Difference between revisions of "Young tableau"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910046.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910046.png" /></td> </tr></table> | ||
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910047.png" /> is the trivial representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910049.png" /> is the Young subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910050.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910053.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910054.png" /> and otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910055.png" /> is the subgroup of permutations on the letters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910056.png" />. | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910047.png" /> is the [[trivial representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910049.png" /> is the Young subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910050.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910053.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910054.png" /> and otherwise <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910055.png" /> is the subgroup of permutations on the letters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910056.png" />. |
The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910057.png" /> acts on the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910059.png" />-tableaux by permuting the labels. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910060.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910061.png" />-tableaux is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910062.png" />-tabloid. The action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910063.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910064.png" />-tableaux induces an action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910065.png" />-tabloids, and extending this linearly over a base field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910066.png" /> gives a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910067.png" /> which is evidently isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910068.png" />. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910069.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910070.png" />. Given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910071.png" />-tableau <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910072.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910073.png" /> be the following element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910074.png" />: | The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910057.png" /> acts on the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910059.png" />-tableaux by permuting the labels. Two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910060.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910061.png" />-tableaux is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910062.png" />-tabloid. The action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910063.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910064.png" />-tableaux induces an action on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910065.png" />-tabloids, and extending this linearly over a base field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910066.png" /> gives a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910067.png" /> which is evidently isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910068.png" />. The dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910069.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910070.png" />. Given a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910071.png" />-tableau <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910072.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910073.png" /> be the following element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099100/y09910074.png" />: |
Revision as of 18:23, 30 November 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=39847