J. Korean Math. Soc. 2005; 42(2): 365-386
Printed March 1, 2005
Copyright © The Korean Mathematical Society.
Jae-Hoon Kwon
University of Seoul
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
2019; 56(5): 1265-1283
2017; 54(1): 227-248
1997; 34(3): 673-693
2000; 37(1): 55-72
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd