Plactic monoid

In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by Donald Knuth (1970) (who called it the tableau algebra), using an operation given by Craige Schensted (1961) in his study of the longest increasing subsequence of a permutation.

It was named the "monoïde plaxique" by Lascoux & Schützenberger (1981), who allowed any totally ordered alphabet in the definition. The etymology of the word "plaxique" is unclear; it may refer to plate techtonics (tectonique des plaques in French), as the action of a generator of the plactic monoid resembles plates sliding past each other in an earthquake.

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Definition

The plactic monoid over some totally ordered alphabet (often the positive integers) is the monoid with the following presentation:

Knuth equivalence

Two words are called Knuth equivalent if they represent the same element of the plactic monoid, or in other words if one can be obtained from the other by a sequence of elementary Knuth transformations.

Several properties are preserved by Knuth equivalence.

Every word is Knuth equivalent to the word of a unique semistandard Young tableau (this means that each row is non-decreasing and each column is strictly increasing). So the elements of the plactic monoid can be identified with the semistandard Young tableaux, which therefore also form a monoid.

Tableau ring

The tableau ring is the monoid ring of the plactic monoid, so it has a Z-basis consisting of elements of the plactic monoid, with the same product as in the plactic monoid.

There is a homomorphism from the plactic ring on an alphabet to the ring of polynomials (with variables indexed by the alphabet) taking any tableau to the product of the variables of its entries.

See also

References