  We report measurements of the $^{115}$In $7p_{1/2}$ and $7p_{3/2}$ scalar and
tensor polarizabilities using two-step diode laser spectroscopy in an atomic
beam. The scalar polarizabilities are one to two orders of magnitude larger
than for lower lying indium states due to the close proximity of the $7p$ and
$6d$ states. For the scalar polarizabilities, we find values (in atomic units)
of $1.811(4) \times 10^5$ $a_0^3$ and $2.876(6) \times 10^5$ $a_0^3$ for the
$7p_{1/2}$ and $7p_{3/2}$ states respectively. We estimate the smaller tensor
polarizability component of the $7p_{3/2}$ state to be $-1.43(18) \times 10^4$
$a_0^3$. These measurements represent the first high-precision benchmarks of
transition properties of such high excited states of trivalent atomic systems.
We also present new ab initio calculations of these quantities and other In
polarizabilities using two high-precision relativistic methods to make a global
comparison of the accuracies of the two approaches. The precision of the
experiment is sufficient to differentiate between the two theoretical methods
as well as to allow precise determination of the indium $7p-6d$ matrix
elements. The results obtained in this work are applicable to other heavier and
more complicated systems, and provide much needed guidance for the development
of even more precise theoretical approaches.
