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MatRep: A Matlab Representation Theory Toolbox (Symmetric Groups)

version 1.0 (57 KB) by Holger I. Meinhardt
The toolbox MatRep provides a set of functions for studying representation in symmetric groups.

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Updated 19 Mar 2017

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The Matlab Representation Theory Toolbox MatRep provides a set of functions (about 50 files) for studying representation in symmetric groups. Notice that this is a very preliminary version of a more comprehensive project, so please use the files with the necessary carefulness. The preliminary character of the toolbox restricts the scope of the toolbox at the moment on tableaux and dimension analysis. Nevertheless, the current toolbox can compute Schensted-Insertion, Jeu de Taquin and their reverse, Hook numbers, Evacuation and much more. In addition, some files require Partitions of an Integer from
http://www.mathworks.com/matlabcentral/fileexchange/12009-partitions-of-an-integer
Since, a manual is at the moment not available, we present now a few examples in order to demonstrate the usage of some interesting commands. Additional information can be retrieved with "help function_name".

Getting Started
###############
I. TABLEAUX:
############
We can, for instance, impose a Schensted-Insertion (RSK-Insertion) on a word array by

Example 1:
##########
Permutation as a word array
>> sig=[1:8;5 7 2 1 4 8 3 6];
>> sig =
1 2 3 4 5 6 7 8
5 7 2 1 4 8 3 6

>> [P,Q]=SchenstedInsertion(sig)
P =

1 3 6
2 4 8
5 7 0

Q =

1 2 6
3 5 8
4 7 0

or alternatively, the permutation given as a word:

Example 2:
##########
>> sig=[5 7 2 1 4 8 3 6];

>> [P,Q,Q1,ld]=SchenstedInsertion(sig)
P =

1 3 6
2 4 8
5 7 0

Q =

1 2 6
3 5 8
4 7 0

Q1 =

1 2 6
3 5 8
4 7 0

ld =

3 3 2

The reverse Schensted-Insertion to get the associated permutation can be computed through
Example 3:
##########
Take tableaux
P =
1 3 6
2 4 8
5 7 0

Q =
1 2 6
3 5 8
4 7 0

>> sig=ReverseSchensted(P,Q)
sig =
5 7 2 1 4 8 3 6

or alternatively, to get a generalized permutation or a word.

Example 4:
##########
Consider the tableaux

gP =
1 4 4
3 5 0
4 7 0
6 0 0

gQ =
1 2 4
3 5 0
6 7 0
8 0 0

>> [sig,wa]=ReverseSchensted(gP,gQ)
sig =
4 6 4 7 5 3 4 1
wa =

1 1 2 2 3 4 4 5
4 6 4 7 5 3 4 1

The Jeu de Taquin on a tableau can be imposed, for instance, by

Example 5:
##########
Take the skew tableau

sT =
0 0 2 5 9
0 3 4 6 0
1 7 0 0 0
8 10 0 0 0
then the rectified Young Tableau is given by
>> [P,lam]=JeuDeTaquin(sT)
P =
1 2 4 5 9
3 6 0 0 0
7 10 0 0 0
8 0 0 0 0

lam =
5 2 2 1

Example 6:
##########
Using a semi-standard skew Tableau T

sT =
0 0 0 0 2
0 0 0 0 6
0 2 4 4 0
2 3 6 0 0
5 5 0 0 0
then the rectified semi-standard Young Tableau is given by

>> [P,lam]=JeuDeTaquin(sT)

[P,lam]=JeuDeTaquin(sT)
P =
2 2 2 4 6
3 4 6 0 0
5 5 0 0 0

lam =
5 3 2

The whole procedure can also be reversed.
Example 7:
##########
Take the standard tableau

T =
1 2 5 9
3 4 6 0
7 10 0 0
8 0 0 0

then the skew Young Tableau is given by

>> [P,lam]=InverseJeuDeTaquin(T)
P =
0 0 0 5
0 2 6 9
1 4 7 0
3 8 10 0

lam =
4 4 3 3

Example 8:
##########
Using a semi-standard skew Tableau T

T =
2 2 2 4 6
3 4 6 0 0
5 5 0 0 0
then the semi-standard skew Young Tableau is given by

>> [P,lam]=InverseJeuDeTaquin(T)

P =
0 0 0 2 2
0 0 3 4 4
2 5 5 6 6

lam =
5 5 5

Next we compute an evacuation tableau.
Example 9:
##########
Take the standard tableau
Q=[1,3,4,7; 2,5,0,0;6,0,0,0]
then the evacuation tableau evQ is given by
>> evQ=Evacuation(Q)
evQ =
1 2 5 6
3 7 0 0
4 0 0 0

Example 10:
###########

Take the arbitrary tableau with shape lam=(5,5,5,4)

>> T=[13 18 10 1 4;17 15 2 3 5;14 16 6 7 11;19 12 8 9 0]

T =
13 18 10 1 4
17 15 2 3 5
14 16 6 7 11
19 12 8 9 0
then we get the corresponding standard Young tableau through

>> P=StandardTableau(T)
P =
1 3 4 13 15
2 5 10 14 16
6 7 11 17 18
8 9 12 19 0

II. DIMENSIONS
##############

We can also compute the dimension of the Schur functor or Weyl module of a vector space V
for the symmetric group S_d by

Examples:
##########
Let dim V = 21, such a space can be obtained from the partition of [3,3,1] of S_7,
and then compute the 3-rd exterior power of V, i.e. Alt^3 V, through

>> D=DimSFuncLamV([1,1,1],21)
D =

1.3300e+03

The 3-rd symmetric power of V, i.e. Sym^3 V, is computed through

>> D=DimSFuncLamV([3],21)
D =

1.7710e+03

Cite As

Holger I. Meinhardt (2019). MatRep: A Matlab Representation Theory Toolbox (Symmetric Groups) (https://www.mathworks.com/matlabcentral/fileexchange/62142-matrep-a-matlab-representation-theory-toolbox-symmetric-groups), MATLAB Central File Exchange. Retrieved .

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