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Let 
$\mathbf {G}$ be a connected reductive algebraic group over an algebraically closed field 
$\Bbbk $ and 
${\mathbf B}$ be a Borel subgroup of 
${\mathbf G}$. In this paper, we completely determine the composition factors of the permutation module 
$\mathbb {F}[{\mathbf G}/{\mathbf B}]$ for any field 
$\mathbb {F}$.
Similarly to the Frobenius–Schur indicator of irreducible characters, we consider higher Frobenius–Schur indicators
$\nu _{p^n}(\chi ) = |G|^{-1} \sum _{g \in G} \chi (g^{p^n})$
for primes p and
$n \in \mathbb {N}$
, where G is a finite group and
$\chi $
is a generalised character of G. These invariants give answers to interesting questions in representation theory. In particular, we give several characterisations of groups via higher Frobenius–Schur indicators.
