Hook-length formula
From Groupprops
Statement
Suppose is an unordered integer partition of a natural number
. We can describe
by means of a Young diagram (also called a Ferrers diagram).
For every box in this Young diagram, define the hook-length at that box as follows:
Hook-length = 1 + (number of boxes directly below it vertically) + (number of boxes directly to the right of it horizontally)
Then, we have the following:
Degree of irreducible representation corresponding to partition = Number of (standard) Young tableaux with shape
= Number of paths in the Young lattice from the trivial partition of
to
=
The product in the denominator is the product, over all boxes in the Young diagram, of the hook-length corresponding to that box.
Particular cases
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Young diagram | Hook lengths | Hook-length formula | Result of formula | Cross-check |
---|---|---|---|---|---|---|
1 | 1 | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | 1 (for unique box) | ![]() |
1 | unique Young tableau: fill box with ![]() |
2 | 2 | ![]() |
2 (for left box), 1 (for right box) | ![]() |
1 | unique Young tableau: 1 in left box, 2 in right box |
2 | 1 + 1 | ![]() |
2 (for top box), 1 (for bottom box) | ![]() |
1 | unique Young tableau: 1 in top box, 2 in bottom box |
3 | 3 | ![]() |
3 (for left box), 2 (for middle box), 1 (for right box) | ![]() |
1 | unique Young tableau: 1 in left, 2 in middle, 3 in right box |
3 | 1 + 1 + 1 | ![]() |
3 (for top box), 2 (for middle box), 1 (for right box) | ![]() |
1 | unique Young tableau: 1 in top, 2 in middle, 3 in bottom box |
3 | 2 + 1 | ![]() |
3 (for top left box), 1 (for right box), 1 (for bottom box) | ![]() |
2 | Two possibilities for the Young tableau: 1 in top left, 2 in right, 3 in bottom 1 in top left, 3 in right, 2 in bottom |
4 | 4 | ![]() |
4,3,2,1 as we move from left-most to right-most box | ![]() |
1 | unique Young tableau: 1,2,3,4 filled left to right |