Spectral sequences

The algebra generated by $\{\theta^i_{j}, R^i_j\}$ is closed under the differential $d$, so we have a subcomplex

$$(\bR\{\theta^i_{j}, R^i_j\}, d) =: (\widetilde{W_n}, d)\subset (C^{\bullet}(\gA_n), d).$$

where

$$\bR\{\theta^i_{j}, R^i_j\}\isom \Lambda^{\bullet}\ggl_n(\bR)^*\tensor S_n(\ggl_n(\bR)^*)$$

The proof uses Hochschild-Serre spectral sequence, which we describe next.