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)?