Tropical plactic algebra, the cloaktic monoid, and semigroup representations

Zur Izhakian

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10 Citations (Scopus)


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 languageEnglish
Pages (from-to)290-366
Number of pages77
JournalJournal of Algebra
Early online date12 Jan 2019
Publication statusPublished - 15 Apr 2019

Bibliographical note

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


  • 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


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