entries in the algebra ${\mathbb C}[t]$ of polynomials with complex coefficients. The degree grading of ${\mathbb C}[t]$ defines a natural grading on ${\mathfrak{sl}}_2[t]$. A finite-dimensional ${\mathbb Z}$-graded ${\mathfrak{sl}}_2[t]$–module is a ${\mathbb Z}$–graded vector space admitting a compatible graded action of ${\mathfrak{sl}}_2[t]$: $$V=\bigoplus_{k\in{\mathbb Z}} V[k],\qquad (a\otimes t^r)V[k]\subset V[k+r]\ \  a\in {\mathfrak{sl}}_2,\ \ r\in{\mathbb Z}_+.$$ Given a ${\mathbb Z}$–graded space $V$ let $\tau_p^* V$ the graded vector space whose $r$–th graded piece is $V[r+p]$.

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The category of finite-dimensional ${\mathbb Z}$–graded ${\mathfrak{sl}}_2[t]$–modules was the central subject of many recent papers (see, for example [@BC15; @BCM12; @BCSV15; @CSSW14; @CV13; @FL99; @KL15]). There is a well–known family of objects in that category which is of particular interest, namely the subclass of fusion products. We recall their description in terms of generators and relations from [@CV13 Section 6]; for a more traditional definition we refer the reader to [@FL99] (see also Section \[551\]). Let $x,h,y$ be the standard basis of ${\mathfrak{sl}}_2$ and set $u^{(r)}:=\frac{1}{r!} u^r$ for $u\in {\mathfrak{sl}}_2[t],\hspace{0,03cm} r\in{\mathbb Z}_+$. For a partition $\xi=(\xi_1\geq \xi_2\geq \cdots \geq \xi_\ell)$ we set $$|\xi|_i:=\sum_{j=i}^{\ell}\xi_j,\ 1\leq i \leq \ell, \ \ |\xi|:=|\xi|_1.$$ The fusion product associated to $\xi$ is the ${\mathfrak{sl}}_2[t]$–module $V(\xi)$ generated by an element $v_{\xi}$ with defining 