On Bardakov's Question on the Linearity of Automorphism Groups of Artin Groups of Finite Type: A Reduction to Six Exceptional Types

Problem 17.16 of the Kourovka Notebook, posed by V. G. Bardakov, asks whether the automorphism group Aut(A) of an Artin group A of finite type is linear, as A is by the theorems of Bigelow, Krammer, Digne and Cohen–Wales; the Notebook records the affirmative answer of Bardakov and Bryukhanov for braid groups. We give a partial answer: Aut(A) is linear for every Artin group A of finite type without components of types E_6, E_7, E_8, F_4, H_3, H_4 (type D_5 relying on an unrefereed preprint), and the question reduces to linearity of Aut(A/Z(A)) for these six irreducible types. First, Aut(A) is linear over C if and only if Aut(A_i/Z(A_i)) is linear over C for every irreducible component A_i of A. The proof assembles known facts (linearity of A, a lifting argument of Charney and Crisp, parabolic closures after Cumplido, Gebhardt, González-Meneses and Wiest, a lemma of Cumplido and Paris); what we did not find in print is the reduction itself, for reducible groups and in the converse direction. Second, for the types A_n, B_n, D_n and I_2(m), Aut(A) has a subgroup of finite index isomorphic to A/Z(A) or to A/Z(A) × Z. This follows, by the argument of Bardakov and Bryukhanov, from the published computations of Aut(A) by Dyer–Grossman, Charney–Crisp, Gilbert–Howie–Metaftsis–Raptis, Crisp–Paris, Bell–Margalit, Soroko, Castel–Paris and Paris–Soroko (for D_5 a preprint of 2026), and may be regarded as folklore. For the six exceptional types, finiteness of Out(A/Z(A)) would suffice; we found no computation of it in print. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-17.16 (Kourovka Notebook, Problem 17.16, V. G. Bardakov, 17th issue, 2010).