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In combinatorics and representation theory, the hook-content formula expresses the number of semistandard Young tableaux with fixed underlying Young diagram and upper bound on its labels in terms of the “hook lengths” and the “content of boxes” of the underlying diagram.
If the dependency in the “contents” is removed, the formula reduces to the hook length formula that counts standard Young tableaux of the given shape.
Both of these formulas equivalently give dimensions of irreps in representation theory:
hook length formula | hook-content formula |
---|---|
number of standard Young tableaux | number of semistandard Young tableaux |
dimension of irreps of Sym(n) | dimension of irreps of SL(n) |
See at hook length formula for more on this.
Given a Young diagram, the hook at any one of its boxes is the collection of boxes to the right and below that box, and including the box itself. We write “” for the length of such a hook, i.e. for the number of boxes it contains. Formally:
(hook length)
Let be a partition/Young diagram. Then for
,
the hook length at is
where denotes the conjugate partition (see there).
Alongside this terminology one says that
is the content of the box .
Consider in the following:
a partition of , ,
equivalently a Young diagram with boxes;
Write:
for the set of semistandard Young tableaux
of shape (i.e. with underlying Young diagram) ,
with labels (i.e. with ).
for the sum of all the labels.
For a partition/Young diagram and , the number of semistandard Young tableaux of shape (hence the value of the Schur polynomial on unit argument) and entries bounded by is:
(Stanley 71, Thm. 15.3, Stanley 99, Thm. 7.21.2)
With the above ingredients, we have the following equalities of polynomials in a variable :
The original statement and proof:
Review:
See also:
Christian Krattenthaler, An Involution Principle-Free Bijective Proof of Stanley’s Hook-Content Formula, Discrete Mathematics and Theoretical Computer Science 3, 1998, 11–32 (hal:hal-00958904, pdf)
Christian Krattenthaler, Another involution principle-free bijective proof of Stanley’s hook-content formula, J. Combin. Theory Ser. A 88 (1999), 66-92 (arXiv:math/9807068)
See also:
Last revised on June 24, 2024 at 04:13:08. See the history of this page for a list of all contributions to it.