On the Combinatorics of Young–capelli Symmetrizers

We deal with the characteristic zero theory of supersymmetric algebras, regarded as bimodules under the action of a pair of general linear Lie superalgebras, as developed by Brini et al. (see [Proc. Natl. Acad. Sci. USA 85 (1988), 1330–1333; Proc. Natl. Acad. Sci. USA 86 (1989), 775–778] for the first version, [Séminaire Lotharingien Combin. 55 (2007), Article B55g, 117 pp.] for the last version; see also Sergeev [Mat. Sb. (N.S.) 123(165) (1984), 422–430; Michigan Math. J. 49 (2001), 113–146] and Cheng and Wang [Compositio Math. 128 (2001), 55–94]). The theory had its roots in the pioneering work of Grosshans, Rota and Stein [Invariant theory and Superalgebras, Amer. Math. Soc., Providence, RI, 1987], Berele and Regev [Bull. Am. Math. Soc. 8 (1983), 337–339; Adv. Math. 64 (1987), 118–175] and Sergeev [Mat. Sb. (N.S.) 123(165) (1984), 422–430]. The basic objects of the theory, i.e., symmetrized bitableaux and Young–Capelli symmetrizers, are defined by means of a superalgebraic extension of Capelli’s method of virtual variables, and the relations between them are proved in the virtual setting, by means of a Triangularity Lemma, a Nondegeneracy Lemma, and the Superstraightening Law. We give a detailed exposition of the foundations of this theory. In doing this, we establish three new propositions on virtual expressions, and give new, elementary combinatorial proofs of the Triangularity Lemma and of the Nondegeneracy Lemma. With these proofs, we complete the process of giving the theory an elementary combinatorial foundation.