  In this paper we study the category of graded modules for the current algebra
associated to $\mathfrak{sl}_2$. The category enjoys many nice properties,
including a tilting theory which was established in previous work of the
authors. We show that the indecomposable tilting modules for
$\mathfrak{sl}_2[t]$ are the exterior powers of the fundamental global Weyl
module and give the filtration multiplicities in the standard and costandard
filtration. An interesting consequence of our result (which is far from obvious
from the abstract definition) is that an indecomposable tilting module admits a
free right action of the ring of symmetric polynomials in finitely many
variables. Moreover, if we go modulo the augmentation ideal in this ring, the
resulting $\mathfrak{sl}_2[t]$-module is isomorphic to the dual of a local Weyl
module.
