Proof of the orthogonal - pin duality

In a previous article in this journal (Bulgarian Journal of Physics v. 48 (2021) p. 390-399) [1], I cited a theorem of dual Fock space representations of an orthogonal Lie algebra and a Pin group, but the space limits of that article as a conference contribution did not allow to include the proof. The present paper serves to render the proof, which is found so far only in a preprint [2], accessible in a journal. For the context of the theorem, which is Theorem 2 below, see [1, 3]. The proof is given in Section 6. Sections 2 - 5 provide necessary preliminaries. I thus specify my terminology in Section 2 and define the fermion Fock space in Section 3. In Section 4, I review the constructions on this space of a number conserving and a commuting number non-conserving representation of orthogonal Lie algebras [4], and in Section 5, I extend the number non-conserving representation to a representation of a Pin group as defined by Atiyah, Bott and Shapiro [5]. Section 7 extends the discussion in [1] by relating my result to contemporary work on Fock space dualities, and Section 8 provides a summary. (author)