A cellular algebra with certain idempotent decomposition

For a cellular algebra $\text{A}$ with a cellular basis $\text{ZC}$, we consider a decomposition of the unit element $1_{\text{A}}$ into orthogonal idempotents (not necessary primitive) satisfying some conditions. By using this decomposition, the cellular basis $\text{ZC}$ can be partitioned into some pieces with good properties. Then by using a certain map $\text{a}$, we give a coarse partition of $\text{ZC}$ whose refinement is the original partition. We construct a Levi type subalgebra $\text{aA}$ of $\text{A}$ and its quotient algebra $\text{oA}$, and also construct a parabolic type subalgebra $\text{tA}$ of $\text{A}$, which contains $\text{aA}$ with respect to the map $\text{a}$. Then, we study the relation of standard modules, simple modules and decomposition numbers among these algebras. Finally, we study the relationship of blocks among these algebras.