The generalized characteristic polynomial, corresponding resolvent and their application

2023/10/26 by A. V. Kosyak, Kosyak, A. V.
Computer Science · Mathematics · #15 #22E65 (5 #26 #40) #Advanced Algebra and Geometry #FOS: Mathematics #Matrix Theory and Algorithms #Representation Theory (math.RT) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2310.17351

openalex publication_date 2023/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduced previously the generalized characteristic polynomial defined by P_C(λ)=\rm det C(λ), where C(λ)=C+\rm diag(λ₁,…,λ_n) for C∈ \rm Mat(n,\mathbb C) and λ=(λ_k)_k=1ⁿ∈ \mathbb Cⁿ and gave the explicit formula for P_C(λ). In this article we define an analogue of the resolvent C(λ)^-1, calculate it and the expression (C(λ)^-1a,a) for a∈ \mathbb Cⁿ explicitly. The obtained formulas and their variants were applied to the proof of the irreducibility of unitary representations of some infinite-dimensional groups.

Related

• The characteristic polynomials of imprimitive groups and affine Coxeter groups
• Persistent components in Canny's Generalized Characteristic Polynomial
• The geometry of polynomial representations in positive characteristic
• Characteristic polynomials and finitely dimensional representations of \mathfraksl(2, ℂ)
• G.N. Chebotarev's "On the Problem of Resolvents"