scholarly journals Invariant theory, tensors and group characters

1944 ◽  
Vol 239 (807) ◽  
pp. 305-365 ◽  
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem ◽  
Orthogonal Group ◽  
Linearly Independent ◽  
Full Development ◽  
Symbolic Method ◽  
The Subject ◽  
Algebraic Forms ◽  
Quadratic Complex ◽  
Groups Of Transformations

In Part I is developed the theory of the tensor as a device for the construction of concomitants. This part includes the specific separation of a complete cogredient tensor of rank r into simple tensors, with a formula indicating the corresponding separation of a mixed tensor; also the corresponding theory in tensors to the Glebsch theory of algebraic forms, and a compact proof of the fundamental theorem that all concomitants under the full linear group can be obtained by the multiplication and contraction of tensors. The general equivalence is demonstrated, so far as elementary applications are concerned, of the method of tensors with the classical symbolic method of invariant theory. The first part forms a foundation for the principal theory of the paper which is developed in Part II. This primarily consists of an analysis of the properties of S -functions which provides methods for predicting the exact number of linearly independent concomitants of each type, of a given set of ground forms. Complementary to this, a method of substitutional analysis based on the tableaux which must be constructed in obtaining a product of S -functions, enables the specific concomitants of each type to be written down. Part III consists of applications to the classical problems of invariant theory. For ternary perpetuants a generating function is obtained which is not only simpler than that given by Young, but is also more general, in so far as it indicates, as well as the covariants, also the mixed concomitants. Extension is made to any number of variables. The complete sets of concomitants, up to degree 5 or 6 m the coefficients, are obtained for the cubic, quartic and quadratic complex in any number of variables. Alternating concomitant types are described and enumerated. A theorem of conjugates is proved which associates the concomitants of one ground form with the concomitants of a ground form of a different type, namely, that which corresponds to the conjugate partition. Some indication is made of the extension of this theory to invariants under restricted groups of transformations, e.g. the orthogonal group, but the full development of this extended theory is to be the subject of another paper.

scholarly journals On invariant theory under restricted groups

1944 ◽  
Vol 239 (809) ◽  
pp. 387-417 ◽  
Keyword(s):  
Invariant Theory ◽  
Orthogonal Group ◽  
Symplectic Group ◽  
Transformation Groups ◽  
Characteristic Analysis ◽  
Linear Complex ◽  
Group Integration ◽  
Group A ◽  
Quadratic Complex

A previous paper, ‘Invariant theory, tensors and group characters’, dealt with invariant theory under the full linear group. In this paper the methods are extended to restricted groups of transformation such as the orthogonal group. A knowledge of the characters of the group is shown to be an essential preliminary to any adequate study of invariants under the group. The characters of the orthogonal and symplectic groups, previously obtained by Schur and Weyl by transcendental methods involving group integration, are here obtained by methods entirely algebraic. Concerning transformation groups with a system of fundamental tensors, a fundamental theorem is proved that every concomitant may be obtained by multiplication and contraction of ground-form tensors, tensor variables, fundamental tensors and the alternating tensor. A characteristic analysis is developed, involving the operation denoted by ®, which enables the numbers and types of the concomitants of any given degree in any system of ground forms to be predicted. The determination of the actual concomitants is also discussed. Application is made for the orthogonal group to the quadratic, the ternary cubic and the quaternary quadratic complex; for the ternary symplectic group, to the quadratic, the linear complex and the quadratic complex. Various applications are also made for intransitive and imprimitive groups of transformation.

scholarly journals ON THE VOLUME CONJECTURE FOR CLASSICAL SPIN NETWORKS

2012 ◽  
Vol 21 (03) ◽  
pp. 1250022 ◽  
Author(s):  
ABDELMALEK ABDESSELAM
Keyword(s):  
Invariant Theory ◽  
Classical Literature ◽  
Spin Networks ◽  
Classical Invariant Theory ◽  
Classical Spin ◽  
Volume Conjecture ◽  
Symbolic Method ◽  
Exact Determination ◽  
Classical Invariant

We prove an upper bound for the evaluation of all classical SU2 spin networks conjectured by Garoufalidis and van der Veen. This implies one half of the analogue of the volume conjecture which they proposed for classical spin networks. We are also able to obtain the other half, namely, an exact determination of the spectral radius, for the special class of generalized drum graphs. Our proof uses a version of Feynman diagram calculus which we developed as a tool for the interpretation of the symbolic method of classical invariant theory, in a manner which is rigorous yet true to the spirit of the classical literature.

scholarly journals The first fundamental theorem of invariant theory for the orthosymplectic super group

2018 ◽  
Vol 327 ◽  
pp. 4-24 ◽  
Author(s):  
P. Deligne ◽  
G.I. Lehrer ◽  
R.B. Zhang
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem

scholarly journals THE SECOND FUNDAMENTAL THEOREM OF INVARIANT THEORY FOR THE ORTHOSYMPLECTIC SUPERGROUP

2019 ◽  
pp. 1-25 ◽  
Author(s):  
G. I. LEHRER ◽  
R. B. ZHANG
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorem ◽  
Brauer Algebra ◽  
Tensor Space ◽  
General Linear Supergroup ◽  
Special Cases ◽  
Linear Description ◽  
Tensor Representations ◽  
Capelli Identities ◽  
Fundamental Theorems

The first fundamental theorem of invariant theory for the orthosymplectic supergroup scheme $\text{OSp}(m|2n)$ states that there is a full functor from the Brauer category with parameter $m-2n$ to the category of tensor representations of $\text{OSp}(m|2n)$ . This has recently been proved using algebraic supergeometry to relate the problem to the invariant theory of the general linear supergroup. In this work, we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. Specifically, we give a linear description of the kernel of the surjective homomorphism from the Brauer algebra to endomorphisms of tensor space, which commute with the orthosymplectic supergroup. The main result has a clear and succinct formulation in terms of Brauer diagrams. Our proof includes, as special cases, new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues, which are independent of the Capelli identities. The results of this paper have led to the result that the map from the Brauer algebra ${\mathcal{B}}_{r}(m-2n)$ to endomorphisms of $V^{\otimes r}$ is an isomorphism if and only if $r<(m+1)(n+1)$ .

scholarly journals A short survey on observability

2019 ◽  
Vol 53 (supl) ◽  
pp. 143-183
Author(s):  
Walter Ferrer Santos
Keyword(s):  
Invariant Theory ◽  
Intermediate Step ◽  
Geometric Invariant ◽  
Related Concept ◽  
Algebraic Subgroup ◽  
Tensor Categories ◽  
Current Survey ◽  
The Subject ◽  
The Rich ◽  
New Applications

The exploration of the notion of observability exhibits transparently the rich interplay between algebraic and geometric ideas in geometric invariant theory. The concept of observable subgroup was introduced in the early 1960s with the purpose of studying extensions of representations from an afine algebraic subgroup to the whole group. The extent of its importance in representation and invariant theory in particular for Hilbert's 14th problem was noticed almost immediately. An important strenghtening appeared in the mid 1970s when the concept of strong observability was introduced and it was shown that the notion of observability can be understood as an intermediate step in the notion of reductivity (or semisimplicity), when adequately generalized. More recently starting in 2010, the concept of observable subgroup was expanded to include the concept of observable action of an afine algebraic group on an afine variety, launching a series of new applications and opening a surge of very interesting activity. In another direction around 2006, the related concept of observable adjunction was introduced, and its application to module categories over tensor categories was noticed. In the current survey, we follow (approximately) the historical development of the subject introducing along the way, the definitions and some of the main results including some of the proofs. For the unproven parts, precise references are mentioned.

scholarly journals Symbolic method in invariant theory

1986 ◽  
Vol 83 (4) ◽  
pp. 844-847 ◽  
Author(s):  
G.-C. Rota ◽  
J. A. Stein
Keyword(s):  
Invariant Theory ◽  
Symbolic Method

Early Colonial Constitutions

1896 ◽  
Vol 10 ◽  
pp. 59-83
Author(s):  
J. P. Wallis
Keyword(s):  
The United States ◽  
Constitutional History ◽  
American Colonies ◽  
Full Development ◽  
Early Colonial ◽  
Mother Country ◽  
Colonial Empire ◽  
History Of ◽  
The Subject ◽  
Colonial Government

Our own constitutional history has been studied, and is still being studied, with all the thoroughness and patient care demanded by so great a subject. On the other hand, the early constitutional history of our colonies has attracted less attention than it deserves. Probably this is because the shadow of a great failure hangs over it; if alluded to at all, it is only to point a moral at the folly which lost the American colonies. And yet a system which, with all its shortcomings, played so large a part in preparing the way for the constitution of the United States, and for our own colonial empire as it at present exists, is certainly worthy of investigation. English colonial government may be said to have been founded in the American colonies, but not to have attained its full development, if it has yet done so, until after their separation. It is, therefore, not surprising that such attention as the subject has recently received should have been devoted to it by American rather than by English writers. I propose to-day to deal with the early forms of government established in the colonies, and to point out how some of them, as in Massachusetts, were calculated from the outset to make for separation, while others, as in Virginia, lent themselves readily to maintaining the connection with the mother country, and are still used in our existing colonies.

VIII.—The Invariant Theory of the Quaternary Quadratic Complex. I. The Prepared System

1929 ◽  
Vol 48 ◽  
pp. 70-91
Author(s):  
H. W. Turnbull
Keyword(s):  
Differential Geometry ◽  
Quadratic Form ◽  
Invariant Theory ◽  
Complex I ◽  
Quadratic Complex ◽  
Image Position ◽  
Quaternary Quadratic Form ◽  
Associated Variables

Projective and differential geometry are in close touch at two places, once because of the fundamental rôle played by a quaternary quadratic form in each,and again through the quadratic in six associated variables,where

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