Capelli's theory, Koszul maps, and superalgebras

Area of Science:

• Lie superalgebras
• Representation theory
• Algebraic combinatorics

Background:

• The study focuses on the enveloping algebra Ukappa(pl(L)) of the general linear Lie superalgebra.
• Understanding the structure and bases of such algebras is crucial in representation theory.

Purpose of the Study:

• To investigate the straightening laws for the enveloping algebra Ukappa(pl(L)).
• To introduce and analyze an isomorphism (Psi) between the supersymmetric algebra Super[L L] and Ukappa(pl(L)).
• To establish a connection between bitableaux in Super[L L] and Young-Capelli bitableaux in Ukappa(pl(L)).

Main Methods:

• Introduction of an isomorphism Psi mapping bitableaux to Young-Capelli bitableaux.
• Demonstration that Psi is the inverse of Koszul's isomorphism.
• Identification of a basis for Ukappa(pl(L)) using costandard determinantal Young-Capelli bitableaux.

Main Results:

• An isomorphism Psi is established between the supersymmetric algebra Super[L L] and the enveloping algebra Ukappa(pl(L)).
• Psi maps bitableaux to Young-Capelli bitableaux, preserving parametrization by Young diagrams.
• The set of costandard determinantal Young-Capelli bitableaux forms a basis for Ukappa(pl(L)).

Conclusions:

• The established isomorphism provides a new perspective on the structure of Ukappa(pl(L)).
• The identified basis exhibits a triangular action on the standard permanental bitableaux basis of Super[L L].
• This work deepens the understanding of algebraic structures in Lie superalgebra representation theory.