schurModule -- creates Schur module from a partition and free module

Synopsis

Usage:

schurModule(lambda,E)
Inputs:
- lambda, a list, a list of numbers representing a partition; e.g. {3,1} stands for 2 rows of length 3 and 1.
- E, a module, a free module
Outputs:
- M, a module, The result of application of the Schur functor associated to lambda to E.

Description

Applies the Schur functor associated to lambda to the free module E. For a detailed definition of the Schur module see p.106 of Fulton "Young Tableaux".

The resulting M comes with cached data M.cache.Schur = {f, finv, AT, ST} where

{"f is a map from exterior_mu E to M;", "finv is a map from M to exterior_mu E;", "AT is a hash table of all tableaux, whose entries increase in every column;", "ST is a hash table of all standard tableaux (tableaux in AT, whose entries nondecrease in every row)." }

Tableaux are represented with objects of class Filling, which is a double list whose entries are lists giving the fillings of the corresponding columns.

i1 : M=QQ^3;

i2 : scan(4, i-> << i+1 << "-th symmetric power of M = " << schurModule({i+1},M) << endl)
                              3
1-th symmetric power of M = QQ
                              6
2-th symmetric power of M = QQ
                              10
3-th symmetric power of M = QQ
                              15
4-th symmetric power of M = QQ

i3 : S = schurModule({3,2,1}, M);

i4 : v = sum(numgens S, i-> (i+1)*S_i) -- an element of S represented by a vector

o4 = | 1 |
     | 2 |
     | 3 |
     | 4 |
     | 5 |
     | 6 |
     | 7 |
     | 8 |

       8
o4 : QQ

i5 : printSchurModuleElement(v, S);
  +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+
1*|0|0|0| 2*|0|0|1| 3*|0|0|2| 4*|0|0|0| 5*|0|0|1| 6*|0|0|2| 7*|0|1|1| 8*|0|1|2| 
  |1|1| |   |1|1| |   |1|1| |   |1|2| |   |1|2| |   |1|2| |   |1|2| |   |1|2| |
  |2| | |   |2| | |   |2| | |   |2| | |   |2| | |   |2| | |   |2| | |   |2| | |
  +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+   +-+-+-+

Caveat

The partition lambda should be a valid nonempty partition.

Ways to use schurModule :

schurModule(List,Module)

For the programmer