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 exteriormu E to M;", "finv is a map from M to exteriormu 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)."
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
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i3 : S = schurModule({3,2,1}, M);
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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
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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| | | +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ |