Alternating and symmetric superpowers of metric generalized Jordan superpairs
Diego Aranda-Orna, Alejandra S. Córdova-Martínez
TL;DR
The paper develops a systematic framework to define and analyze alternating and symmetric (super)powers of metric generalized Jordan superpairs by transferring classical tensor-power constructions from Lie supermodules through the Faulkner correspondence. It introduces both general and restricted tensor superproducts/powers, provides explicit formulas for bilinear forms and triple products in the MGJSP setting, and establishes automorphism-structure results (with kernels isomorphic to $\mathrm{SL}_n$ or $\boldsymbol{\mu}_n$ depending on parity). It also gives concrete decompositional insights by relating simple Jordan pairs of types II and III to exterior/symmetric squares of type I, up to tensor shifts and similarities. Collectively, these results unify and extend exterior and symmetric power constructions to the super Jordan framework, enabling Schur–Weyl-type dualities and broader representation-theoretic applications in superalgebraic contexts.
Abstract
The aim of this paper is to define and study the constructions of alternating and symmetric (super)powers of metric generalized Jordan (super)pairs. These constructions are obtained by transference via the Faulkner construction. The construction of tensor (super)products for metric generalized Jordan (super)pairs is revisited. We always assume that the characteristic of the base field $\mathbb{F}$ is different from $2$; in case of positive characteristic, sometimes we require that the characteristic is large enough to allow nondegeneracy of certain bilinear forms.