Remarks on the Capelli identities for reducible modules

2019 ◽  
Author(s):  
Tôru Umeda
Keyword(s):  
Capelli Identities

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 Rank ideals and Capelli identities

2004 ◽  
Vol 273 (2) ◽  
pp. 686-699 ◽  
Author(s):  
Victor Protsak
Keyword(s):  
Capelli Identities

ALGEBRAS SATISFYING CAPELLI IDENTITIES

1982 ◽  
Vol 18 (1) ◽  
pp. 125-144 ◽  
Author(s):  
Ju P Razmyslov
Keyword(s):  
Capelli Identities

Capelli identities and representations of finite type

1994 ◽  
Vol 22 (14) ◽  
pp. 5733-5744 ◽  
Author(s):  
P. Yu Razmyslov ◽  
K.A. Zuhrilin
Keyword(s):  
Finite Type ◽  
Capelli Identities

scholarly journals On the Proof of the Capelli Identities

2008 ◽  
Vol 51 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Tôru Umeda
Keyword(s):  
Capelli Identities
Sign in / Sign up

Export Citation Format

Share Document