} = {\bf  32_2} \oplus {\bf  32_0} \oplus {\bf  32_{-2}} \nonumber \\
& {\bf  (220,2)} = {\bf  220_1} \oplus {\bf  220_{-1}} \\
& {\bf  (12,2)} = {\bf  12_1} \oplus {\bf  12_{-1}} \quad ,\nonumber\end{aligned}$$ where the subscript denotes the $\mathbb{R}^+$ weight. The representation ${\bf  32_2}$ corresponds to the RR fluxes, the ${\bf  220_1}$ corresponds to the NS-NS fluxes and the $ {\bf  352_0}$ corresponds to the $P$ fluxes. This can be seen by decomposing each of the $SO(6,6)$ representations in terms of $GL(6,\mathbb{R})$, which we do in detail now.

It is straightforward to see how this decomposition works for the case of the RR fluxes. The ${\bf 32}$ is the spinorial representation $\theta_\alpha$ of $SO(6,6)$, and according to the convention that one chooses for the chirality of this spinor, one has the two possible decompositions $$\theta_\alpha  \rightarrow \left\{ \begin{array}{ll} F_a \ \  F_{abc} \ \  F_{abcde} \ \  \ \ \  \ \ \ \ \ \ ({\rm IIB})
        \\
        \\
         F \ \  F_{ab} \ \  F_{abcd}  \ \ F_{abcdef} \ \ \ ({\rm IIA})\end{array} \right.   \quad , \label{decompositionofRRfluxes}$$ corresponding to the RR fluxes of odd rank in the IIB theory and of even rank in the IIA theory. Obviously this representation contains only geometric fluxes. The T-duality rule given in eq. maps a given flux in one theory to a flux in the other theory. All the components in eq. are connected by chains of T-duality transformations.

The next representation is the ${{\bf 220 }}$, which is the representation $\theta_{MNP}$ of $SO(6,6)$ with three antisymmetrised vector indices. This corresponds to the NS-NS fluxes, and indeed the embedding tensor decomposes under $GL(6,\mathbb{R})$ as $$\theta_{MNP} \ \rightarrow \ H_{abc} \quad f_{ab}^c \quad Q_{a}^{bc} \quad R^{abc} \quad .\label{decompositionofNSfluxes}$$ The T-duality rule given in eq. connects the different components in the equation above, but in this case not all the components can be reached by chains of T-dualities starting for instance from a 