Universal characters from the MacDonald identities

The main result of [Kostant, Invent. Math. 158 (1) (2004) 181–226, arXiv:math.GR/0309232] shows that a sequence of complexes associated with the MacDonald identities gives a sequence of universal characters which extends and generalises some representations of the exceptional series given in [Deligne, C. R. Acad. Sci. Paris Sér. Math. 322(4) (1996) 321–326]. This proves the conjectures in [Macfarlane, Pfeiffer, J. Phys. A 36 (2003) 230 5–2317. arxiv.math-ph/0208014].In this paper, we use a twisted version of this definition to define a sequence of complexes for the first row of the Freudenthal magic square.We also calculate the dimensions of these characters for the special linear groups and for the symplectic groups. This gives two new explicit expressions for the D’Arcais polynomials.