$GL_n$-decomposition of the Schur complex $S_r%\wedge^2 \varphi)$

Bull. Korean Math. Soc. 2003 Vol. 40, No. 1, 29-51 Published online March 1, 2003

Eun J. Choi, Young H. Kim, Hyoung J. Ko, and Seoung J. Won Yonsei University, Yonsei University, Yonsei University, Yonsei University

Abstract : In this paper we construct a natural filtration associated to the plethysm $S_r(\wedge^2 \varphi)$ over arbitrary commutative ring $R$. Let $\varphi :~G\longrightarrow F$ be a morphism of finite free $R-$modules. We construct the natural filtration of $S_r (\wedge^2 \varphi )$ as a $GL(F)\times GL(G)-$ complex such that its associated graded complex is $\sum_{\lambda \in \Omega_r ^-} L_{2\lambda} \varphi $, where $\Omega_r^-$ is a set of partitions such that $|\lambda|= r $ and $2\lambda$ is a partition of which $i$-th term is $2\lambda_i$. Specializing our result, we obtain the filtrations of $S_r (\wedge^2 F)$ and $ D_r (D_2 G)$.