Y-proper graded cocharacters of upper triangular matrices of order m graded by the m -tuple ϕ=(0,0,1,…,m−2)ϕ=(0,0,1,…,m−2)

Let F   be a field of characteristic 0. We consider the algebra UTm(F)UTm(F) of upper triangular matrices of order m   endowed with an elementary ZmZm-grading induced by the m  -tuple ϕ=(0,0,1,…,m−2)ϕ=(0,0,1,…,m−2), then we compute its Y  -proper graded cocharacter sequence and we give the explicit formulas for the multiplicities in the case m=2,3,4,5m=2,3,4,5.