Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8896273 | Journal of Algebra | 2018 | 33 Pages |
Abstract
Let OSp(V) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension (m|2n). Lehrer and Zhang showed that there is a surjective algebra homomorphism Frr:Br(mâ2n)âEndOSp(V)(Vâr), where Br(mâ2n) is the Brauer algebra of degree r with parameter mâ2n. The second fundamental theorem of invariant theory in this setting seeks to describe the kernel KerFrr of Frr as a 2-sided ideal of Br(mâ2n). In this paper, we show that KerFrrâ 0 if and only if râ¥rc:=(m+1)(n+1), and give a basis and a dimension formula for KerFrr. We show that KerFrr as a 2-sided ideal of Br(mâ2n) is generated by KerFrcrc for any râ¥rc, and we provide an explicit set of generators for KerFrcrc. These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra Endosp(V)(Vâr) over the orthosymplectic Lie superalgebra osp(V) to be isomorphic to Br(mâ2n).
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Yang Zhang,