. Structure of chevalley groups over rings via universal localization. Ams proceedings of the american mathematical society. Special linear groups are simply connected chevalley groups of type a l. Diameters of chevalley groups over local rings oren dinai abstract. Armand borel, properties and linear representations of chevalley groups, seminar on algebraic groups and related finite groups the institute for advanced study, princeton, n. Classical chevalley groups fall into this category and it follows from the results of 8 that k 1 are nilpotent for chevalley groups of types c l and d l. Pdf the article contains a survey of the recent authors results on the structure of a chevalley group gr over a ring r.
Diameters of chevalley groups over local rings springerlink. Elementary calculations article pdf available in acta applicandae mathematicae 451 may 1999 with 161 reads how we measure reads. In the present paper, which is a sequel to our paper 52, we show that a slight modi cation of. K 1 of chevalley groups are nilpotent, journal of pure and. The isomorphisms of symplectic groups over local rings, acta math. Automorphisms of chevalley groups of type f4 over local rings.