Molecules of an affine FPF W-graph and a labelled row-Beissinger reconstruction
Abstract
Kazhdan--Lusztig W-graphs encode the cell structure of Hecke algebras, while their bidirected connected components are called molecules. In finite type~A, the Robinson--Schensted correspondence describes cells and molecules, and Beissinger's row insertion constructs the common tableau associated with an involution. In affine type~A, the affine matrix-ball construction (AMBC) assigns an affine permutation a pair of tabloids together with a dominant weight, and Marberg introduced affine fixed-point-free (FPF) W-graphs indexed by affine FPF involutions. We classify and enumerate the molecules of Marberg's m-type affine FPF W-graph Γnm and show that molecules with the same AMBC shape have isomorphic underlying simple bidirected graphs. We also prove that labelled complete-cycle row-Beissinger truncations recover the full AMBC datum of an affine FPF involution.
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BibTeX
@misc{zhang2026molecules,
title = {{Molecules of an affine FPF $W$-graph and a labelled row-Beissinger reconstruction}},
author = {Yifeng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.03792},
eprint = {2608.03792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03792},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Molecules of an affine FPF W-graph and a labelled row-Beissinger reconstruction
AU - Yifeng Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2608.03792
UR - https://arxiv.org/abs/2608.03792
AB - Kazhdan--Lusztig W-graphs encode the cell structure of Hecke algebras, while their bidirected connected components are called molecules. In finite type~A, the Robinson--Schensted correspondence describes cells and molecules, and Beissinger's row insertion constructs the common tableau associated with an involution. In affine type~A, the affine matrix-ball construction (AMBC) assigns an affine permutation a pair of tabloids together with a dominant weight, and Marberg introduced affine fixed-point-free (FPF) W-graphs indexed by affine FPF involutions. We classify and enumerate the molecules of Marberg's m-type affine FPF W-graph Γ_n^m and show that molecules with the same AMBC shape have isomorphic underlying simple bidirected graphs. We also prove that labelled complete-cycle row-Beissinger truncations recover the full AMBC datum of an affine FPF involution.
ER -