Ordinary and ℤ₂-Graded Cocharacters ofUT₂(E)

Abstract

Let E be the infinite dimensional Grassmann algebra over a field F of characteristic 0. In this article we consider the algebra R of 2 × 2 matrices with entries in E and its subalgebra G, which is one of the minimal algebras of polynominal identity (PI) exponent 2. We compute firstly the Hilbert series of G and, as a consequence, we compute its cocharacter sequence. Then we find the Hilbert series of R, using the tool of proper Hilbert series, and we compute its cocharacter sequence. Finally we describe explicitely the 2 -graded cocharacters of R.

Algebraic properties of codimension series of PI-algebras

“…Partial cases of the corollary follow from other results, e.g. Drensky [7] and Petrogradsky [22] for U k (K) and U k (E), Stoyanova-Venkova [32] and Centrone [5] for the T-ideal generated by [x 1 , x 2 , x 3 ][x 4 , x 5 ].…”

Computing with rational symmetric functions and applications to invariant theory and PI-algebras

“…The multiplicities of U 2 (E) were determined by Centrone [21]. In both cases the results were obtained using the Young rule only, without the MacMahon partition analysis.…”

Cocharacters of $UT_n(E)$

“…The explicit form of the multiplicities in the cocharacter sequence of a PI-algebra is known for few cases. Among them are the infinite dimensional Grassmann algebra E (Olsson and Regev [41]), the 2 × 2 matrix algebra M 2 (F ) (Formanek [23] and Drensky [18]), the algebra U T 2 (F ) of 2 × 2 upper triangular matrices (Mishchenko et al [40], based on the approach of Berele and Regev [10], see also [20]), the tensor square E ⊗ E of the Grassmann algebra (Popov [43], Carini and Di Vincenzo [13]), the algebra U T 2 (E) of 2 × 2 upper triangular matrices with entries from the Grassmann algebra E (Centrone [14]), the algebra U T n (F ) of n × n upper triangular matrices (Boumova and Drensky [12]), the algebra R p,q (F ) of upper block triangular (p + 2q) × (p + 2q) when p and q are small values (Drensky and Kostadinov [22]).…”

“…Then we compute the double Hilbert series of E and, as a consequence, we build up an algorithm with output the (k, l)-multiplicity series of U T n (E). In the spirit of [14] we compute the (2, 3)-multiplicity series of U T 2 (E), which contains all multiplicities of the cocharacter sequence of U T 2 (E) and finally we compute the (1, 1)-multiplicity series of U T 3 (E).…”