  The Bernstein-Sato polynomial, or the $b$-function, is an important invariant
of singularities of hypersurfaces that is difficult to compute in general. We
describe a few different results towards computing the $b$-function of the
Vandermonde determinant $\xi$.
  We use a result of Opdam to produce a lower bound for the $b$-function of
$\xi$. This bound proves a conjecture of Budur, Musta\c{t}\u{a}, and Teitler
for the case of finite Coxeter hyperplane arrangements, proving the Strong
Monodromy Conjecture in this case.
  In our second set of results, we show the duality of two
$\mathcal{D}$-modules, and conclude that the roots of the $b$-function of $\xi$
are symmetric about $-1$. We then use some results about jumping coefficients
to prove an upper bound for the $b$-function of $\xi$, and finally we
conjecture a formula for the $b$-function of $\xi$.
