Vertical Versus Diagonal Dimensional Reduction for $p$-branes

In addition to the double-dimensional reduction procedure that employs world-volume Killing symmetries of $p$-brane supergravity solutions and acts diagonally on a plot of $p$ versus spacetime dimension $D$, there exists a second procedure of ``vertical'' reduction. This reduces the transverse-space dimension via an integral that superposes solutions to the underlying Laplace equation. We show that vertical reduction is also closely related to the recently-introduced notion of intersecting $p$-branes. We illustrate this with examples, and also construct a new $D=11$ solution describing four intersecting membranes, which preserves $1/16$ of the supersymmetry. Given the two reduction schemes plus duality transformations at special points of the scalar modulus space, one may relate most of the $p$-brane solutions of relevance to superstring theory. We argue that the maximum classifying duality symmetry for this purpose is the Weyl group of the corresponding Cremmer-Julia supergravity symmetry $E_{r(+r)}$. We also discuss a separate class of duality-invariant $p$-branes with $p=D-3$.