Matching polytopes and Specht modules

The (fractional) matching polytope of a graph G is the space of all assignments of nonnegative weights to the edges of G such that the sum of the weights of the edges incident to any vertex is at most 1. We will first describe some recurrences for calculating the volume of this polytope. Then, in the case when G is a tree, we will provide a combinatorial interpretation of this volume via certain representations of the symmetric group called Specht modules. Finally, we will interpret this result from the viewpoint of symmetric functions and the geometry of the Grassmannian.