On $C$-bases, partition pairs and filtrations for induced or restricted Specht modules

We obtain alternative explicit Specht filtrations for the induced and the restricted Specht modules in the Hecke algebra of the symmetric group (defined over the ring $A=\mathbb{Z}[q^{1\mathrm{/}2},q^{-1\mathrm{/}2}]$ where $q$ is an indeterminate) using $C$-bases for these modules. Moreover, we provide a link between a certain $C$-basis for the induced Specht module and the notion of pairs of partitions.