scholarly journals Partition algebras Pk(n) with 2k>n and the fundamental theorems of invariant theory for the symmetric group Sn

2018 ◽  
Vol 99 (1) ◽  
pp. 194-224
Author(s):  
Georgia Benkart ◽  
Tom Halverson
Keyword(s):  
Symmetric Group ◽  
Invariant Theory ◽  
Partition Algebras ◽  
Fundamental Theorems

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

On the Disjoint Product of Irreducible Representations of the Symmetric Group

1949 ◽  
Vol 1 (2) ◽  
pp. 166-175 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Symmetric Group ◽  
Linear Group ◽  
Invariant Theory ◽  
Schur Functions ◽  
Irreducible Representations ◽  
Invariant Matrices ◽  
Disjoint Product

The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).

scholarly journals Spin invariant theory for the symmetric group

2011 ◽  
Vol 215 (7) ◽  
pp. 1569-1581 ◽  
Author(s):  
Jinkui Wan ◽  
Weiqiang Wang
Keyword(s):  
Symmetric Group ◽  
Invariant Theory

scholarly journals Ringel Duals of Brauer Algebras via Super Groups

2019 ◽  
Author(s):  
Kevin Coulembier
Keyword(s):  
Invariant Theory ◽  
Positive Characteristic ◽  
General Linear ◽  
Complex Numbers ◽  
Brauer Algebra ◽  
Algebra For All ◽  
Walled Brauer Algebra ◽  
Brauer Algebras ◽  
Algebraic Proofs ◽  
Fundamental Theorems

Abstract We prove that the Brauer algebra, for all parameters for which it is quasi-hereditary, is Ringel dual to a category of representations of the orthosymplectic super group. As a consequence we obtain new and algebraic proofs for some results on the fundamental theorems of invariant theory for this super group over the complex numbers and also extend them to some cases in positive characteristic. Our methods also apply to the walled Brauer algebra in which case we obtain a duality with the general linear super group, with similar applications.

On a Theorem of Osima and Nagao

1954 ◽  
Vol 6 ◽  
pp. 125-127 ◽  
Author(s):  
J. S. Frame ◽  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
Fundamental Theorems

If we define the weight b of a Young diagram containing n nodes to be the number of removable p-hooks where n = a + bp, then three fundamental theorems stand out in the modular representation theory of the symmetric group Sn.

scholarly journals FREE QUANTUM ANALOGUES OF THE FIRST FUNDAMENTAL THEOREMS OF INVARIANT THEORY

2004 ◽  
Vol 47 (2) ◽  
pp. 297-303
Author(s):  
Julien Bichon
Keyword(s):  
Linear Group ◽  
Invariant Theory ◽  
Hopf Algebras ◽  
General Linear Group ◽  
Polynomial Function ◽  
Free Algebras ◽  
Function Algebras ◽  
Primary 16W30 ◽  
Fundamental Theorems

AbstractWe formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial function algebras on matrices are replaced by free algebras, while the universal cosovereign Hopf algebras play the role of the general linear group.AMS 2000 Mathematics subject classification: Primary 16W30

scholarly journals First fundamental theorems of invariant theory for quantum supergroups

2019 ◽  
Vol 6 (3) ◽  
pp. 928-976
Author(s):  
Gustav I. Lehrer ◽  
Hechun Zhang ◽  
Ruibin Zhang
Keyword(s):  
Invariant Theory ◽  
Fundamental Theorems
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