### Abstract

Original language | English |
---|---|

Pages (from-to) | 290-366 |

Number of pages | 77 |

Journal | Journal of Algebra |

Volume | 524 |

Early online date | 12 Jan 2019 |

DOIs | |

Publication status | Published - 15 Apr 2019 |

### Fingerprint

### Keywords

- Idempotent semiring
- tropical plactic algebra
- tropical matrix algebra
- colored weighted digraph
- semigroup identity
- forward semigroup
- plactic monoid
- cloaktic monoid
- semigroup representation
- young tableau
- configuration tableau
- symmetric group
- Semigroup identities
- Tropical plactic algebra
- Young tableaux
- Semigroup representations
- Colored weighted digraphs
- Idempotent semirings
- Plactic monoid
- Symmetric group
- Cloaktic monoid
- Forward semigroup
- Configuration tableaux
- Tropical matrix algebra
- RANK
- IDENTITIES
- MATRICES

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Tropical plactic algebra, the cloaktic monoid, and semigroup representations.** / Izhakian, Zur.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Tropical plactic algebra, the cloaktic monoid, and semigroup representations

AU - Izhakian, Zur

N1 - The research of the author has been sported by the Research Councils UK (EPSRC), grant no EP/N02995X/1.

PY - 2019/4/15

Y1 - 2019/4/15

N2 - A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid Pn. This algebra manifests a natural framework for accommodating representations of Pn, or equivalently of Young tableaux, and its moderate coarsening -- the cloaktic monoid Kn and the co-cloaktic coKn. The faithful linear representations of Kn and coKn by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that Kn and coKn admit all the semigroup identities satisfied by n×n triangular tropical matrices, which holds also for P3.

AB - A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid Pn. This algebra manifests a natural framework for accommodating representations of Pn, or equivalently of Young tableaux, and its moderate coarsening -- the cloaktic monoid Kn and the co-cloaktic coKn. The faithful linear representations of Kn and coKn by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that Kn and coKn admit all the semigroup identities satisfied by n×n triangular tropical matrices, which holds also for P3.

KW - Idempotent semiring

KW - tropical plactic algebra

KW - tropical matrix algebra

KW - colored weighted digraph

KW - semigroup identity

KW - forward semigroup

KW - plactic monoid

KW - cloaktic monoid

KW - semigroup representation

KW - young tableau

KW - configuration tableau

KW - symmetric group

KW - Semigroup identities

KW - Tropical plactic algebra

KW - Young tableaux

KW - Semigroup representations

KW - Colored weighted digraphs

KW - Idempotent semirings

KW - Plactic monoid

KW - Symmetric group

KW - Cloaktic monoid

KW - Forward semigroup

KW - Configuration tableaux

KW - Tropical matrix algebra

KW - RANK

KW - IDENTITIES

KW - MATRICES

UR - http://www.scopus.com/inward/record.url?scp=85061035851&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/tropical-plactic-algebra-cloaktic-monoid-semigroup-representations

U2 - 10.1016/j.jalgebra.2018.12.014

DO - 10.1016/j.jalgebra.2018.12.014

M3 - Article

VL - 524

SP - 290

EP - 366

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -