Multiplication Formulas Of Quantum (?)-Schur Superalgebras

This thesis mainly generalizes the geometric realization of quantum Schur algebras of type A due to Beilinson-Lusztig-Mac Pherson to the quantum (?)-Schur superalgebra of type B/C.In the framework of Hecke algebra,using the algebraic method of combina-torics of Weyl groups,some ordered and compact multiplication formulas of quantum(?)-Schur superalgebras are derived by studying the properties of specific double cosets.Furthermore,We give an application of these multiplication formulas —semisimplicity criteria of quantum (?)-Schur superalgebras.The thesis is divided into four parts.The first part briefly introduces the properties of Weyl group,(super)parabolic group and double coset.The second part introduces the quantum (?)-Schur superalgebra,which is the automorphism algebra of a direct sum of permutation modules of the type B/C Hecke algebra,and the quantum (?)-Schur duality.The third part gives specific multiplication formulas in quantum (?)-Schur superalgebra.The fourth part uses these multiplication formulas to give the semisimplicity criteria of quantum (?)-Schur superalgebras.