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Tableaux

Definition: Young Diagram

A Young diagram of size n is a sequence \lambda_1 \geq \lambda_2 \geq \cdots of non- negative integers such that \sum_i \lambda_i = n. We denote this by \lambda and draw it as a collection of unit boxes where each \lambda_i is the length of a row.

The length l(\lambda)=sup\{k \geq 1 | \lambda_k >0\} is the number of rows in \lambda. We denote the set of all Young diagrams by \mathbb{Y}, including the empty diagram \emptyset=(0,0, \ldots).

Below we see the Young diagram [4,3,2] on the left with its transpose on the right.

YD

Definition: Young Tableau

A semi-standard Young tableau of shape \lambda and rank N is a filling of boxes of a Young diagram \lambda with numbers from 1 to N such that the numbers strictly increase along the columns and weakly increase along the rows.

Definition: Domino Diagram

A domino diagram is a Young diagram that can be tiled by combinations of 2x1 and 1x2 rectangles (which we call dominoes)

Definition: Domino Tableau

A domino tableau with shape \lambda and rank N is a filling of dominoes of a tiling of a domino diagram \lambda with numbers from 1 to N such that the numbers strictly increase along the columns and weakly increase along the rows.

Definition: Schur Polynomial

For a given Young Diagram \lambda, we defined the corresponding Schur polynomial by
\[</p>

<p>s_{\lambda}(X) = \sum_{P: sh(P)=\lambda} {\textbf{X}}^{weight(P)},</p>

<p> \]
If the content of P has c_1 1s, c_2 2s, ... , c_n ns, then {\textbf{x}}^{weight(P)} := \Pi_{i=1}^n x_i^{c_i}.