Crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape
Abstract
We prove commutation relations between the ordinary type A crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape. We describe the involution by Boolean fibers with fixed maximum entries. It commutes with every raising operator except the one indexed by the entry changed by the involution. We characterize commutation at this exceptional raising operator and for lowering operators away from the two colors involving that entry using the box where the involution acts and reduced signatures. We construct a bijection from these tableaux to decorated five-vertex states that preserves weight and excess, subject to a restriction on nontrivial bumps determined by the inner partition. Rotation with alphabet reversal yields the dual statements.
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BibTeX
@misc{shimazaki2026crystal,
title = {{Crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape}},
author = {Tatsushi Shimazaki},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11233},
eprint = {2609.11233},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11233},
}
RIS
TY - UNPB
TI - Crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape
AU - Tatsushi Shimazaki
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.11233
UR - https://arxiv.org/abs/2609.11233
AB - We prove commutation relations between the ordinary type A crystal operators and a sign-reversing involution on semistandard set-valued tableaux of skew shape. We describe the involution by Boolean fibers with fixed maximum entries. It commutes with every raising operator except the one indexed by the entry changed by the involution. We characterize commutation at this exceptional raising operator and for lowering operators away from the two colors involving that entry using the box where the involution acts and reduced signatures. We construct a bijection from these tableaux to decorated five-vertex states that preserves weight and excess, subject to a restriction on nontrivial bumps determined by the inner partition. Rotation with alphabet reversal yields the dual statements.
ER -