Intersecting principal Bruhat ideals and grades of simple modules

We prove that the grades of simple modules indexed by boolean permutations, over incidence algebra symmetric group with respect to Bruhat order, are given Lusztig's $\mathbf{a}$-function. Our arguments combinatorial, and include a description intersection two principal order ideals when at least one permutation is boolean. An important object in our work reduced word written as minimally many runs consecutive integers, step argument shows this minimal quantity equal length second row permutation's shape under Robinson-Schensted correspondence. also module above-mentioned perfect if only its index longest element parabolic subgroup.Mathematics Subject Classifications: 20F55, 06A07, 05E15