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Remarks on the shuffle elements of Iwahori--Hecke algebras
Abstract
Let Hn(q) be the generic Iwahori--Hecke algebra associated to the symmetric group Sn of degree n and let 𝒴n,i be the sum of standard basis of Hn(q) with Coxeter length i for 1≤ i≤ binomn2. We show that 𝒴n,1 and 𝒴n,2 are noncommutative whenever n≥ 4, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of Hn(q) in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by 𝒴n,i in Hn(q) is q-symmetric for all i.
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BibTeX
@misc{zhao2026remarks,
title = {{Remarks on the shuffle elements of Iwahori--Hecke algebras}},
author = {Deke Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11345},
eprint = {2609.11345},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11345},
}
RIS
TY - UNPB
TI - Remarks on the shuffle elements of Iwahori--Hecke algebras
AU - Deke Zhao
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.11345
UR - https://arxiv.org/abs/2609.11345
AB - Let H_n(q) be the generic Iwahori--Hecke algebra associated to the symmetric group S_n of degree n and let 𝒴_n,i be the sum of standard basis of H_n(q) with Coxeter length i for 1≤ i≤ binomn2. We show that 𝒴_n,1 and 𝒴_n,2 are noncommutative whenever n≥ 4, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of H_n(q) in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by 𝒴_n,i in H_n(q) is q-symmetric for all i.
ER -