REVIEWS 79taken up in the exercises and historical notes.The notes are up to date.M. L. Wage's recent example of a normal space Z with Ind Z = 0, Z X …
REVIEWS 79taken up in the exercises and historical notes.The notes are up to date.M. L. Wage's recent example of a normal space Z with Ind Z = 0, Z X Z normal, and Ind(Z X Z) > 0 is mentioned as well as J. Walsh's infinite-dimensional compact metric space with no finite-dimensional subsets.(This is an improvement of D. Henderson's example which had no closed finite-dimensional subsets.)From this book the student gets a good idea where dimension theory stands today.The lack of research questions indicates that this area may have passed its most fruitful period of research.The few remaining questions don't hold much prospect of giving us significantly new insights.New theorems and interesting examples will continue to appear, but it is unlikely that anything will arise to alter our basic perceptions of this theory.As with most theories which reach this state of maturity new ideas simply cannot find a place in the old theory.They must begin their life as a new theory and require a new classification.Dimension theory is an excellent text giving us traditional dimension theory as it stands today.It presents all the essential features of interest to the general topologist without being compulsive.We have here a text that will probably be up to date for a considerable time.
Our object is to indicate how lrge classes of finite simple groups, specifically
Our object is to indicate how lrge classes of finite simple groups, specifically
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of …
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)
Our aim here is to study this property. The exclusion of the vector 0 is not essential in what follows. It turns out that many of the other properties of …
Our aim here is to study this property. The exclusion of the vector 0 is not essential in what follows. It turns out that many of the other properties of root systems of Lie algebras depend only on property P and are thus shared by a much wider class of vector systems. In the statements to follow it is assumed that all vectors considered come from a real Euclidean space Vof finite dimension n. 1.1 Let S be finite and have property P, and let A be a subset with property P. For each x in S, assume that at most one (resp. exactly one, at least one) of x and -x is in A. Then there is an ordering of the space V such that A is contained in (resp. A is, A contains) the set of positive elements of S. Harish-Chandra [6, Lemma 4] proves the second part of this result for Lie algebra root systems, however, using nontrivial properties of Lie algebras. Borel and Hirzebruch [2, pp. 471-473] give a geometric proof of the second and third parts, but then revert to Lie algebra techniques to prove the first part, all for Lie algebra root systems. All of these authors make a somewhat stronger assumption on A than property P, namely, if x, y e A and x + y e S, then x + y e A. Our proof of 1.1 depends on a preliminary result which may have some independent interest. 1.2. Let B be finite and have property P, and assume that x e B implies -x q B. Then ther-e is an ordering of the space V such that all elements of B are positive. The real numbers of the form k 112 (k, I positive integers) show that the assumption of finiteness in 1.1 (or 1.2) can not be dropped. It can, however, be weakened thus. 1.1'. In 1.1 replace the assumption of finiteness by the assumption that 0 is not a point of accumulation of S. In 1.2 we can go further, in terms of a weakening of property P.
The purpose of this note is to give a proof of the simplicity of certain ''Lie groups" considered in [2].The main feature of the present development is the proof of …
The purpose of this note is to give a proof of the simplicity of certain ''Lie groups" considered in [2].The main feature of the present development is the proof of Lemma 2 below: it is superior to the corresponding proof given in [2], because no assumption on the number of elements of the base field is required, and is very much shorter than the one given by Chevalley [1] for the direct analogues, over arbitrary fields, of the simple (complex) Lie groups.Thus it turns out that the groups E\{q 2 ) with q g 4, and Dl(q 3 ) with q ^ 3, to which the proof in (2) is not applicable, are simple.Assuming the notations of [1] and [2] to be in effect, we shall prove:
By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it …
By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.
Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a …
Let us define a reflection to be a unitary transformation, other than the identity, which leaves fixed, pointwise, a (reflecting) hyperplane, that is, a subspace of deficiency 1, and a reflection group to be a group generated by reflections. Chevalley (1) (and also Coxeter (2) together with Shephard and Todd (4)) has shown that a reflection group G, acting on a space of n dimensions, possesses a set of n algebraically independent (polynomial) invariants which form a polynomial basis for the set of all invariants of G.
Let { p, q, r } be the regular 4-dimensional poly tope for which each face is a { p, q } and each vertex figure is a { q, …
Let { p, q, r } be the regular 4-dimensional poly tope for which each face is a { p, q } and each vertex figure is a { q, r }, where { p, q }, for example, is the regular polyhedron with p-gonal faces, q at each vertex. A Petrie polygon of { p, q } is a skew polygon made up of edges of { p, q } such that every two consecutive sides belong to the same face, but no three consecutive sides do. Then a Petrie polygon of { p, g, r } is defined by the property that every three consecutive sides belong to a Petrie polygon of a bounding {p, q}, but no four do. Let h Pqr be the number of sides of such a polygon, and g p,q,r the order of the group of symmetries of {p, g, r}.
The aim of this paper is two-fold: first, to extend the results of (4) to the exceptional finite Lie groups recently discovered by Chevalley (1), and, secondly, to give a …
The aim of this paper is two-fold: first, to extend the results of (4) to the exceptional finite Lie groups recently discovered by Chevalley (1), and, secondly, to give a construction which works simultaneously for the groups A n , B n , C n , D n , E n , F 4 and G 2 (in the usual Lie group notation), and which depends only on intrinsic structural properties of these groups.
1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each …
1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each group acting on a vector space of a finite number of elements (2; 3).
This paper is a result of an investigation into general methods of determining the irreducible characters of GL( n, q ), the group of all non-singular linear substitutions with marks …
This paper is a result of an investigation into general methods of determining the irreducible characters of GL( n, q ), the group of all non-singular linear substitutions with marks in GF( q ), and of the related groups, SL( n, q ), PGL( n, q ), PSL( n, q ), the corresponding group of determinant unity, projective group, projective group of determinant unity, respectively.
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).
By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).
The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrar
The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrar
Article Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. was published on January 1, 1907 in the journal Journal für die reine und angewandte Mathematik (volume 1907, …
Article Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. was published on January 1, 1907 in the journal Journal für die reine und angewandte Mathematik (volume 1907, issue 132).
Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes …
Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of …
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)
1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each …
1. Introduction. There are five well-known, two-parameter families of simple finite groups: the unimodular projective group, the symplectic group,1 the unitary group,2 and the first and second orthogonal groups, each group acting on a vector space of a finite number of elements (2; 3).
Ce travail apporte quelques renseignements sur la torsion du groupe de cohomologie entiere d'un groupe de Lie compact connexe G, que nous appellerons, suivant Γusage, la torsion de G, sur …
Ce travail apporte quelques renseignements sur la torsion du groupe de cohomologie entiere d'un groupe de Lie compact connexe G, que nous appellerons, suivant Γusage, la torsion de G, sur les sous-groupes commutatifs de G, et met ces deux questions en relation.Nous nous interesserons en particulier aux rapports qui existent entre la ^-torsion (p nombre premier) et les sous-groupes commutatifs de type (p, ,p), que nous appellerons ici les [^>]-sous-groupes.Ce travail a ete resume dans une Note au Bull.Amer.Math.Soc.66 (I960),.pp.285-288.En nous appuyant sur quelques remarques concernant les ί/-espaces dont la cohomologie entiere est de type fini, faites dans le §1, et sur le Theoreme V de [14], on verra que le groupe simplement connexe exceptionnel E έ n'a pas de ^-torsion lorsque p = 5, i = 6, 7 et p = 7, i = 7, 8, et Ton determinera aussi if* (E 6 ; Z 3 ), H* (E 8 ; Z 5 ), (Theor.2.2,2.3).Compte tenu de resultats connus [5,8,14], on pourra alors indiquer les nombres premiers intervenant dans les coefficients de torsion de tous les groupes simples et simplement connexes (2.5) et Γon verifiera (2.6) le theoreme suivant, conjecture dans [7]: THEOREME A. Supposons G, simple et simplement connexe, et soit p un nombre premier ne divisant pas les coefficients de la plus grande racine de G, exprimee comme somme de racines simples.Alors G na pas de p-torsion.Dans un groupe de Lie compact connexe G il existe des [/>]-sous-groupes evidents, les elements d'ordre p d'un tore maximal, mais il peut y en avoir d'autres, (pour p = 2, les matrices diagonales de SO (n), (n > 3) par exemple).Cependant, d'apres [9, XII,5,3, 5.4], si G n'a pas de p-torsion, tout [/>]-sousgroupe // fait partie d'un tore, ce qui precise un resultat de [11] affirmant que,.sous Γhypothese faite, le rang de H est au plus egal a la dimension des tores maximaux de G. Cette derniere condition est evidemment necessaire pour que H fasse partie d'un tore de G, mais elle n'est pas suffisante en general.En effet, on demontrera: (i) Pour que le groupe fundamental TΓ^G) de G soit sans p-torsion, il faut et il suffit que tout [/>]-sous-groupe de rang deux soit contenu dans un
0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring …
0.Introduction. 1.Let G be a group of linear transformations on a finite dimensional real or complex vector space X.Assume X is completely reducible as a G-module.Let 5 be the ring of all complexvalued polynomials on X, regarded as a G-module in the obvious way, and let JC5 be the subring of all G-invariant polynomials on X.Now let J + be the set of all ƒ £ J having zero constant term and let HQS be any graded subspace such that S=J + S+H is a G-module direct sum.It is then easy to see that
By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it …
By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.
The purpose of this note is to give a proof of the simplicity of certain ''Lie groups" considered in [2].The main feature of the present development is the proof of …
The purpose of this note is to give a proof of the simplicity of certain ''Lie groups" considered in [2].The main feature of the present development is the proof of Lemma 2 below: it is superior to the corresponding proof given in [2], because no assumption on the number of elements of the base field is required, and is very much shorter than the one given by Chevalley [1] for the direct analogues, over arbitrary fields, of the simple (complex) Lie groups.Thus it turns out that the groups E\{q 2 ) with q g 4, and Dl(q 3 ) with q ^ 3, to which the proof in (2) is not applicable, are simple.Assuming the notations of [1] and [2] to be in effect, we shall prove:
We will be concerned in this paper with the irreducible characters of the Weyl groups of type E6, E7, and E8. Character tables for these groups have been given by …
We will be concerned in this paper with the irreducible characters of the Weyl groups of type E6, E7, and E8. Character tables for these groups have been given by J. S. Frame [6, 7]. In all cases, the characters are rationalvalued. It is fairly easy to show that each character has a real splitting field. The object of this paper is to show that each character has a rational representation. Other authors have shown that the same results hold for other irreducible Weyl groups. A. Young [13] studied the families of groups of type A, and B, as permutation groups. He devised a method for constructing all representations and showed that they could be constructed over the rational field Q. W. Specht [11] did the same for the family of groups of type D. This result is easily shown for the group of type G2, which is dihedral of order 12. T. Kondo [9] presented a character table for the group of type F4 and showed that each character has a rational representation. Our present results complete the proof that any representation of a Weyl group of a finite dimensional semi-simple complex Lie algebra is equivalent to a rational representation. Our methods are character-theory methods and we restate our results in terms of the Schur index. Let X be an irreducible character of a group G and let F be a field. The Schur index m,(X) of X over F is the smallest positive integer m such that mX is a character afforded by an F(X)-representation of G.
where the A{ are rational integers, is called an integral linear recurrence of order k. Given such a linear recurrence and an integer c, one would like to know for …
where the A{ are rational integers, is called an integral linear recurrence of order k. Given such a linear recurrence and an integer c, one would like to know for what n does f(n) — c? In a very few particular instances (e.g. see [2], [6]) this question has been answered, but in general the question is very difficult. A less exacting problem is the determination of upper and lower bounds on the number, M{c), of distinct n for which f(n) = c. We shall call M(c) the multiplicity of c in the recurrence. Much work has been done by C. L. Siegel [4], K. Mahler [3], Morgan Ward [9], [10], [11] and others concerning the multiplicity of 0 and the pattern of the appearance of 0 in the recurrence. Quite often from information on the multiplicity of 0 in one recurrence one can infer a bound on the multiplicity of all integers in another recurrence. However, as much of the information available concerning the zeros of a recurrence is for recurrences satisfying special conditions on the Ai9 one cannot always ascertain in this way whether M(c) is bounded. Define the multiplicity of a recurrence as the least upper bound of the M(c), as c ranges over the integers; and say that the multiplicity of the recurrence is strictly infinite if for some integer c, M(c) is infinite. We are interested in examining the following questions: ( I ) When is the multiplicity of a recurrence finite? When infinite? (II) If the multiplicity of a recurrence is finite, what is it or at least what is an upper bound for it? (III) Can the multiplicity of a recurrence be infinite and not strictly infinite ? Here, we confine our attention to recurrences of order 2. In the direction of the above questions, there is a conjecture that for a recurrence of order 2 either the multiplicity is strictly infinite or it is bounded above by 5. We are unable to resolve this conjecture, but we do obtain reasonably satisfactory answers to the questions for all recurrences of order 2 having (A19 A2) = 1.