Relative singular value decomposition and applications to LS-category

Abstract

Let $Sp(n)$ be the symplectic group of quaternionic $(n\times n)$-matrices. For any $1\le k\le n$, an element $A$ of $Sp(n)$ can be decomposed in $A=\left[\begin{array}{cc}\alpha & T\\ \beta & P\end{array}\right]$ with $P$ a $(k\times k)$-matrix. In this work, starting from a singular value decomposition of $P$, we obtain what we call a relative singular value decomposition of $A$. This feature is well adapted for the study of the quaternionic Stiefel manifold $X_{n,k}$, and we apply it to the determination of the Lusternik-Schnirelmann category of $Sp(k)$ in $X_{2k-j,k}$, for $j=0,1,2$

18 pages