Since the establishment of the quantum Schur–Weyl duality in Jimbo (Lett. Math. Phys. 11, 247–252, 1986), the duality pair \((\textbf{U}(\mathfrak {gl}_n),\varvec{\mathcal {H}}(\mathfrak S_r))\) of type A has been extended to the duality pairs \((\textbf{U}^\jmath (n),\varvec{\mathcal {H}}(B_r))\) and \((\textbf{U}^\imath (n),\varvec{\mathcal {H}}(C_r))\) in the Hecke algebra series in Bao and Wang (Astérisque 402, vii+134, 2018), where \(\textbf{U}^\jmath (n),\textbf{U}^\imath (n)\) are i-quantum groups arising from certain quantum symmetric pairs. The quantum Schur algebra associated with the pair \((\textbf{U}(\mathfrak {gl}_n),\varvec{\mathcal {H}}(\mathfrak S_r))\) has a nice and simple presentation; see Doty and Giaquinto (2002). In this paper, we tackle the presentation problem for the i-quantum Schur algebras associated with the duality pair \((\textbf{U}^\imath (n),\varvec{\mathcal {H}}(C_r))\) . Such a q-Schur algebra is called the hyperoctahedral q-Schur algebras in Green (J. Algebra 192, 418–438, 1997). See Bhattacharya (2026) for the \(\textbf{U}^\jmath (n)\) case. Building on the explicit epimorphism \(\phi _{n,r}^\imath \) from the i-quantum group \(\textbf{U}^\imath (n)\) to the hyperoctahedral q-Schur algebras \(\mathcal {S}^\imath (n,r)\) (see Du and Wu, Pacific J. Math. 320(1), 61–101, 2022), we compute the kernel of \(\phi _{n,r}^\imath \) in terms of generators. This results in a presentation for \(\mathcal {S}^\imath (n,r)\) with defining relations which include not only the Doty–Giaquinto’s diagonal relations but also some tridiagonal relations.
