We obtain inductive and enumerative formulas for the multiplicities of the weights of the spin module for the Clifford algebra of a Levi subalgebra in a complex semisimple Lie algebra. Our formulas involve only matrices and tableaux, and our techniques combine linear algebra, Lie theory, and combinatorics. Moreover, this suggests a relationship with complex nilpotent orbits. The case of the special linear Lie algebra $\mathfrak{sl}(n,{\mathbb C})$ is emphasized.

• 05A17 - Combinatorial aspects of partitions of integers
• 15A66 - Clifford algebras, spinors
• 15B30 - Matrix Lie algebras
• 17B10 - Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)