Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra

Abstract

Qiu, Guan, and Yu conjectured that in the even-order Kaleidoscope Yang--Baxter equation every word $\Lambda T^{k_1}\Lambda \cdots T^{k_M}\Lambda$, built from a square-zero matrix $\Lambda$ and integer powers of a shift $T$ of order $N$, vanishes once $M\ge N/2$. We prove this for every even $N\ge 4$ and every complex parameter $u$ with $u\ne 0$ and $u^N\ne 1$, and show that the bound is sharp. For $M=N/2-1$ we determine exactly which words are nonzero; each of them has rank two. After block reduction, the factors associated with the intervening powers of $T$ lower a common flag of length $N/2$ by at least one step. The spectral idempotents of $T$ then give a presentation of the algebra generated by $T$ and $\Lambda$ by paths between levels, in which two paths compose only when the level changes monotonically. This algebra is independent of $u$, has dimension $N^2-N+2$, and is a pullback of two incidence algebras. Its radical is generated by $\Lambda$ and has nilpotency index $N/2+1$, and the algebra has infinite global dimension and infinite representation type. Replacing powers of $T$ by arbitrary functions of $T$ gives exact rank formulas and, for $N\ge 6$, rank-one elements in the top radical power. Finally, four conditions on a filtered pair of operators over an arbitrary field recover the threshold and the radical; none of them can be omitted.

arXiv comment19 pages, 0 figures. Revised exposition, notation and references; clarified several proof steps; added a proof of non-monomiality and an incidence-algebra interpretation