Diagrammatic Okada monoid and cellularity of the Okada algebra
Abstract
It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from SN to SN−1. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the Bratelli diagram of a family of algebras {ON(X,Y)}N ≥ 0 by Okada in 1994. In this paper, we first realize the Okada algebra ON(X,Y) and the associated monoid ON using a labelled version of non-crossing arc-diagrams appearing in the description of the Temperley-Lieb algebra and Jones monoid. We establish, for general parameters (X,Y), that the dimension of the Okada algebra ON(X,Y) is N!, noting that Okada proved this result only in the semisimple case. We interpret a natural bijection between permutations and labelled arc-diagrams as an incarnation of Fomin's version of the Robinson-Schensted correspondence associated to the Young-Fibonacci lattice. The arc-diagram formalism allow us to probe the structure of the Okada monoid and algebra. In particular we prove that the Okada monoid is a regular, aperiodic *-monoid and we describe its Green relations and order. These results allow us to construct a cellular basis of the Okada algebra and to show that {ON(X,Y)}N ≥ 0 forms a coherent tower of cellular algebras in the sense of Goodman and Graber. We present some conjectures expressing the Gram determinant of the invariant bilinear form attached to each cell module in terms of Okada's clone Schur functions. We conclude the paper by presenting two follow-up, ongoing projects along with a series of questions pushing further the analogy between the symmetric groups and the Okada algebras.
Related
BibTeX
@misc{hivert2026diagrammatic,
title = {{Diagrammatic Okada monoid and cellularity of the Okada algebra}},
author = {Florent Hivert and Jeanne Scott},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01440},
eprint = {2609.01440},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05795536v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Diagrammatic Okada monoid and cellularity of the Okada algebra
AU - Florent Hivert
AU - Jeanne Scott
PY - 2026
DA - 2026/09/01
PB - HAL
DO - 10.48550/arxiv.2609.01440
UR - https://hal.science/hal-05795536v1
AB - It is well known that the Young lattice is the Bratelli diagram of the symmetric groups, expressing how irreducible representations restrict from S_N to S_N−1. In 1975, Stanley discovered a similar lattice called the Young-Fibonacci lattice which was identified as the Bratelli diagram of a family of algebras {O_N(X,Y)}_N ≥ 0 by Okada in 1994. In this paper, we first realize the Okada algebra O_N(X,Y) and the associated monoid O_N using a labelled version of non-crossing arc-diagrams appearing in the description of the Temperley-Lieb algebra and Jones monoid. We establish, for general parameters (X,Y), that the dimension of the Okada algebra O_N(X,Y) is N!, noting that Okada proved this result only in the semisimple case. We interpret a natural bijection between permutations and labelled arc-diagrams as an incarnation of Fomin's version of the Robinson-Schensted correspondence associated to the Young-Fibonacci lattice. The arc-diagram formalism allow us to probe the structure of the Okada monoid and algebra. In particular we prove that the Okada monoid is a regular, aperiodic *-monoid and we describe its Green relations and order. These results allow us to construct a cellular basis of the Okada algebra and to show that {O_N(X,Y)}_N ≥ 0 forms a coherent tower of cellular algebras in the sense of Goodman and Graber. We present some conjectures expressing the Gram determinant of the invariant bilinear form attached to each cell module in terms of Okada's clone Schur functions. We conclude the paper by presenting two follow-up, ongoing projects along with a series of questions pushing further the analogy between the symmetric groups and the Okada algebras.
ER -