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In combinatorics a (semi-)standard Young tableau is a labelling of the boxes of a Young diagram with positive natural numbers (a Young tableau) satisfying extra conditions, at the minimum that labels do not decrease to the right and do increase downwards.
The number of (semi-)standard Young tableau of given shape (underlying Young diagram) govern various objects in the representation theory of the symmetric- and general linear group.
(Young diagram)
For a positive natural number, a Young diagram with boxes is a partition of , hence a sequence of weakly decreasing positive natural numbers
whose sum is :
Given a Young diagram/partition (Def. ), a semistandard Young tableau of shape
is an indexed set of positive natural numbers of the following form (a Young tableau):
such that these labels are:
weakly increasing to the right/along rows;
strictly increasing downwards/along columns:
Beware that in a semistandard Young tableau (Def. ):
the label in the top left box is not required to be 1;
in some applications (e.g. (7) below), though not in general, the labels are in addition required to be bounded .
(standard Young tableau)
A semistandard Young tableau (Def. ) of boxes (1) is called a standard Young tableau if every element of appears exactly once as a label.
In particular, in a standard Young tableau (Def.) we have that:
not just the columns but also the rows are strictly increasing (cf. (2)),
the first label of a standard Young tableau must be (cf. Rem. ).
The following is a semistandard Young tableau that does happen to start with 1 but is not a standard Young tableau:
Given a partition , we write
for the set of semistandard Young tableaux (Def. ) whose underlying Young diagram (i.e. forgetting its labels) is given by .
Given moreover a natural number we write
for the subset on those semistandard Young tableau whose labels are bounded by in that .
Given a semistandard Young tableau , we write for the monomial which contains one factor of the variable for each occurrence of in the Young tableau:
For and a partition of , the corresponding Schur polynomial is equal to the sum over the monomials (6) associated with all semistandard Young tableau of shape (4):
(Sagan 01, Def. 4.4.1, review in Sagan Enc., p. 1)
This means that Schur polynomials in a finite set of variables count semistandard Young tableau (of fixed shape and) with bounded labels (5):
See at hook length formula.
The hook-content formula expresses the number of semistandard Young-Tableau with
shape (a partition),
labels in ,
sum of labels
through the following identification of the generating function for numbers of semistandard Young tableaux (ssYT) as a polynomial in a variable :
where
See at hook-content formula for more.
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) |
We discuss formulas for the number
of standard Young tableaux with boxes and rows.
Asymptotically for large this is (Regev 81, (F.4.5.1)):
Later this appears again as a conjecture in Kotěšovec 13:
(The first line is the expression conjectured in Kotěšovec 13, under the brace we use the translation formula, and the second line shows agreement with Regev 81 (F.4.5.1) as in (8).)
This may further be re-expressed (Kotěšovec 13, p. 2) in terms of the Barnes G-function (see there):
Using the large- asymptotic expansion (here) of the Barnes G-function
inside the large- asymptotic expansion (10), we obtain the doubly asymptotic expansion of the number of standard Yound tableaux with boxes and rows:
Exact formulas for small :
where
are the Catalan numbers;
are the Motzkin numbers.
This is attributed in Gouyou-Beauchamps 89 to Regev 81.
For :
due to Gouyou-Beauchamps 89.
Textbook account:
Survey in the context of Schur functions:
Specifically for standard Young tableaux:
See also the references at Young tableau.
Amitai Regev, Asymptotic values for degrees associated with strips of young diagrams, Advances in Mathematics Volume 41, Issue 2, August 1981, Pages 115-136 (doi:10.1016/0001-8708(81)90012-8)
Dominique Gouyou-Beauchamps, Standard Young Tableaux of Height 4 and 5, Europ. J. Combinatorics (1989) 10, 69-82 (doi:10.1016/S0195-6698(89)80034-4, pdf)
Marilena Barnabei, Flavio Bonetti, and Matteo Silimbani: Combinatorial properties of the numbers oftableaux of bounded height (arXiv:0803.2112, pdf)
Václav Kotěšovec, Asymptotic of Young tableaux of bounded height, 2013 (pdf, pdf)
Last revised on November 29, 2022 at 09:46:04. See the history of this page for a list of all contributions to it.