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Given a partition/Young diagram with boxes/of :
the hook length formula expresses both
the number of standard Young tableaux of shape ,
the dimension of the irreducible representation of Sym(n) labelled by (i.e. of the Specht module )
in terms of the lengths of all “hooks” inside the Young diagram;
similarly, the hook-content formula expresses both
the number of semistandard Young tableau of shape ;
the dimension of the irreducible representation of SU(n)
in terms of the length of all hooks and the “content” of all of the boxes.
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) |
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).
(numbers of (semi-)standard Young tableaux)
Given a partition , and a positive natural number , consider
the number of standard Young tableaux:
the number of standard Young tableaux with bounded entries :
of shape .
(hook length formula for standard Young tableaux)
Given a partition (Young diagram) of (boxes), the number (1) of standard Young tableaux of shape equals the factorial of over the product of the hook lengths (Def. ) at all the boxes of :
This is due to Frame, Robinson & Thrall 54. Textbook accounts include Stanley 99, Cor. 7.21.6, Sagan 01 Thm. 3.10.2.
The dimension of the irrep of the symmetric group that is labelled by a given Young diagram (the Specht module , see at representation theory of the symmetric group) equals the number of standard Young tableaux of shape
(e.g. Sagan, Thm. 2.6.5)
and hence is also given by the hook length formula (3):
This is actually the statement of Frame, Robinson, & Thrall 54, Thm. 1. Textbook accounts include James 78, Thm. 20.1.
(hook length formula for semi-standard Young tableaux)
Given
a partition (Young diagram) of (boxes),
the number (2) of semi-standard Young tableaux of shape and entries (hence the value of the Schur polynomial at ) is:
Here
is also called the content of the box , whence (5) is also called a hook-content formula (“hook length and box content”):
The dimension of the irrep of the general linear group that is labelled by a given Young diagram (see at representation theory of the general linear group), is also given by the hook-content formula (5):
This appears as Sternberg 94 (C.27)
The original proof is due to:
Textbook accounts:
G. D. James, Thm. 20.1 in: The Representation Theory of the Symmetric Groups, Lecture Notes in Mathematics, volume 682, Springer 1978 (doi:10.1007/BFb0067708, pdf)
Richard Stanley, Cor. 7.21.6 in: Enumerative combinatorics 2, Cambridge University Press (1999, 2010) (doi:10.1017/CBO9780511609589, webpage)
Bruce Sagan, Thm. 3.10.2 & Thm. 2.6.5 in: The symmetric group, Springer 2001 (doi:10.1007/978-1-4757-6804-6, pdf)
Further review:
Alex Ghorbani, Section 4.4 of: Applications of representation theory to combinatorics (pdf)
Yufei Zhao, Section 4.4. of Young Tableaux and the Representations of the Symmetric Group (pdf, pdf)
Shiyue and Andrew, Young Tableaux and Probability, 2019 (pdf)
See also:
Alternative proofs:
Jean-Christophe Novelli, Igor Pak, Alexander V. Stoyanovskii, A direct bijective proof of the hook-lengthformula, Discrete Mathematics and Theoretical Computer Science1, 1997, 53–67 (pdf)
Kenneth Glass, Chi-Keung Ng, A Simple Proof of the Hook Length Formula, The American Mathematical Monthly Vol. 111, No. 8 (Oct., 2004), pp. 700-704 (jstor:4145043)
Jason Bandlow, An elementary proof of the hook formula, The Electronic Journal of Combinatorics 15 (2008) (pdf)
Bruce Sagan, Probabilistic proofs of the hook length formulas involving trees, Séminaire Lotharingien de Combinatoire 61A (2009) (pdf)
Generalizations:
Ionuţ Ciocan, Fontanine Matjaž, Konvalinka, Igor Pak, The weighted hook length formula, Journal of Combinatorial Theory, Series A Volume 118, Issue 6, August 2011, Pages 1703-1717 (doi:10.1016/j.jcta.2011.02.004)
Alejandro Morales, Igor Pak, Greta Panova, Hook formulas for skew shapes I. q-analogues and bijections, Journal of Combinatorial Theory Series A 154 (2018), pp 350–405 (arXiv:1512.08348)
The original proof:
Review:
Richard Stanley, Thm. 7.21.2 in: Enumerative combinatorics 2, Cambridge University Press (1999, 2010) (doi:10.1017/CBO9780511609589, webpage)
Textbook accounts:
Review for Sym(n):
Review for SU(n):
Sarah Peluse, Section 1 of: Irreducible representations of with prime power degree, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d (pdf)
Some Notes on Young Tableaux as useful for irreps of (pdf)
Section 2 of: Group Theory primer (pdf)
Last revised on January 30, 2024 at 11:27:43. See the history of this page for a list of all contributions to it.