## Wiring Diagrams

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 $S_{(4,3,1)}V$ (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 $V^{\otimes 8}$ of the tensor product of $V$ 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):

$v_1 \otimes v_2 \otimes v_3 \mapsto v_1 \otimes v_2 \otimes v_3 + v_2 \otimes v_1 \otimes v_3$

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

$v_1 \otimes v_2 \otimes v_3 \mapsto v_1 \otimes v_2 \otimes v_3 - v_3 \otimes v_2 \otimes v_1$

A wiring diagram is a picture of this process. The lines coming in at the top are the three factors of $V$ in $V^{\otimes 3}$. 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)?