## Young Tableaux

We want to come up with some nice sets for our symmetric group to act on. Our first step in this direction is to define a “Young tableau”.

If is a partition of , we define a Young tableau of shape to be an array of numbers. We start with the Ferrers diagram of the partition , and we replace the dots with the numbers to in any order. Clearly, there are Young tableaux of shape if .

For example, if , the Ferrers diagram is

We see that , and so there are Young tableaux of shape . They are

We write for the entry in the place. For example, the last tableau above has , , and .

We also call a Young tableau of shape a “-tableau”, and we write . We can write a generic -tableau as .

[…] cousins to Young tableaux, Young tabloids give us another set on which our symmetric group will […]

Pingback by Young Tabloids « The Unapologetic Mathematician | December 11, 2010 |

[…] Action on Tableaux and Tabloids We’ve introduced Young tableaux and Young tabloids. We’ve also said that they carry symmetric group actions, but we never […]

Pingback by The Action on Tableaux and Tabloids « The Unapologetic Mathematician | December 13, 2010 |

[…] Young tableau thus contains all numbers on the single row, so they’re all row-equivalent. There is only […]

Pingback by Permutation Representations from Partitions « The Unapologetic Mathematician | December 14, 2010 |

[…] and be Young tableaux of shape and , respectively. If for each row, all the entries on that row of are in different […]

Pingback by The Dominance Lemma « The Unapologetic Mathematician | December 20, 2010 |

[…] Young tableau with shape gives us two subgroups of , the “row-stabilizer” and the […]

Pingback by Row- and Column-Stabilizers « The Unapologetic Mathematician | December 22, 2010 |

[…] is the column-stabilizer of a Young tableau . If has columns , then . Letting run over is the same as letting run over for each from to […]

Pingback by Polytabloids « The Unapologetic Mathematician | December 23, 2010 |

[…] is the submodule of the Young tabloid module spanned by the polytabloids where runs over the Young tableaux of shape […]

Pingback by Specht Modules « The Unapologetic Mathematician | December 27, 2010 |

[…] let and are two Young tableaux of shapes and , respectively, where and . If — where is the group algebra element […]

Pingback by Corollaries of the Sign Lemma « The Unapologetic Mathematician | December 31, 2010 |

[…] say that a Young tableau is “standard” if its rows and columns are all increasing sequences. In this case, we […]

Pingback by Standard Tableaux « The Unapologetic Mathematician | January 5, 2011 |

[…] notions of Ferrers diagrams and Young tableaux, and Young tabloids carry over right away to compositions. For instance, the Ferrers diagram of the […]

Pingback by Compositions « The Unapologetic Mathematician | January 6, 2011 |

[…] predictably enough, certain Garnir elements we’re particularly interested in. These come from Young tableaux, and will be useful to us as we move […]

Pingback by Garnir Elements from Tableaux « The Unapologetic Mathematician | January 17, 2011 |

[…] as a quick use of this concept, think about how to fill a Ferrers diagram to make a standard Young tableau. It should be clear that since is the largest entry in the tableau, it must be in the rightmost […]

Pingback by Inner and Outer Corners « The Unapologetic Mathematician | January 26, 2011 |

[…] Young Tableaux And now we have another generalization of Young tableaux. These are the same, except now we allow repetitions of the […]

Pingback by Generalized Young Tableaux « The Unapologetic Mathematician | February 2, 2011 |