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 $ΛT^{k_1}Λ\cdots T^{k_M}Λ$, built from a square-zero matrix $Λ$ and integer powers of a shift $T$ of order $N$, vanishes once $M\geq N/2$. We prove this for every even $N\geq4$ and every complex parameter $u$ with $u\neq0$ and $u^N\neq1$, 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 $Λ$ 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 $Λ$ 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 $N\geq6$, 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.