Lusztig's asymptotic algebra and a cell factorisation for the second lowest two-sided cell in type B̃n
Abstract
We determine Lusztig's asymptotic algebra for the second lowest two-sided cell~Δ1 in the equal parameter extended affine Weyl group B̃n for all n≥ 2, realising it as a full matrix algebra over the representation ring of Sp2n−2(CC)× C2 with C2 the group of order~$2$. Our result is an explicit isomorphism of based rings, determining the image of each basis element tw of the asymptotic algebra. As a consequence we verify that Lusztig's original conjecture from 1989 on the structure of the asymptotic algebra holds for this cell. A key ingredient in our proof is a refinement of Shi's description of~Δ1, giving an explicit parametrisation of the elements of this cell (a \textit{cell factorisation}). This cell factorisation simultaneously describes all intersections of left and right cells, with uniqueness and length-additivity of expressions. Applications include a classification of the involutions in Δ1, and a calculation of the length generating functions for Δ1 and the right cells in~Δ1.
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BibTeX
@misc{parkinson2026lusztigs,
title = {{Lusztig's asymptotic algebra and a cell factorisation for the second lowest two-sided cell in type $\tildeB_n$}},
author = {James Parkinson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25771},
eprint = {2610.25771},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25771},
}
RIS
TY - UNPB
TI - Lusztig's asymptotic algebra and a cell factorisation for the second lowest two-sided cell in type B̃_n
AU - James Parkinson
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.25771
UR - https://arxiv.org/abs/2610.25771
AB - We determine Lusztig's asymptotic algebra for the second lowest two-sided cell~Δ_1 in the equal parameter extended affine Weyl group B̃_n for all n≥ 2, realising it as a full matrix algebra over the representation ring of Sp_2n−2(CC)× C_2 with C_2 the group of order~$2$. Our result is an explicit isomorphism of based rings, determining the image of each basis element t_w of the asymptotic algebra. As a consequence we verify that Lusztig's original conjecture from 1989 on the structure of the asymptotic algebra holds for this cell. A key ingredient in our proof is a refinement of Shi's description of~Δ_1, giving an explicit parametrisation of the elements of this cell (a \textit{cell factorisation}). This cell factorisation simultaneously describes all intersections of left and right cells, with uniqueness and length-additivity of expressions. Applications include a classification of the involutions in Δ_1, and a calculation of the length generating functions for Δ_1 and the right cells in~Δ_1.
ER -