Difference between revisions of "Young tableau"
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| − | ''of order | + | ''of order $m$'' |
| − | A [[Young diagram|Young diagram]] of order | + | A [[Young diagram|Young diagram]] of order $m$ in whose cells the different numbers $1,\ldots,m$ have been inserted in some order, e.g. |
| − | + | $$ | |
| − | + | \fbox{5,7,9,4|8,2,1|3|6} | |
| − | + | $$ | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| + | A Young tableau is called standard if in each row and in each column the numbers occur in increasing order. The number of all Young tableau for a given Young diagram $t$ of order $m$ is equal to $m!$ and the number of standard Young tableaux is | ||
| + | $$ | ||
| + | \frac{m!}{\prod\lambda_{ij}} | ||
| + | $$ | ||
| + | where the product extends over all the cells $c_{ij}$ of $t$ and $\lambda_{ij}$ denotes the length of the corresponding hook. | ||
====Comments==== | ====Comments==== | ||
Revision as of 04:14, 2 June 2016
of order $m$
A Young diagram of order $m$ in whose cells the different numbers $1,\ldots,m$ have been inserted in some order, e.g. $$ \fbox{5,7,9,4|8,2,1|3|6} $$
A Young tableau is called standard if in each row and in each column the numbers occur in increasing order. The number of all Young tableau for a given Young diagram $t$ of order $m$ is equal to $m!$ and the number of standard Young tableaux is $$ \frac{m!}{\prod\lambda_{ij}} $$
where the product extends over all the cells $c_{ij}$ of $t$ and $\lambda_{ij}$ denotes the length of the corresponding hook.
Comments
In Western literature the phrase Ferrers diagram is also used for a Young diagram. In the Russian literature the phrase "Young tableau" ( "Yunga tablitsa" ) and "Young diagram" ( "Yunga diagramma" ) are used precisely in the opposite way, with "tablitsa" referring to the pictorial representation of a partition and "diagramma" being a filled-in "tablitsa" .
Let 
denote a partition of 
(
, 
, 
) as well as its corresponding Young diagram, its pictorial representation. Let 
be a second partition of 
. A 
-tableau of type 
is a Young diagram 
with its boxes filled with 
's, 
's, etc. For a semi-standard 
-tableau of type 
the labelling of the boxes is such that the rows are non-decreasing (from left to right) and the columns are strictly increasing (from top to bottom). E.g.
 |
is a semi-standard 
-tableau of type 
. The numbers 
of semi-standard 
-tableaux of type 
are called Kostka numbers.
To each partition 
of 
there are associated two "natural" representations of 
, the symmetric group on 
letters: the induced representation 
and the Specht module 
. The representation 
is:
 |
where 
is the trivial representation of 
and 
is the Young subgroup of 
determined by 
, 
, where 
if 
and otherwise 
is the subgroup of permutations on the letters 
.
The group 
acts on the set of all 
-tableaux by permuting the labels. Two 
-tableaux are equivalent if they differ by a permutation of their labels keeping the sets of indices in each row set-wise invariant. An equivalence class of 
-tableaux is a 
-tabloid. The action of 
on 
-tableaux induces an action on 
-tabloids, and extending this linearly over a base field 
gives a representation of 
which is evidently isomorphic to 
. The dimension of 
is 
. Given a 
-tableau 
, let 
be the following element of 
:
 |
where 
is the column-stabilizer of 
, i.e. the subgroup of 
of all permutations that leave the labels of the columns of 
set-wise invariant.
The Specht module, 
, of 
is the submodule of 
spanned by all the elements 
, where 
is the tabloid of 
and 
is a 
-tableau. Over a field of characteristic zero the Specht modules give precisely all the different absolutely-irreducible representations of 
. By Young's rule, the number of times that the Specht module 
over 
occurs (as a composition factor) in 
is equal to the Kostka number 
. If 
is the Young symmetrizer of a 
-tableau 
, then the Specht module defined by the underlying diagram is isomorphic to the ideal 
of 
. This is also (up to isomorphism) the representation denoted by 
in Representation of the symmetric groups. Cf. Majorization ordering for a number of other results involving partitions, Young diagrams and tableaux, and representations of the symmetric groups.
References
| [a1] | D. Knuth, "The art of computer programming" , 3 , Addison-Wesley (1973) |
Young tableau. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_tableau&oldid=11610