Y-proper graded cocharacters and codimensions of upper triangular matrices of size 2, 3, 4

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

“…We want to mention that there is a simple formula (in the spirit of Proposition 7.4 below) which allows to calculate the graded cocharacter sequence knowing the Y -proper graded cocharacters. We finish with a comparison between the experimental results obtained in [5] and we observe that the multiplicities obtained when dealing with the grading induced by ψ are asymptotically greater than in the case the grading is induced by φ.…”

“…In [5] the authors studied the Y -proper graded cocharacters of U T m (F ) equipped with the grading studied by Di Vincenzo and Vasilovsky. We recall that the latter grading over U T m (F ) is a Z m -grading induced by the m-tuple ψ = (0, 1, .…”

“…We start with the following combinatorial lemmas whose proofs repeat word by word those of Lemmas 4.1 and 4.2 of [5].…”

“…Recently in [1] Boumova and Drensky gave an easy algorithm which calculates the generating function of the cocharacter sequence of U T m (F ). In [5] Centrone and Cirrito gave a combinatorial method in order to compute the exact value of the Y -proper graded cocharacter sequence of U T m (F ) endowed with the grading induced by that studied by Di Vincenzo in [8] and by Vasilovsky in [23] and they showed the exact value of the multiplicities for upper-triangular matrices of small size. We recall that the latter grading is an elementary grading over U T m (F ) induced by the m-tuple ψ = (0, 1, .…”

“…, m − 2) and we ask what happens to the Y -proper graded cocharacter sequence. As well as in [5] we give a combinatorial method in order to compute the Y -proper cocharacter sequence of U T m (F ). The paper is organized as follows: the first section is introductory and furnishes all the tools for the study of graded polynomial identities and related structures.…”