Table of Young

The tables of Young are objects Combinatoire S which play a big role in Théorie of the representations of the groups and in the theory of the symmetrical functions. They make it possible in particular to build the irreducible representations symmetrical Groupe, like those of the linear general Groupe on the body of the complex .

Definition

Diagram of Young

A diagram of Young, or diagram To shoe, is to some extent a chart of a Partition of an entirety. It consists of a whole of boxes justified on the left and in bottom, and the number of boxes of each line corresponds to the elements of the associated partition. The image on the right watch the diagram associated with the partition (3,3,1).

Table of Young

A table of Young with values in is a diagram of Young of which the boxes are filled by entireties ranging between 1 and m, with the constraint which the lines must be increasing in the broad sense, and increasing columns with the direction strict. The partition associated with the subjacent diagram is called the form table.

Plaxic monoid

Algorithm of Schensted

The algorithm of Schensted makes it possible to insert a succession of entireties in a table, so as to obtain a new table. This algorithm allows:

to provide the unit with the tables of a Law of composition interns, while inserting successivment the elements of a table in another.
to induce a Relation of equivalence on the whole of the finished continuations of entireties (cad of the words on the alphabet). Indeed, being taken such an action pursuant, it is possible to insert it in the empty table: one obtains a table which is called the P-symbol continuation. Two continuations will be equivalent if they are the same P-symbol.

Relations of Knuth

Starting from the study of this relation of equivalence on the words length 3, Donald Knuth defined rewriting rules on the whole of the words on. These rewriting rules also induce a relation of equivalence, and Knuth showed that it coincides with the relation of Schensted. An important consequence of this theorem is that the law of composition which rises from the algorithm of Schensted has all the necessary properties to give to the whole of the tables a structure of Monoïde: the plaxic monoid .

Applications

Representations of the symmetrical group

The tables of Young make it possible to simply calculate the Symétriseurs of Young.

Representation of GL (E)

If E is a

\ mathbb {C}

-espace vectorial of dimension m, and

\ lambda

a partition, one defines the module of Schur

E^ {\ lambda}

as being the

\ mathbb {C}

-espace vectorial whose base is formed by the whole of the tables of Young of form

\ lambda

and with value in. Knowing that it is possible to identify a table of Young with values in with a polynomial of

\ mathbb {C} I, J \ Leq m

, there exists a natural action of GL (E) on the tables of Young by simple matric multiplication. The modules of Schur are thus representations of GL (E). One can show that any irreducible polynomial representation of GL (E) is isomorphous with a single module of Schur.

Symmetrical functions

The Character S of the modules of Schur (as representations of GL (E)) are symmetrical polynomials called Polynômes of Schur . The tables of Young provide an elegant means to express these polynomials. In addition, there exists a purely combinative rule which calls upon the tables of Young, and who allows to break up the product of two polynomials of Schur. This implies in particular that the tables make it possible to break up the tensorial product of two irreducible representations of GL (E) all in all direct of representation irreducible.

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