Presenting mirabolic quantum Schur algebras ${\mathcal{M}\mathcal{S}}_v(2,d)$

Let $v$ be an indeterminate. We give a presentation of the mirabolic quantum Schur algebra ${\mathcal{M}\mathcal{S}}_v(2,d)$ in terms of generators and defining relations. More precisely, we determine the kernel of the natural epimorphism from Rosso’s mirabolic quantum ${\mathfrak{s}\mathfrak{l}}_2$, ${\mathbf{U}}_v^{\mathrm{m}\mathrm{i}\mathrm{r}}(2)=⟨e,f,k^{\pm 1},\mathrm{\ell }⟩$, onto ${\mathcal{M}\mathcal{S}}_v(2,d)$. Setting $P_d(k)={\prod }_{r=0}^d(k-v^{2r-d})$ and $Q_d(k)={\prod }_{r=1}^d(k-v^{2r-d})$, we prove ${\mathcal{M}\mathcal{S}}_v(2,d)\cong {\mathbf{U}}_v^{\mathrm{m}\mathrm{i}\mathrm{r}}(2)\mathrm{/}⟨P_d(k),(\mathrm{\ell }-1)Q_d(k)⟩$. We further determine the split Wedderburn decomposition of this quotient and provide an equivalent presentation in terms of weight idempotents.