Abstract
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.
Original language | English |
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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 |
Bibliographical note
The research of the author has been sported by the Research Councils UK (EPSRC), grant no EP/N02995X/1.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