Gradings on the algebra of upper triangular matrices of size three

Abstract Let UT 3 ( F ) be the algebra of 3 × 3 upper triangular matrices over a field F . On UT 3 ( F ) , up to isomorphism, there are at most five non-trivial elementary gradings and we study the graded polynomial identities for such gradings. In case F is of characteristic zero we give a complete description of the space of multilinear graded identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . We finally compute the multiplicities in the graded cocharacter sequence for every elementary G -grading on UT 3 ( F ) .

Ordinary and graded cocharacter of the Jordan algebra of 2x2 upper triangular matrices

Abstract Let F be a field of characteristic zero and U J 2 ( F ) be the Jordan algebra of 2 × 2 upper triangular matrices over F . In this paper we give a complete description of the space of multilinear graded and ordinary identities in the language of Young diagrams through the representation theory of a Young subgroup of S n . For every Z 2 -grading of U J 2 ( F ) we compute the multiplicities in the graded cocharacter sequence and furthermore we compute the ordinary cocharacter.

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

Abstract Let F be a field of characteristic 0. We consider the algebra UT m ( F ) of upper triangular matrices of order m endowed with an elementary Z m -grading induced by the m-tuple 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 , 5 .

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

Abstract Let F be a field of characteristic 0. We consider the upper triangular matrices with entries in F of size 2, 3 and 4 endowed with the grading induced by that of Vasilovsky. In this paper we give explicit computation for the multiplicities of the Y -proper graded cocharacters and codimensions of these algebras.

Group graded algebras and multiplicities bounded by a constant

AbstractLet G be a finite group and A a G-graded algebra over a field of characteristic zero. When A is a PI-algebra, the graded codimensions of A are exponentially bounded and one can study the corresponding graded cocharacters via the representation theory of products of symmetric groups. Here we characterize in two different ways when the corresponding multiplicities are bounded by a constant.

Storia e didattica delle equazioni di secondo grado: Un caso di studio

Il presente lavoro riguarda lo studio delle equazioni di secondo grado da un punto di vista storico e didattico e si intende rivolto ai docenti e agli alunni di una classe seconda di un liceo scientifico. Si precisa che l’evoluzione storica può indicare le graduali tappe che devono essere percorse nell’itinerario didattico, per poter avviare gli allievi alla disciplina considerata e quindi, nel caso trattato, all’uso del simbolismo algebrico proprio delle equazioni di secondo grado e ad un processo di astrazione sempre più massiccio. Dal punto di vista dello studente, invece, lo studio dell’evoluzione storica di un concetto può essergli di stimolo per fargli comprendere come ogni concetto m…

On Graded Cocharacters and Corresponding Multiplicities