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 tectonics ("tectonique des plaques" in French), as elementary relations that generate the equivalence allow conditional commutation of generator symbols: they can sometimes slide across each other (in apparent analogy to tectonic plates), but not freely.

Definition

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

The generators are the letters of the alphabet
The relations are the elementary Knuth transformations yzx ≡ yxz whenever x < y ≤ z and xzy ≡ zxy whenever x ≤ y < z.

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.

If a word is a reverse lattice word, then so is any word Knuth equivalent to it.
If two words are Knuth equivalent, then so are the words obtained by removing their rightmost maximal elements, as are the words obtained by removing their leftmost minimal elements.
Knuth equivalence preserves the length of the longest nondecreasing subsequence, and more generally preserves the maximum of the sum of the lengths of k disjoint non-decreasing subsequences for any fixed k.

Correspondence with semistandard Young tableaux

Multiplying the element with tableau form (38)(1257) with the generator 4, illustrated using Young tableau notation:
• Using the plactic relations, (1257)*4 = 5*(1247)
• (38)*5 = 8*(35), so (5) replaces (8) in the second row
• (8) creates the third row
• The product then has the tableau form (8)(35)(1247)

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) over the same ordered alphabet, where the tableau may be read by rows or by columns. So the elements of the plactic monoid can be identified with the semistandard Young tableaux, which therefore also form a monoid.

Multiplying the word of a semistandard Young tableau to the left with a generator is equivalent to Schensted insertion into the Young tableau. In row order, the word of the tableau is equivalent to a product of increasingly longer nondecreasing sequences of generators. The new generator may be inserted in its proper place by either appending it if it is larger, and otherwise by repeatedly applying the plactic relations to move the out of sequence element to the next row. In the latter case, the out of order element replaces the leftmost entry larger than it in each row, and the displaced element is then inserted in the next row.

Since Schensted insertion preserves Young tableaux, this gives an inductive proof that elements of the plactic monoid can be written in a standard form corresponding to a Young tableau, and the construction defines a natural product of semistandard tableaux.

Jeu de Taquin

Two skew Young Tableaux are Jeu de taquin equivalent if and only if their word readings are Knuth equivalent, i.e. correspond to equivalent elements of the plactic group. This gives an alternative definition of the plactic group product directly in terms of Young tableaux. Two tableaux may be multiplied by drawing them both around an empty rectangle to form a skew tableau, and using Jeu de taquin slides to rectify it.

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, corresponding to the abelianization of the plactic semigroup.

Growth

The generating function of the plactic monoid on an alphabet of size n is

\Gamma (t)={\frac {1}{(1-t)^{n}}}{\frac {1}{(1-t^{2})^{n(n-1)/2}}}\

showing that there is polynomial growth of dimension ${\frac {n(n+1)}{2}}$ .

References

Duchamp, Gérard; Krob, Daniel (1994), "Plactic-growth-like monoids", Words, languages and combinatorics, II (Kyoto, 1992), World Sci. Publ., River Edge, NJ, pp. 124–142, MR 1351284, Zbl 0875.68720
Fulton, William (1997), Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, ISBN 978-0-521-56144-0, MR 1464693, Zbl 0878.14034
Knuth, Donald E. (1970), "Permutations, matrices, and generalized Young tableaux", Pacific Journal of Mathematics, 34 (3): 709–727, doi:10.2140/pjm.1970.34.709, ISSN 0030-8730, MR 0272654
Lascoux, Alain; Leclerc, B.; Thibon, J-Y., "The Plactic Monoid", archived from the original on 2011-07-18 {{citation}}: Missing or empty |title= (help)
Littelmann, Peter (1996), "A plactic algebra for semisimple Lie algebras", Advances in Mathematics, 124 (2): 312–331, doi:10.1006/aima.1996.0085, ISSN 0001-8708, MR 1424313
Lascoux, Alain; Schützenberger, Marcel-P. (1981), "Le monoïde plaxique" (PDF), Noncommutative structures in algebra and geometric combinatorics (Naples, 1978), Quaderni de La Ricerca Scientifica, vol. 109, Rome: CNR, pp. 129–156, MR 0646486
Lothaire, M. (2011), Algebraic combinatorics on words, Encyclopedia of Mathematics and Its Applications, vol. 90, With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.), Cambridge University Press, ISBN 978-0-521-18071-9, Zbl 1221.68183
Schensted, C. (1961), "Longest increasing and decreasing subsequences", Canadian Journal of Mathematics, 13: 179–191, doi:10.4153/CJM-1961-015-3, ISSN 0008-414X, MR 0121305, S2CID 247197258
Schützenberger, Marcel-Paul (1997), "Pour le monoïde plaxique", Mathématiques, Informatique et Sciences Humaines (140): 5–10, doi:10.4000/msh.2764, ISSN 0995-2314, MR 1627563