Symmetry and Anti-Symmetry

The shapes in the picture below might look more at home in a game of Tetris than in maths research. However, these shapes made of squares are useful tools to encode parts of a space. They are called Young Tableaux and were invented by the  Cambridge mathematician Alfred Young at the start of the 20th Century. Given some space, we break it up into pieces corresponding to the different “Tetris blocks”. IMG_0400 Given some vector space V, the space V \otimes V is made up of elements of the form a \otimes b where a and b are in V. For example, if the elements of V are represented by vectors then the elements of V \otimes V can be represented by matrices. To understand V \otimes V we break it down into smaller pieces. The map f: a \otimes b \mapsto b \otimes a swaps a and b. This map gives us our pieces:

The symmetric part S^2 V is the things that stay the same when we apply the map f. For example, a \otimes b + b \otimes a

and

The antisymmetric part \Lambda^2 V is the things that get multiplied by “-1” when we apply the map f. For example, a \otimes b - b \otimes a.

We write this decomposition as V \otimes V = S^2 V \oplus \Lambda^2 V. Anything can be written as a sum of a symmetric thing and an anti-symmetric thing. In the Young Tableaux, boxes in the same row indicate symmetry and boxes in the same column indicate antisymmetry. The total number of boxes is the number of copies of V. Here we have 2 copies of V in V \otimes V and the Young Tableaux are:

S2V and Lambda2VMore generally if, instead of V \otimes V, we have V \otimes V \otimes \cdots \otimes V (with d copies of V) then we use the same idea to decompose the space. In this case, however, it’s not so simple that everything can be written as a symmetric thing plus an antisymmetric thing. We require more spaces that are symmetric in some ways and antisymmetric in others. This gives us a Young Tableau with a more complicated arrangement of boxes, as in the picture above.

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