The stylic monoid

• Published in: Semigroup forum Vol. 105; no. 1; pp. 1 - 45
• Main Authors: Abram, A., Reutenauer, C.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2022; Springer Nature B.V
• Subjects: Algebra; Algorithms; Automorphisms; Mathematics; Mathematics and Statistics; Monoids; Quotients; Research Article; Partitions; trivial; Plactic monoid; Evacuation; Stylic monoid; Standard immaculate tableaux; Tableaux; order
• ISSN: 0037-1912; 1432-2137
• DOI: 10.1007/s00233-022-10285-3

The free monoid A ∗ on a finite totally ordered alphabet A acts at the left on columns, by Schensted left insertion. This defines a finite monoid, denoted Styl ( A ) and called the stylic monoid. It is canonically a quotient of the plactic monoid. Main results are: the cardinality of Styl ( A ) is equal to the number of partitions of a set on | A | + 1 elements. We give a bijection with so-called N -tableaux, similar to Schensted’s algorithm, explaining this fact. Presentation of Styl ( A ) : it is generated by A subject to the plactic (Knuth) relations and the idempotent relations a 2 = a , a ∈ A . The canonical involutive anti-automorphism on A ∗ , which reverses the order on A , induces an involution of Styl ( A ) , which similarly to the corresponding involution of the plactic monoid, may be computed by an evacuation-like operation (Schützenberger involution on tableaux) on so-called standard immaculate tableaux (which are in bijection with partitions). The monoid Styl ( A ) is J -trivial, and the J -order of Styl ( A ) is graded: the co-rank is given by the number of elements in the N -tableau. The monoid Styl ( A ) is the syntactic monoid for the the function which associates to each word w ∈ A ∗ the length of its longest strictly decreasing subword.