Speaker
Description
Let $\widehat{\mathfrak g}$ be an affine Lie algebra of type $C_\ell^{(1)}$, $\ell\geq2$. In [CMPP] it is conjectured that for every integrable highest weight $\widehat{\mathfrak g}$-module $L(k_0,\dots,k_\ell)$ there is a Rogers-Ramanujan type combinatorial identity, that is, the infinite product $\prod_{k_0,\dots,k_\ell}(q)$ obtained from the Weyl-Kac character formula is the generating function for $(k_0,\dots,k_\ell)$-admissible partitions $\pi$ with parts in the array $\mathcal N_{2\ell+1}$ consisting of $w=2\ell+1$ rows of
odd, even, $\dots$, odd, even, odd $\quad$ positive integers,
and satisfying certain initial and difference conditions. By erasing the last row in $\mathcal N_{2\ell+1}$ we obtain the array $\mathcal N_{2\ell}$ consisting of $w=2\ell$ rows, and partitions $\pi'$ with parts in $\mathcal N_{2\ell}$, and with the inherited initial and difference conditions, have (conjecturally) the generating function obtained by erasing some factors in $\prod_{k_0,\dots,k_\ell}(q)$, but with no obvious connection to representation theory.
By erasing further the last row in $\mathcal N_{2\ell}$ one obtains the array $\mathcal N_{2\ell-1}$ consisting of $w=2\ell-1$ rows, and by erasing further some factors in $\prod_{k_0,\dots,k_\ell}(q)$, we are back to the $C_{\ell-1}^{(1)}$ case. However, if we erase the first row in $\mathcal N_{2\ell}$, we obtain the array $\mathcal N_{2\ell-1}^\text{odd}$ for which the generating functions of admissible partitions get to be very strange.
