An elementary proof of the first fundamental theorem of vector invariant theory

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 $r<(m+1)(n+1)$ .

Download Full-text

On Hilbert polynomial of certain determinantal ideals

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171291000157 ◽

1991 ◽

Vol 14 (1) ◽

pp. 155-162 ◽

Cited By ~ 2

Author(s):

Shrinivas G. Udpikar

Keyword(s):

Polynomial Ring ◽

Invariant Theory ◽

Fundamental Theorem ◽

Hilbert Function ◽

Hilbert Polynomial ◽

Determinantal Ideals ◽

The Ideal

LetX=(Xij)be anm(1)bym(2)matrix whose entriesXij,1≤i≤m(1),1≤j≤m(2); are indeterminates over a fieldK. LetK[X]be the polynomial ring in thesem(1)m(2)variables overK. A part of the second fundamental theorem of Invariant Theory says that the idealI[p+1]inK[X], generated by(p+1)by(p+1)minors ofXis prime. More generally in [1], Abhyankar defines an idealI[p+a]inK[X], generated by different size minors ofXand not only proves its primeness but also calculates the Hilbert function as well as the Hilbert polynomial of this ideal. The said Hilbert polynomial is completely determined by certain integer valued functionsFD(m,p,a). In this paper we prove some important properties of these integer valued functions.

Download Full-text

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

Communications in Mathematical Physics ◽

10.1007/s00220-016-2731-7 ◽

2016 ◽

Vol 349 (2) ◽

pp. 661-702 ◽

Cited By ~ 11

Author(s):

G. I. Lehrer ◽

R. B. Zhang

Keyword(s):

Invariant Theory ◽

Fundamental Theorem

Download Full-text

Invariant theory, tensors and group characters

Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences ◽

10.1098/rsta.1944.0001 ◽

1944 ◽

Vol 239 (807) ◽

pp. 305-365 ◽

Cited By ~ 70

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.

Download Full-text

The Cellular Second Fundamental Theorem of Invariant Theory for Classical Groups

International Mathematics Research Notices ◽

10.1093/imrn/rny079 ◽

2018 ◽

Vol 2020 (9) ◽

pp. 2626-2683

Author(s):

Christopher Bowman ◽

John Enyang ◽

Frederick M Goodman

Keyword(s):

Invariant Theory ◽

Fundamental Theorem ◽

Symmetric Groups ◽

Classical Groups ◽

Tensor Space ◽

Integral Bases ◽

Diagram Algebras ◽

Brauer Algebras

Abstract We construct explicit integral bases for the kernels and the images of diagram algebras (including the symmetric groups, orthogonal and symplectic Brauer algebras) acting on tensor space. We do this by providing an axiomatic framework for studying quotients of diagram algebras.

Download Full-text