This week I have been looking at more pictures of Young Tableaux. These combinations of squares are useful tools to understand the symmetry of a space. They are widely used both in pure maths and physics, and are deceptively complicated!

This picture shows (there are 4 boxes in the first row, three in the second and 1 in the third). There are eight blocks in total, so it lives in the space of the tensor product of with itself eight times. The arrangement of blocks tells us **how many** symmetrizations and anti-symmetrizations occur, and to know **where** the symmetrizations occur we label the boxes.

We symmetrize in the factors that are in the same row (1 and 2):

anti-symmetrize in the factors that are in the same column (1 and 3):

A wiring diagram is a picture of this process. The lines coming in at the top are the three factors of in . The black boxes indicate anti-symmetrization and the white boxes are symmetrization. The picture above is from J. M. Landsberg’s book “Tensors: Geometry and Applications”. Our example above is (a) in the picture: there is a black box which the first and third strands pass through and a white box for the first and second strands. Can you work out whats going on in (b)-(f)?

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