relations: $$\begin{gathered}
\label{demreldes}(x\otimes {\mathbb C}[t])v_{\xi}=0,\ \ (h\otimes f) v_{\xi}= |\xi| f(0)v_{\xi},\ \  \  (y\otimes 1)^{|\xi|+1}v_{\xi}=0,\\ \label{demrel2a}  
(x\otimes t)^{(p)}(y\otimes 1)^{(r+p)}v_{\xi}=0,\ \ r,p\in {\mathbb N},\ r+p\geq 1+rk+\sum_{j\geq k+1}\xi_j\ \ \text{for some $k\in {\mathbb N}$.}\end{gathered}$$ It turns out that many other well–known families of representations belong to the class of fusion products. For example, Demazure modules occur in irreducible integrable representations of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and are parametrized by tuples $(m,n)\in {\mathbb N}\times {\mathbb Z}_+$, where the integer $m$ is called the level. We denote such a module by $D(m,n)$. If $n_0,n_1\in{\mathbb Z}_+$ are such that $n_0<m$ and $n=n_1m+n_0$, then the fusion product $V(\xi(m,n))$ associated to the partition $\xi(m,n):=(m^{n_1},n_0)$ is isomorphic to $D(m,n)$ (see [@CV13 Theorem 2]). Hence Demazure modules can be categorized into the family of fusion products, but the class of fusion products is generically much bigger. Nevertheless, there is a beautiful result saying that fusion products admit a filtration by Demazure modules under a suitable condition on the partition. The following proposition was proved in [@CSSW14 Theorem 3.3].

\[existflag\] Let $\xi=(\xi_1\geq \xi_2\geq \cdots \geq \xi_{\ell})$ a partition and $m\in{\mathbb N}$. The module $V(\xi)$ admits a filtration of level $m$, i.e., there exists a decreasing sequence of graded submodules 