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''of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990702.png" />''
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$#C+1 = 32 : ~/encyclopedia/old_files/data/Z099/Z.0909070 Young diagram
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A graphical representation of a partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990703.png" /> of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990704.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990705.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990706.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990707.png" />). The Young diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990708.png" /> consists of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y0990709.png" /> cells, arranged in rows and columns in such a way that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907010.png" />-th row has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907011.png" /> cells, where the first cell in each row lies in one (the first) column. E.g., the partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907012.png" /> of 20 is represented by the Young diagram (cf. the diagram on the left).
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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/y099070/y09907013.png" /></td> </tr></table>
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''of order  $  m $''
  
The transposed Young diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907014.png" /> corresponds to the conjugate partition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907016.png" /> is the number of cells in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907017.png" />-th column of the Young diagram. Thus, in the example given above the conjugate partition will be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907018.png" />.
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A graphical representation of a partition  $\lambda = ( \lambda _ {1} \dots \lambda _ {r} ) $
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of a natural number $m$ (where  $  \lambda _ {i} \in \mathbf Z $,
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$  \lambda _ {1} \geq  \dots \geq  \lambda _ {r} > 0 $,
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$  \sum \lambda _ {i} = m $).  
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The Young diagram  $t_\lambda  $
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consists of  $  m $ cells, arranged in rows and columns in such a way that the $  i $-
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th row has  $  \lambda _ {i} $
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cells, where the first cell in each row lies in one (the first) column. E.g., the partition  $  ( 6, 5, 4, 4, 1) $
 +
of 20 is represented by the Young diagram (cf. the diagram on the left).
  
Each cell of a Young diagram defines two sets of cells, known as a hook and a skew-hook. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907019.png" /> be the cell situated in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907020.png" />-th row and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907021.png" />-th column of a given Young diagram. The hook <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907022.png" /> corresponding to it is the set consisting of all cells <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907026.png" />, while the skew-hook is the least connected set of border cells including the last cell of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907027.png" />-th row and the last cell of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907028.png" />-th column. E.g. for the Young diagram chosen on the left, the hook and skew-hook corresponding to the cell <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907029.png" /> have the shape shown in the centre and on the right of the figure.
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<pre style="font-family: monospace;color:black">
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⎛ ┌───┬───┬───┬───┬───┬───┐  ┌───┬───┬───┬───┬───┬───┐  ┌───┬───┬───┬───┬───┬───┐ ⎞
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⎜ │  │  │  │  │  │  │  │  │  │  │  │  │  │  │  │  │  │  │  │  │ ⎟
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⎜ ├───┼───┼───┼───┼───┼───┘  ├───┼───┼───┼───┼───┼───┘  ├───┼───┼───┼───┼───┼───┘ ⎟
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⎜ │  │  │  │  │  │      │  │ ▧ │ ▧ │ ▧ │ ▧ │      │  │  │  │ x │ x │    ⎟
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⎜ ├───┼───┼───┼───┼───┘      ├───┼───┼───┼───┼───┘      ├───┼───┼───┼───┼───┘    ⎟
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⎜ │  │  │  │  │          │  │ ▧ │  │  │          │  │  │  │ x │        ⎟
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⎜ ├───┼───┼───┼───┤          ├───┼───┼───┼───┤          ├───┼───┼───┼───┤        ⎟
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⎜ │  │  │  │  │          │  │ ▧ │  │  │          │  │ x │ x │ x │        ⎟
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⎜ ├───┼───┴───┴───┘          ├───┼───┴───┴───┘          ├───┼───┴───┴───┘        ⎟
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⎜ │  │                      │  │                      │  │                    ⎟
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⎝ └───┘                    , └───┘                    , └───┘                    ⎠
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</pre>
  
The length of a hook (respectively, a skew-hook) is understood to be the number of its cells. The length of the hook <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907030.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907031.png" />. By removing from a Young diagram a skew-hook of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907032.png" /> one obtains a Young diagram of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/y/y099/y099070/y09907033.png" />. The height of a hook (respectively, a skew-hook) is understood to be the number of rows over which the hook (skew-hook) is distributed.
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The transposed Young diagram  $  t _  \lambda  ^  \prime  $
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corresponds to the conjugate partition  $  \lambda  ^  \prime  = ( \lambda _ {1}  ^  \prime  \dots \lambda _ {s}  ^  \prime  ) $,  
 +
where  $  \lambda _ {j}  ^  \prime  $
 +
is the number of cells in the $  j $-
 +
th column of the Young diagram. Thus, in the example given above the conjugate partition will be  $  ( 5, 4, 4, 4, 2, 1) $.
  
The language of Young diagrams and Young tableaux (cf. [[Young tableau|Young tableau]]) is applied in the [[Representation of the symmetric groups|representation of the symmetric groups]] and in the [[Representation of the classical groups|representation of the classical groups]]. It was proposed by A. Young (cf. ).
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Each cell of a Young diagram defines two sets of cells, known as a hook and a skew-hook. Let  $  c _ {ij} $
 
+
be the cell situated in the $  i $-
====References====
+
th row and the  $  j $-
<table><TR><TD valign="top">[1a]</TD> <TD valign="top"> A. Young, "On quantitative substitutional analysis" ''Proc. London Math. Soc.'' , '''33''' (1901) pp. 97–146 {{MR|0049156}} {{ZBL|32.0157.02}} </TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> A. Young, "On quantitative substitutional analysis" ''Proc. London Math. Soc.'' , '''34''' (1902) pp. 361–397 {{MR|0049156}} {{ZBL|33.0158.03}} </TD></TR></table>
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th column of a given Young diagram. The hook  $  h _ {ij} $
 +
corresponding to it is the set consisting of all cells  $  c _ {il} $,
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$  l \geq  j $,
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and  $  c _ {kj} $,
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$  k \geq  i $,
 +
while the skew-hook is the least connected set of border cells including the last cell of the $  i $-
 +
th row and the last cell of the  $  j $-
 +
th column. E.g. for the Young diagram chosen on the left, the hook and skew-hook corresponding to the cell  $  c _ {22} $
 +
have the shape shown in the centre and on the right of the figure.
  
 +
The length of a hook (respectively, a skew-hook) is understood to be the number of its cells. The length of the hook  $  h _ {ij} $
 +
is  $  \lambda _ {ij} = \lambda _ {i} + \lambda _ {j}  ^  \prime  - i - j + 1 $.
 +
By removing from a Young diagram a skew-hook of length  $  p $
 +
one obtains a Young diagram of order  $  m - p $.
 +
The height of a hook (respectively, a skew-hook) is understood to be the number of rows over which the hook (skew-hook) is distributed.
  
 +
The language of Young diagrams and Young tableaux (cf. [[Young tableau|Young tableau]]) is applied in the [[Representation of the symmetric groups|representation of the symmetric groups]] and in the [[Representation of the classical groups|representation of the classical groups]]. It was proposed by A. Young (cf. ).
  
 
====Comments====
 
====Comments====
Line 22: Line 68:
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Kerber, G.D. James, "The representation theory of the symmetric group" , Addison-Wesley (1981) {{MR|0644144}} {{ZBL|0491.20010}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Kerber, "Algebraic combinatorics via finite group actions" , B.I. Wissenschaftsverlag Mannheim (1991) {{MR|1115208}} {{ZBL|0726.05002}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976) {{MR|0557013}} {{ZBL|0371.10001}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979) {{MR|0553598}} {{ZBL|0487.20007}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1a]</TD> <TD valign="top"> A. Young, "On quantitative substitutional analysis" ''Proc. London Math. Soc.'' , '''33''' (1901) pp. 97–146 {{MR|0049156}} {{ZBL|32.0157.02}} </TD></TR>
 +
<TR><TD valign="top">[1b]</TD> <TD valign="top"> A. Young, "On quantitative substitutional analysis" ''Proc. London Math. Soc.'' , '''34''' (1902) pp. 361–397 {{MR|0049156}} {{ZBL|33.0158.03}} </TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Kerber, G.D. James, "The representation theory of the symmetric group" , Addison-Wesley (1981) {{MR|0644144}} {{ZBL|0491.20010}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Kerber, "Algebraic combinatorics via finite group actions" , B.I. Wissenschaftsverlag Mannheim (1991) {{MR|1115208}} {{ZBL|0726.05002}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976) {{MR|0557013}} {{ZBL|0371.10001}} </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979) {{MR|0553598}} {{ZBL|0487.20007}} </TD></TR></table>

Latest revision as of 19:39, 9 November 2023


of order $ m $

A graphical representation of a partition $\lambda = ( \lambda _ {1} \dots \lambda _ {r} ) $ of a natural number $m$ (where $ \lambda _ {i} \in \mathbf Z $, $ \lambda _ {1} \geq \dots \geq \lambda _ {r} > 0 $, $ \sum \lambda _ {i} = m $). The Young diagram $t_\lambda $ consists of $ m $ cells, arranged in rows and columns in such a way that the $ i $- th row has $ \lambda _ {i} $ cells, where the first cell in each row lies in one (the first) column. E.g., the partition $ ( 6, 5, 4, 4, 1) $ of 20 is represented by the Young diagram (cf. the diagram on the left).

⎛ ┌───┬───┬───┬───┬───┬───┐  ┌───┬───┬───┬───┬───┬───┐  ┌───┬───┬───┬───┬───┬───┐ ⎞
⎜ │   │   │   │   │   │   │  │   │   │   │   │   │   │  │   │   │   │   │   │   │ ⎟
⎜ ├───┼───┼───┼───┼───┼───┘  ├───┼───┼───┼───┼───┼───┘  ├───┼───┼───┼───┼───┼───┘ ⎟
⎜ │   │   │   │   │   │      │   │ ▧ │ ▧ │ ▧ │ ▧ │      │   │   │   │ x │ x │     ⎟
⎜ ├───┼───┼───┼───┼───┘      ├───┼───┼───┼───┼───┘      ├───┼───┼───┼───┼───┘     ⎟
⎜ │   │   │   │   │          │   │ ▧ │   │   │          │   │   │   │ x │         ⎟
⎜ ├───┼───┼───┼───┤          ├───┼───┼───┼───┤          ├───┼───┼───┼───┤         ⎟
⎜ │   │   │   │   │          │   │ ▧ │   │   │          │   │ x │ x │ x │         ⎟
⎜ ├───┼───┴───┴───┘          ├───┼───┴───┴───┘          ├───┼───┴───┴───┘         ⎟
⎜ │   │                      │   │                      │   │                     ⎟
⎝ └───┘                    , └───┘                    , └───┘                     ⎠

The transposed Young diagram $ t _ \lambda ^ \prime $ corresponds to the conjugate partition $ \lambda ^ \prime = ( \lambda _ {1} ^ \prime \dots \lambda _ {s} ^ \prime ) $, where $ \lambda _ {j} ^ \prime $ is the number of cells in the $ j $- th column of the Young diagram. Thus, in the example given above the conjugate partition will be $ ( 5, 4, 4, 4, 2, 1) $.

Each cell of a Young diagram defines two sets of cells, known as a hook and a skew-hook. Let $ c _ {ij} $ be the cell situated in the $ i $- th row and the $ j $- th column of a given Young diagram. The hook $ h _ {ij} $ corresponding to it is the set consisting of all cells $ c _ {il} $, $ l \geq j $, and $ c _ {kj} $, $ k \geq i $, while the skew-hook is the least connected set of border cells including the last cell of the $ i $- th row and the last cell of the $ j $- th column. E.g. for the Young diagram chosen on the left, the hook and skew-hook corresponding to the cell $ c _ {22} $ have the shape shown in the centre and on the right of the figure.

The length of a hook (respectively, a skew-hook) is understood to be the number of its cells. The length of the hook $ h _ {ij} $ is $ \lambda _ {ij} = \lambda _ {i} + \lambda _ {j} ^ \prime - i - j + 1 $. By removing from a Young diagram a skew-hook of length $ p $ one obtains a Young diagram of order $ m - p $. The height of a hook (respectively, a skew-hook) is understood to be the number of rows over which the hook (skew-hook) is distributed.

The language of Young diagrams and Young tableaux (cf. Young tableau) is applied in the representation of the symmetric groups and in the representation of the classical groups. It was proposed by A. Young (cf. ).

Comments

A Young diagram is also known as a Ferrers diagram in the West.

References

[1a] A. Young, "On quantitative substitutional analysis" Proc. London Math. Soc. , 33 (1901) pp. 97–146 MR0049156 Zbl 32.0157.02
[1b] A. Young, "On quantitative substitutional analysis" Proc. London Math. Soc. , 34 (1902) pp. 361–397 MR0049156 Zbl 33.0158.03
[a1] A. Kerber, G.D. James, "The representation theory of the symmetric group" , Addison-Wesley (1981) MR0644144 Zbl 0491.20010
[a2] A. Kerber, "Algebraic combinatorics via finite group actions" , B.I. Wissenschaftsverlag Mannheim (1991) MR1115208 Zbl 0726.05002
[a3] G.E. Andrews, "The theory of partitions" , Addison-Wesley (1976) MR0557013 Zbl 0371.10001
[a4] I.G. Macdonald, "Symmetric functions and Hall polynomials" , Clarendon Press (1979) MR0553598 Zbl 0487.20007
How to Cite This Entry:
Young diagram. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_diagram&oldid=24174
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article