# Relative singular value decomposition and applications to LS-category

## TL;DR

This paper introduces a relative singular value decomposition for quaternionic symplectic matrices and applies it to determine the Lusternik-Schnirelmann category of certain symplectic groups within quaternionic Stiefel manifolds.

## Contribution

It develops a new relative singular value decomposition technique tailored for quaternionic symplectic matrices and uses it to compute LS-category in specific geometric contexts.

## Key findings

- Derived a new relative singular value decomposition for $Sp(n)$ matrices.
- Applied the decomposition to compute LS-category of $Sp(k)$ in certain Stiefel manifolds.
- Provided explicit LS-category values for cases $j=0,1,2$.

## Abstract

Let $Sp(n)$ be the symplectic group of quaternionic $(n\times n)$-matrices. For any $1\leq k\leq n$, an element $A$ of $Sp(n)$ can be decomposed in $A= \begin{bmatrix} \alpha&T\cr \beta&P \end{bmatrix}$ 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 $j= 0,\,1,\,2$