On Semi-Invariants of a Matrix

2020/09/25 by Amir Jafari, Jafari, Amir, Amin Najafi Amin +1
Mathematics · #15A72 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.AG #msc:15A72

paper · pdf · doi:10.48550/arxiv.2009.12426

A major revise, with a lot of new material

arxiv created 2021/09/11 · arxiv updated 2021/09/14

Abstract

For an algebraically closed field K of characteristic zero and a non-singular matrix A∈ GL_n(K), a semi-invariant polynomial of A is defined to be a polynomial p(x)=p(x₁,…,x_n) with coefficients in K such that p(xA)=λp(x) for some λ∈ K. In this article, we classify all semi-invariant polynomials of A in terms of a canonically constructed basis that will be made precise in the text.

Related

• On generating the ring of matrix semi-invariants
• Semi-characteristic polynomials, ϕ-modules and skew polynomials
• Polynomial degree bounds for matrix semi-invariants
• Characterization of polynomials by their invariance properties
• The images of non-commutative polynomials evaluated on 2× 2 matrices over an arbitrary field