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 ΛTk1Λ⋯ TkMΛ, built from a square-zero matrix Λ and integer powers of a shift T of order N, vanishes once M≥ N/2. We prove this for every even N≥4 and every complex parameter u with u≠0 and uN≠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 Λ by paths between levels, in which two paths compose only when the level changes monotonically. This algebra is independent of u, has dimension N2−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≥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.
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BibTeX
@misc{wang2026exact,
title = {{Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra}},
author = {Qihang Wang and Zhiyuan Yao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.39262},
eprint = {2609.39262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39262},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang-Baxter Algebra
AU - Qihang Wang
AU - Zhiyuan Yao
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.39262
UR - https://arxiv.org/abs/2609.39262
AB - Qiu, Guan, and Yu conjectured that in the even-order Kaleidoscope Yang--Baxter equation every word ΛT^k_1Λ⋯ T^k_MΛ, built from a square-zero matrix Λ and integer powers of a shift T of order N, vanishes once M≥ N/2. We prove this for every even N≥4 and every complex parameter u with u≠0 and u^N≠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 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≥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.
ER -