Journal of the
Korean Mathematical Society JKMS

Let $\mathscr{L}$ be the free Lie superalgebra generated by a $\mathbb{Z}_2$-graded vector space $V$ over $\mathbb{C}$. Suppose that $\frak{g}$ is a Lie superalgebra $\frak{gl}(m,n)$ or $\mathfrak{q}(n)$. We study the $\frak{g}$-module structure on the $k$th homogeneous component $\mathscr{L}_k$ of $\mathscr{L}$ when $V$ is the natural representation of $\frak{g}$. We give the multiplicities of irreducible representations of $\frak{g}$ in $\mathscr{L}_k$ by using the character of $\mathscr{L}_k$. The multiplicities are given in terms of the character values of irreducible (projective) representations of the symmetric groups.

Keywords: free lie superalgebra, representation, character, general linear Lie superalgebra

MSC numbers: Primary 17B01, 17B10; Secondary 05E05