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
2019 ◽  
pp. 1-25 ◽  
Author(s):  
G. I. LEHRER ◽  
R. B. ZHANG

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


1991 ◽  
Vol 14 (1) ◽  
pp. 155-162 ◽  
Author(s):  
Shrinivas G. Udpikar

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.


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.


2018 ◽  
Vol 2020 (9) ◽  
pp. 2626-2683
Author(s):  
Christopher Bowman ◽  
John Enyang ◽  
Frederick M Goodman

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.


2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Martin Rubey ◽  
Bruce W. Westbury

International audience An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant theory states that all invariants are expressible in terms of a finite number among them, whereas a second main theorem determines the relations between those basic invariants.Here we present a transparent, combinatorial proof of a second fundamental theorem for the defining representation of the symplectic group $Sp(2n)$. Our formulation is completely explicit and provides a very precise link to $(n+1)$-noncrossing perfect matchings, going beyond a dimension count. As a corollary, we obtain an instance of the cyclic sieving phenomenon. Un problème important de la théorie des invariantes est de décrire le sous espace d’une puissance tensorielle d’une représentation invariant à l’action du groupe. Suivant la classique de Weyl, le théorème fondamental premier pour la représentation standard du groupe sympléctique dit que tous les invariants peuvent être exprimés entre un nombre fini d’entre eux. Par ailleurs, un théorème fondamental second détermine les relations entre ces invariants basiques.Ici, nous présentons une preuve transparente d’un théorème fondamental second pour la représentation standard du groupe sympléctique $Sp(2n)$. Notre formulation est complètement explicite et elle fournit un lien très précis avec les couplages parfaits $(n+1)$ -noncroissants, plus précis qu’un dénombrement de la dimension. Comme corollaire nous exhibons un phénomène de crible cyclique.


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