On the graded identities and cocharacters of the algebra of 3 × 3 matrices

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Abstract

Let M_2,1(F) be the algebra of 3×3 matrices over an algebraically closed field F of characteristic zero with non-trivial ℤ_2-grading. We study the graded identities of this algebra through the representation theory of the hyperoctahedral group ℤ_2 ∼ S_n. After splitting the space of multilinear polynomial identities into the sum of irreducibles under the ℤ_2 ∼ S_n-action, we determine all the irreducible ℤ_2 ∼ S_n-characters appearing in this decomposition with non-zero multiplicity. We then apply this result in order to study the graded cocharacter of the Grassmann envelope of M_2,1(F). Finally, using the representation theory of the general linear group, we determine all the graded polynomial identities of the algebra M_2,1(F) up to degree 5.
Lingua originaleEnglish
pagine (da-a)55-75
Numero di pagine21
RivistaLinear Algebra and Its Applications
Volume384
Stato di pubblicazionePublished - 2004

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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