  Hermite reciprocity refers to a series of natural isomorphisms involving
compositions of symmetric, exterior, and divided powers of the standard
$SL_2$-representation. We survey several equivalent constructions of these
isomorphisms, as well as their recent applications to Green's Conjecture on
syzygies of canonical curves. The most geometric approach to Hermite
reciprocity is based on an idea of Voisin to realize certain multilinear
constructions cohomologically by working on a Hilbert scheme of points. We
explain how in the case of ${\bf P}^1$ this can be reformulated in terms of
cohomological properties of Schwarzenberger bundles. We then proceed to study
these bundles from several perspectives:
  We show that their exterior powers have supernatural cohomology, arising as
special cases of a construction of Eisenbud and Schreyer.
  We recover basic properties of secant varieties $\Sigma$ of rational normal
curves (normality, Cohen-Macaulayness, rational singularities) by considering
their desingularizations via Schwarzenberger bundles, and applying the
Kempf-Weyman geometric technique.
  We show that Hermite reciprocity is equivalent to the self-duality of the
unique rank one Ulrich module on the affine cone $\widehat{\Sigma}$ of some
secant variety, and we explain how for a Schwarzenberger bundle of rank $k$ and
degree $d\ge k$, Hermite reciprocity can be viewed as the unique (up to
scaling) non-zero section of $(Sym^k\mathcal{E})(-d+k-1)$.

Algebraic Geometry;Commutative Algebra;Representation Theory