The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices

Abstract

For $2\times 2$ matrices $A_1,\dots ,A_k$, write $[A_1,\dots ,A_k]$ for the left-normed iterated commutator $[\dots [[A_1,A_2],A_3],\dots ,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k\ge 2$ over any field of characteristic $\ne 2$, and as four independently-established results rather than one bundled claim: $I_k$ has codimension 2; $I_k$ has exactly 3 minimal generators; $R_k/I_k$ is Cohen-Macaulay; and $I_k$ is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: $I_k=P_2\cap \cdots \cap P_k$ is an irredundant primary decomposition into exactly $k-1$ primes, following an explicit recursive block-involvement pattern. The proof identifies $I_k$ as the ideal of $2\times 2$ minors of an explicit $2\times 3$ matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.

arXiv comment29 pages. Ancillary files include the SageMath and Macaulay2 scripts cited in the text, with their full output transcripts, and a certificate recording for each reference what was checked and how. Parts of the results are formally verified in Lean 4 / mathlib; the paper states precisely which. Prepared with disclosed AI assistance, as described in the paper