Diamond Cone for $${{\mathfrak {sl}}}(m,n)$$

In this paper, we study the shape and reduced algebra for Lie superalgebra $${\mathfrak {sl}}(m,n)$$ . We define quasistandard Young tableaux, they give a combinatorial basis algebra, called diamond cone. There is natural bijective map from set of semistandard tableaux with $$\lambda $$ to smaller shape: cone compatible stratification algebra.