  We study the cohomology of the free loop space of $SU(n+1)/T^n$, the simplest
example of a complete flag manifolds and an important homogeneous space.
Through this enhanced analysis we reveal rich new combinatorial structures
arising in the cohomology algebra of the free loop spaces. We build new theory
to allow for the computation of $H^*(\Lambda(SU(n+1)/T^{n});\mathbb{Z})$, a
significantly more complicated structure than other known examples. In addition
to our theoretical results, we explicitly implement a novel integral Gr\"obner
basis procedure for computation. This procedure is applicable to any
Leray-Serre spectral sequence for which the cohomology of the base space is the
quotient of a finitely generated polynomial algebra. The power of this
procedure is illustrated by the explicit calculation of $H^*(
\Lambda(SU(4)/T^3);\mathbb{Z})$. We also provide a python library with examples
of all procedures used in the paper.
