Extended T-systems, Q matrices and T-Q relations for $s\ell(2)$ models at roots of unity

The mutually commuting $1\times n$ fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity $q=e^{i\lambda}$ with crossing parameter $\lambda=\frac{(p'-p)\pi}{p'}$ a rational fraction of $\pi$. The $1\times n$ transfer matrices of the dense loop model analogs, namely the logarithmic minimal models ${\cal LM}(p,p')$, are similarly considered. For these $s\ell(2)$ models, we find explicit closure relations for the $T$-system functional equations and obtain extended sets of bilinear $T$-system identities. We also define extended $Q$ matrices as linear combinations of the fused transfer matrices and obtain extended matrix $T$-$Q$ relations. These results hold for diagonal twisted boundary conditions on the cylinder as well as $U_q(s\ell(2))$ invariant/Kac vacuum and off-diagonal/Robin vacuum boundary conditions on the strip. Using our extended $T$-system and extended $T$-$Q$ relations for eigenvalues, we deduce the usual scalar Baxter $T$-$Q$ relation and the Bazhanov-Mangazeev decomposition of the fused transfer matrices $T^{n}(u+\lambda)$ and $D^{n}(u+\lambda)$, at fusion level $n=p'-1$, in terms of the product $Q^+(u)Q^-(u)$ or $Q(u)^2$. It follows that the zeros of $T^{p'-1}(u+\lambda)$ and $D^{p'-1}(u+\lambda)$ are comprised of the Bethe roots and complete $p'$ strings. We also clarify the formal observations of Pronko and Yang-Nepomechie-Zhang and establish, under favourable conditions, the existence of an infinite fusion limit $n\to\infty$ in the auxiliary space of the fused transfer matrices. Despite this connection, the infinite-dimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations.