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By embedding the Hecke algebra Hq of type D into the Hecke algebra Hₐ, ₁ of type B with unequal parameters (q, 1), the q-Schur algebras Sq (n, r) of type D is naturally defined as the endomorphism algebra of the tensor space with the Hq-action restricted from the Hₐ, ₁-action that defines the (q, 1) -Schur algebra S^ₐ, ₁ (n, r) of type B. We investigate the algebras S^ₐ, ₁ (n, r) and Sq (n, r) both algebraically and geometrically and describe their standard bases, dimension formulas and weight idempotents. Most importantly, we use the geometrically derived two sets of the fundamental multiplication formulas in S^ₐ, ₁ (n, r) to derive multi-sets (9 sets in total!) of the fundamental multiplication formulas in Sq (n, r).
Du et al. (Thu,) studied this question.