Computing Multiplicities in the Sign Trace Cocharacters of M 2,1(F)

In Regev (Linear Multilinear Algebra 21:1–28, 1987), Regev applied the representation theory of the general Lie superalgebra to generalize the theory of trace identities as developed by Procesi and Razmyslov. Regev showed that certain cocharacters arising from sign trace identities were given by $$\displaystyle \sum _{\lambda \in H(k,l;n)} \chi _\lambda \otimes \chi _\lambda $$ where χλ ⊗ χλ denotes the Kronecker product of the irreducible character of the symmetric group associated with the partition λ with itself and H(k, l;n) denotes the set of partitions of nλ = (λ1 ≥ λ2 ≥… ≥ λn) such that λk+1 ≤ l. In case of k = 2, l = 1, we show how to compute some multiplicities which occur in the expansion of the cocharacter in terms of irreducible characters by using the reduced notation Scharf et al. (J Phys A Math Gen 26:7461–7478, 1993).