Tableaux
Definition: Young Diagram
A Young diagram of size n is a sequence of non- negative integers such that
. We denote this by
and draw it as a collection of unit boxes where each
is the length of a row.
The length is the number of rows in
. We denote the set of all Young diagrams by
, including the empty diagram
.
Below we see the Young diagram [4,3,2] on the left with its transpose on the right.
Definition: Young Tableau
A semi-standard Young tableau of shape and rank N is a filling of boxes of a Young diagram
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 and rank
is a filling of dominoes of a tiling of a domino diagram
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 , we defined the corresponding Schur polynomial by
If the content of P has 1s,
2s, ... ,
ns, then
.