Irreducible Characters of the Symmetric Group
Author
Bernd Günther
Title
Irreducible Characters of the Symmetric Group
Description
Computation of irreducible representations and their characters of the symmetric group
Category
Educational Materials
Keywords
group characters, group representations, symmetric group, Young tableaux, integer partitions
URL
http://www.notebookarchive.org/2021-06-7v6u9hx/
DOI
https://notebookarchive.org/2021-06-7v6u9hx
Date Added
2021-06-17
Date Last Modified
2021-06-17
File Size
406.35 kilobytes
Supplements
Rights
Redistribution rights reserved
Download
Open in Wolfram Cloud
Characters of the Symmetric Group
Characters of the Symmetric Group
In[]:=
Needs["IrrCharSymGrp`",$UserDocumentsDirectory<>"/Wolfram Mathematica/IrrCharSymGrp.m"]
This notebook explains the use the package “IrrCharSymGrp” for computations involving characters and representations of the symmetric group. Although elementary in presentation style, it is not intended as textbook on representation theory. For that purpose, we recommend any of the following books or papers:
[1] B.E. Sagan: The Symmetric Group. Springer GTM 201, 2nd ed. 2001.
[2] H. Boerner: Darstellungen von Gruppen. Springer Grundlehren 74, 2nd ed. 1967.
[3] W. Ledermann: Introduction to Group Characters. Cambridge Universitiy Press, 2nd ed. 1987.
[4] A.M. Vershik, A.Yu. Okounkov: A New Approach to the Representation Theory of the Symmetric Groups II. J. Math. Sci. 131 (2005) 5471-5494. doi=10.1007/s10958-005-0421-7.
[5] A. Kerber: Applied Finite Group Actions. Springer Algorithms and Combinatorics 19, 2nd ed. 1999.
[1] B.E. Sagan: The Symmetric Group. Springer GTM 201, 2nd ed. 2001.
[2] H. Boerner: Darstellungen von Gruppen. Springer Grundlehren 74, 2nd ed. 1967.
[3] W. Ledermann: Introduction to Group Characters. Cambridge Universitiy Press, 2nd ed. 1987.
[4] A.M. Vershik, A.Yu. Okounkov: A New Approach to the Representation Theory of the Symmetric Groups II. J. Math. Sci. 131 (2005) 5471-5494. doi=10.1007/s10958-005-0421-7.
[5] A. Kerber: Applied Finite Group Actions. Springer Algorithms and Combinatorics 19, 2nd ed. 1999.
Irreducible Characters
Conjugacy classes in the symmetric group are uniquely determined by cycle decomposition type of permutations, i.e. by the lengths of the disjoint cycles composing them, thus by a list of integers adding up to . Such a list of integers, by convention listed in decreasing order, is called a partition of . Hence conjugacy classes in and irreducible characters of are in one to one correspondence to the partitions of . We obtain the character table by the function:
n
S
n
n
n
S
n
In[]:=
?CharTblSymGrp
Out[]=
Symbol | |
CharTblSymGrp[n_Integer] returns the character table of the symmetric group. Each character is stored as a row vector. | |
It can be visualized for instance as follows:
In[]:=
groupdim=6;TableForm[CharTblSymGrp[groupdim],TableHeadings{IntegerPartitions[groupdim],IntegerPartitions[groupdim]}]
Out[]//TableForm=
{6} | {5,1} | {4,2} | {4,1,1} | {3,3} | {3,2,1} | {3,1,1,1} | {2,2,2} | {2,2,1,1} | {2,1,1,1,1} | {1,1,1,1,1,1} | |
{6} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
{5,1} | -1 | 0 | -1 | 1 | -1 | 0 | 2 | -1 | 1 | 3 | 5 |
{4,2} | 0 | -1 | 1 | -1 | 0 | 0 | 0 | 3 | 1 | 3 | 9 |
{4,1,1} | 1 | 0 | 0 | 0 | 1 | -1 | 1 | -2 | -2 | 2 | 10 |
{3,3} | 0 | 0 | -1 | -1 | 2 | 1 | -1 | -3 | 1 | 1 | 5 |
{3,2,1} | 0 | 1 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 16 |
{3,1,1,1} | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 2 | -2 | -2 | 10 |
{2,2,2} | 0 | 0 | -1 | 1 | 2 | -1 | -1 | 3 | 1 | -1 | 5 |
{2,2,1,1} | 0 | -1 | 1 | 1 | 0 | 0 | 0 | -3 | 1 | -3 | 9 |
{2,1,1,1,1} | 1 | 0 | -1 | -1 | -1 | 0 | 2 | 1 | 1 | -3 | 5 |
{1,1,1,1,1,1} | -1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 |
In[]:=
?ExtCharTblSymGrp
Out[]=
Symbol | |
ExtCharTblSymGrp[n_Integer] returns the extended character table of the symmetric group, i.e. the multiplicities of primitive roots of unity. | |
In[]:=
printCType[l_]:=ToExpression[StringJoin@@((ToString[Part[#,1]]<>"^"<>ToString[Part[#,2]]<>" ")&/@l),InputForm,HoldForm];
In[]:=
printNType[n_Integer]:=ToExpression[(("1^"<>ToString[n])),InputForm,HoldForm];
In[]:=
?ExtCharTblSymGrp
Out[]=
Symbol | |
ExtCharTblSymGrp[n_Integer] returns the extended character table of the symmetric group, i.e. the multiplicities of primitive roots of unity. | |
In[]:=
n=5;TableForm[Map[printCType,ExtCharTblSymGrp[n],{2}],TableDepth2,TableHeadings{If[Length[#]n,printNType[n],printCType[DeleteCases[Sort[Tally[#]],{1,_Integer}]]]&/@IntegerPartitions[n],If[Length[#]n,printNType[n],printCType[DeleteCases[Sort[Tally[#]],{1,_Integer}]]]&/@IntegerPartitions[n]}]
Out[]//TableForm=
1 5 | 1 4 | 1 2 1 3 | 1 3 | 2 2 | 1 2 | 5 1 | |
1 5 | 1 1 | 1 1 | 1 1 | 1 1 | 1 1 | 1 1 | 1 1 |
1 4 | 1 5 | 1 1 1 2 1 4 | 1 1 1 2 1 3 | 2 1 1 3 | 2 1 2 2 | 3 1 1 2 | 4 1 |
1 2 1 3 | 1 1 1 5 | 1 1 2 2 1 4 | 1 1 1 3 1 6 | 1 1 2 3 | 3 1 2 2 | 3 1 2 2 | 5 1 |
1 3 | 2 1 1 5 | 1 1 1 2 2 4 | 1 1 1 2 1 3 1 6 | 2 1 2 3 | 2 1 4 2 | 3 1 3 2 | 6 1 |
2 2 | 1 1 1 5 | 2 1 1 2 1 4 | 1 2 1 3 1 6 | 1 1 2 3 | 3 1 2 2 | 2 1 3 2 | 5 1 |
1 2 | 1 5 | 1 1 1 2 1 4 | 1 1 1 2 1 6 | 2 1 1 3 | 2 1 2 2 | 1 1 3 2 | 4 1 |
5 1 | 1 1 | 1 2 | 1 2 | 1 1 | 1 1 | 1 2 | 1 1 |
An individual character value, i.e. an entry in the character table, can be more easily computed by the function:
In[]:=
?CharacterSymGrp
Out[]=
Symbol | |
CharacterSymGrp[λ_?DecrPartitionQ,ρ_?DecrPartitionQ] returns the value of the irreducible character λ χ | |
In[]:=
CharacterSymGrp[{3,2,1},{3,3}]
Out[]=
-2
The character scalar product
One of the troubles in representation theory is to keep track of which scalar product to use in which context. For characters, the inverses of the sizes of centralizers must be used as weights (notice that up to a factor of the group order this coincides with the size of the conjugacy class). Thus in matrix form the scalar product is defined by
In[]:=
groupdim=6;MatrixForm[DiagonalMatrix[(1/CycleZ[#])&/@IntegerPartitions[groupdim]]]
Out[]//MatrixForm=
1 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 18 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 6 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 18 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 48 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 16 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 48 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 720 |
In[]:=
?CycleZ
Out[]=
Symbol | |
CycleZ[ρ_?DecrPartitionQ] returns the size of the centralizer class of an element of cycle type ρ. | |
In[]:=
?CharacterScalarProduct
Out[]=
Symbol | |
CharacterScalarProduct[f_List,g_List,groupRank_Integer] returns the scalar product of the class vectors f and g. | |
In[]:=
groupdim=6;CharacterScalarProduct[{0,1,0,0,0,0,0,0,0,0,0},{0,1,0,0,0,0,0,0,0,0,0},groupdim]
Out[]=
1
5
The character table is orthogonal with respect to this scalar product, meaning that the rows are orthogonal (the columns are not):
In[]:=
groupdim=6;MatrixForm[CharTblSymGrp[groupdim].DiagonalMatrix[(1/CycleZ[#])&/@IntegerPartitions[groupdim]].Transpose[CharTblSymGrp[groupdim]]]
Out[]//MatrixForm=
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Irreducible representations of the symmetric group are difficult to construct. In contrast it is easy to write down a sufficiently rich family of compound representations, whose characters are given here:
In[]:=
?CompoundYoungChar
Out[]=
Symbol | |
CompoundYoungChar[λ_?DecrPartitionQ] returns the composite Young character corresponding to the partition λ. | |
In[]:=
groupdim=6;TableForm[CompoundYoungChar/@IntegerPartitions[groupdim],TableHeadings{IntegerPartitions[groupdim],IntegerPartitions[groupdim]}]
Out[]//TableForm=
{6} | {5,1} | {4,2} | {4,1,1} | {3,3} | {3,2,1} | {3,1,1,1} | {2,2,2} | {2,2,1,1} | {2,1,1,1,1} | {1,1,1,1,1,1} | |
{6} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
{5,1} | 0 | 1 | 0 | 2 | 0 | 1 | 3 | 0 | 2 | 4 | 6 |
{4,2} | 0 | 0 | 1 | 1 | 0 | 1 | 3 | 3 | 3 | 7 | 15 |
{4,1,1} | 0 | 0 | 0 | 2 | 0 | 0 | 6 | 0 | 2 | 12 | 30 |
{3,3} | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 4 | 8 | 20 |
{3,2,1} | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 0 | 4 | 16 | 60 |
{3,1,1,1} | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 24 | 120 |
{2,2,2} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 6 | 6 | 18 | 90 |
{2,2,1,1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | 24 | 180 |
{2,1,1,1,1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 24 | 360 |
{1,1,1,1,1,1} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 720 |
The complete set of irreducible representations can be obtained therefrom by reduction, as can be seen by checking the multiplicities of the irreducible characters in these compound characters using our scalar product:
In[]:=
groupdim=6;MatrixForm[CharTblSymGrp[groupdim].DiagonalMatrix[(1/CycleZ[#])&/@IntegerPartitions[groupdim]].Transpose[CompoundYoungChar/@IntegerPartitions[groupdim]]]
Out[]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 2 | 1 | 2 | 3 | 2 | 3 | 4 | 5 |
0 | 0 | 1 | 1 | 1 | 2 | 3 | 3 | 4 | 6 | 9 |
0 | 0 | 0 | 1 | 0 | 1 | 3 | 1 | 3 | 6 | 10 |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 3 | 5 |
0 | 0 | 0 | 0 | 0 | 1 | 2 | 2 | 4 | 8 | 16 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 4 | 10 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 5 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 9 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 5 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Notice the triangular structure of this matrix, which means that irreducible components of our representations can be derived by Schmidt’s orthogonalization procedure. The multiplicities are called Kostka numbers:
In[]:=
?KostkaMatrix
Out[]=
Symbol | |
KostkaMatrix[n_Integer] returns the matrix of Kostka numbers of rank n. | |
In[]:=
groupdim=6;MatrixForm[KostkaMatrix[groupdim]]
Out[]//MatrixForm=
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0 | 1 | 1 | 2 | 1 | 2 | 3 | 2 | 3 | 4 | 5 |
0 | 0 | 1 | 1 | 1 | 2 | 3 | 3 | 4 | 6 | 9 |
0 | 0 | 0 | 1 | 0 | 1 | 3 | 1 | 3 | 6 | 10 |
0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 2 | 3 | 5 |
0 | 0 | 0 | 0 | 0 | 1 | 2 | 2 | 4 | 8 | 16 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 | 4 | 10 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 2 | 5 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 9 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 5 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
The Coxeter relations
Notice that the symmetric group is defined by the generators =(i,i+1) for , i.e. the transpositions of neighboring elements, and the relations
n
S
τ
i
1≤i<n
2
τ
i
(
1
)3
()
τ
i
τ
i+1
(
2
)τ
i
τ
j
τ
j
τ
i
(
3
)They are known as Coxeter relations, although Coxeter himself attributes them to Moore. This means that it is perfectly legitimate to define a representation of by dreaming up a list of matrices ,..., showing that they satisfy the Coxeter relations with replaced by and designating as the image of .
n
S
A
1
A
n
τ
i
A
i
A
i
τ
i
In[]:=
?CoxeterTest
Out[]=
Symbol | |
CoxeterTest[ynr_] applied to the matrices of Young's natural representation checks whether these matrices satisfy Coxeter's relations, as they must. So unless you tamper with the definitions this function should always return TRUE. | |
We have coded this test in the function “CoxeterTest”, and of course any representation must satisfy this test (I yet have to adapt the description text accordingly). We might try e.g. =
, all :
A
i
0 | 1 |
1 | 0 |
i
In[]:=
groupdim=6;CoxeterTest[ConstantArray[{{0,1},{1,0}},groupdim]]
Out[]=
True
So these matrices do indeed define a representation. Of course you notice right away that it is reducible, because the two lines with slope and are invariant one dimensional subspaces.
o
45
o
135
If you have a representation defined this way, then the image of any permutation can be constructed as follows:
π∈
n
S
In[]:=
?NTranspDecomp
Out[]=
Symbol | |
NTranspDecomp[pi_?PermutationListQ] represents pi as product of transpositions of immediate neighbors. An entry value of k in the returned list denotes the transposition (k,k+1).Attention: Permutations are multiplied right to left like right operators, not like functions! | |
In[]:=
NTranspDecomp[{1,2,6,3,4,5}]
Out[]=
{3,4,5}
Use the function “NTranspDecomp” to expand a permutation in list notation in a product of transpositions, in our example as . Then the image of this permutation under our representation is .
τ
3
τ
4
τ
5
A
3
A
4
A
5
Young’s natural representation
Let’s take a look at the representation matrices. The nice thing about the natural representation is that its entries are integers. There are as many different irreducible representations of as there are partitions of .
n
S
n
In[]:=
?YoungsNaturalReprValue
Out[]=
Symbol | |
YoungsNaturalReprValue[λ_?DecrPartitionQ,pi_?PermutationListQ] is the matrix assigned to permutation π by Young's natural representation corresponding to partition λ. | |
In[]:=
MatrixForm[YoungsNaturalReprValue[{4,2},{1,2,6,3,4,5}]]
Out[]//MatrixForm=
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | -1 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
Like any representation, the matrices are orthogonal with respect to the adequate scalar product, which in our case is given by:
In[]:=
?InvariantYMetric
Out[]=
Symbol | |
InvariantYMetric[λ_?DecrPartitionQ] is the scalar product invariant under Young's natural presentation corresponding to the integer partition λ. | |
In[]:=
MatrixForm[InvariantYMetric[{4,2}]]
Out[]//MatrixForm=
4 | 2 | 2 | 1 | 1 | 1 | -1 | 1 | 0 |
2 | 4 | 2 | 2 | 1 | 1 | 0 | 1 | -1 |
2 | 2 | 4 | 1 | 2 | 2 | 1 | 2 | 1 |
1 | 2 | 1 | 4 | 0 | 2 | 1 | 1 | -1 |
1 | 1 | 2 | 0 | 4 | 1 | 2 | 1 | 2 |
1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 1 |
-1 | 0 | 1 | 1 | 2 | 2 | 4 | 1 | 2 |
1 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 |
0 | -1 | 1 | -1 | 2 | 1 | 2 | 2 | 4 |
In this context orthogonality means that the representation matrices leave the scalar product invariant:
In[]:=
MatrixForm[YoungsNaturalReprValue[{4,2},{1,2,6,3,4,5}].InvariantYMetric[{4,2}].Transpose[YoungsNaturalReprValue[{4,2},{1,2,6,3,4,5}]]]
Out[]//MatrixForm=
4 | 2 | 2 | 1 | 1 | 1 | -1 | 1 | 0 |
2 | 4 | 2 | 2 | 1 | 1 | 0 | 1 | -1 |
2 | 2 | 4 | 1 | 2 | 2 | 1 | 2 | 1 |
1 | 2 | 1 | 4 | 0 | 2 | 1 | 1 | -1 |
1 | 1 | 2 | 0 | 4 | 1 | 2 | 1 | 2 |
1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 1 |
-1 | 0 | 1 | 1 | 2 | 2 | 4 | 1 | 2 |
1 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 2 |
0 | -1 | 1 | -1 | 2 | 1 | 2 | 2 | 4 |
The output equals the metric:
In[]:=
YoungsNaturalReprValue[{4,2},{1,2,6,3,4,5}].InvariantYMetric[{4,2}].Transpose[YoungsNaturalReprValue[{4,2},{1,2,6,3,4,5}]]==InvariantYMetric[{4,2}]
Out[]=
True
The images of the generating transpositions are given by:
In[]:=
?YoungsNaturalRepresentation
Out[]=
Symbol | |
YoungsNaturalRepresentation[λ_?DecrPartitionQ] computes the matrices of Young's natural representation of the symmetric group corresponding to the integer partition λ by transforming the seminormal representation. The function returns the images of the transpositions of immediate neighbors, listed in order of the transposed elements. The matrices are supposed to operate from the right on row vectors. | |
In[]:=
MatrixForm/@YoungsNaturalRepresentation[{4,2}]
Out[]=
,
,
,
,
1 | 0 | 0 | 0 | -1 | 0 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | -1 | 0 | 0 | 0 | 1 |
0 | 0 | 1 | 0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 1 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
1 | 0 | 0 | 0 | 0 | -1 | 1 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 | 1 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | -1 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | -1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | -1 | 0 | 1 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
1 | 0 | 0 | 0 | -1 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
Of course they must satisfy the Coxeter relations:
In[]:=
CoxeterTest[YoungsNaturalRepresentation[{4,2}]]
Out[]=
True
Also, its character should equal the corresponding entry in our character table above. The test is encoded in:
In[]:=
?YnrCharacterTest
Out[]=
Symbol | |
YnrCharacterTest[ynr_,λ_] applied to the matrices of Young's natural representation corresponding to the integer partition λ computes the character and compares it to the relevant entry in the character table. So unless you tamper with the definitions this function should always return TRUE. A complete test would be for instance: testPartition=RandomPartition[5];testYnr=YoungsNaturalRepresentation[testPartition];CoxeterTest[testYnr]&&YnrCharacterTest[testYnr,testPartition] | |
In[]:=
YnrCharacterTest[YoungsNaturalRepresentation[{4,2}],{4,2}]
Out[]=
True
A note on efficiency: the function “YoungsNaturalReprValue” calls the time consuming part “YoungsNaturalRepresentation” as subprocedure. So if you want to use a particular representation during a computation more than once, it is best to call “YoungsNaturalRepresentation” just once, store the result and then derive individual representation matrices by aid of “NTranspDecomp”.
The seminormal representation
Its entries are rational, and its matrices are orthogonal with respect a diagonal scalar product (though still not the standard one).
In[]:=
?YoungsSeminormalReprValue
Out[]=
Symbol | |
YoungsSeminormalReprValue[λ_?DecrPartitionQ,pi_?PermutationListQ] is the matrix assigned to permutation π by Young's seminormal representation corresponding to partition λ. | |
In[]:=
MatrixForm[YoungsSeminormalReprValue[{4,2},{1,2,6,3,4,5}]]
Out[]//MatrixForm=
- 1 4 | - 5 16 | 0 | 5 6 | 0 | 0 | 0 | 0 | 0 |
- 1 3 | 1 36 | - 8 27 | - 2 27 | 0 | 64 81 | 0 | 0 | 0 |
1 | - 1 12 | - 1 9 | 2 9 | 0 | 8 27 | 0 | 0 | 0 |
0 | - 1 3 | - 4 9 | - 1 9 | 0 | - 4 27 | 0 | 2 3 | 0 |
0 | 0 | 0 | 0 | - 1 3 | 0 | 8 9 | 0 | 0 |
0 | 1 | - 1 6 | 1 3 | 0 | - 1 18 | 0 | 1 4 | 0 |
0 | 0 | 0 | 0 | - 1 2 | 0 | - 1 6 | 0 | 3 4 |
0 | 0 | 1 | 0 | 0 | 1 3 | 0 | 1 2 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | - 1 3 | 0 | - 1 2 |
In[]:=
?YoungsSeminormalRepresentation
Out[]=
Symbol | |
YoungsSeminormalRepresentation[λ_?DecrPartitionQ] computes the matrices of Young's seminormal representation of the symmetric group corresponding to the integer partition λ. The function returns the images of the transpositions of immediate neighbors, listed in order of the transposed elements. The matrices are supposed to operate from the right on row vectors. | |
In[]:=
MatrixForm/@YoungsSeminormalRepresentation[{4,2}]
Out[]=
,
,
,
,
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | - 1 2 | 0 | 3 4 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 1 2 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | - 1 2 | 3 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 1 2 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | - 1 2 | 3 4 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 2 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | - 1 3 | 8 9 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 1 3 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | - 1 3 | 0 | 8 9 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 1 3 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 |
- 1 4 | 15 16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | 1 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | - 1 2 | 0 | 3 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | - 1 2 | 0 | 3 4 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 2 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1 2 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | - 1 3 | 0 | 8 9 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | - 1 3 | 0 | 0 | 8 9 | 0 | 0 | 0 |
0 | 1 | 0 | 1 3 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | - 1 3 | 0 | 8 9 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 1 3 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 1 3 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
In[]:=
CoxeterTest[YoungsSeminormalRepresentation[{4,2}]]
Out[]=
True
In[]:=
YnrCharacterTest[YoungsSeminormalRepresentation[{4,2}],{4,2}]
Out[]=
True
The appropriate invariant scalar product is as follows:
In[]:=
MatrixForm[ivm=DiagonalMatrix[NormSquareOfTableau/@First/@WeakLeftBruhatGraph[{4,2}]]]
Out[]//MatrixForm=
5 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 4 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 2 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 2 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 2 3 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 9 16 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 3 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
In[]:=
MatrixForm[YoungsSeminormalReprValue[{4,2},{1,2,6,3,4,5}].ivm.Transpose[YoungsSeminormalReprValue[{4,2},{1,2,6,3,4,5}]]]
Out[]//MatrixForm=
5 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 4 9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 2 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 2 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 2 3 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 9 16 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 3 4 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 4 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
If you want genuine orthogonal representation matrices (the “normal representation”), then you can simply divide out the square roots of the diagonal elements, but then your representation is no longer rational.
Cite this as: Bernd Günther, "Irreducible Characters of the Symmetric Group" from the Notebook Archive (2021), https://notebookarchive.org/2021-06-7v6u9hx
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