Description
Builds the $GL(V)$-equivariant inclusion $\Psi_Q \colon S_\lambda V \to S_\nu V \otimes S_\mu V$ associated to the LR tableau
Q. The map is constructed by iterating row-Pieri inclusions $S_{\lambda^{(a)}} V \to \mathrm{Sym}^{\nu_a} V \otimes S_{\lambda^{(a-1)}} V$ in the order $a = r, r-1, \ldots, 1$, then projecting the $\mathrm{Sym}$ tensor side onto $S_\nu V$ via straightening.
The columns of the output are indexed by
standardTableaux(n, lambda) (or the corresponding Filling / WeylFilling basis if a non-default
Convention is chosen). The rows are indexed by pairs $(T_\nu, T_\mu)$ in lex order: row $i \cdot \#S_\mu + j$ corresponds to $(T_\nu = N\nu\#i, T_\mu = N\mu\#j)$ where
Nnu = standardTableaux(n, nu) and
Nmu = standardTableaux(n, mu padded to length #lambda). When
Convention => "Filling" is used, the equivariant change of basis
pmToFillingMatrix is applied on each side, and the rank is preserved.
i1 : shapes = ({2,1}, {1}, {1,1}); -- a multiplicity-1 example
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i2 : Q = (lrTableaux shapes)#0;
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i3 : Mrow = lrMap(shapes, Q, 3, Convention => "Row")
o3 = | 3 0 0 0 0 0 0 0 |
| 0 0 3/2 0 0 0 0 0 |
| 0 0 0 0 3/2 0 0 0 |
| 0 3 0 0 0 0 0 0 |
| 0 0 0 3/2 0 0 0 0 |
| 0 0 0 0 0 3/2 0 0 |
| 0 0 0 3/2 -3/2 0 0 0 |
| 0 0 0 0 0 0 3 0 |
| 0 0 0 0 0 0 0 3/2 |
9 8
o3 : Matrix QQ <-- QQ
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i4 : shapes = ({2,1}, {1}, {1,1});
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i5 : Q = (lrTableaux shapes)#0;
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i6 : Mfilling = lrMap(shapes, Q, 3, Convention => "Filling")
o6 = | 3 0 0 0 0 0 0 0 |
| 0 3 0 0 0 0 0 0 |
| 0 0 2 0 1 0 0 0 |
| 0 0 0 3 0 0 0 0 |
| 0 0 1 0 2 0 0 0 |
| 0 0 0 0 0 3 0 0 |
| 0 0 -1 0 1 0 0 0 |
| 0 0 0 0 0 0 3 0 |
| 0 0 0 0 0 0 0 3 |
9 8
o6 : Matrix QQ <-- QQ
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i7 : shapes = ({2,1}, {1}, {1,1});
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i8 : Q = (lrTableaux shapes)#0;
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i9 : Mweyl = lrMap(shapes, Q, 3, Convention => "Weyl")
o9 = | 3 0 0 0 0 0 0 0 |
| 0 0 3/2 0 0 0 0 0 |
| 0 0 0 0 3/2 0 0 0 |
| 0 3 0 0 0 0 0 0 |
| 0 0 0 3/2 0 0 0 0 |
| 0 0 0 0 0 3/2 0 0 |
| 0 0 0 3/2 -3/2 0 0 0 |
| 0 0 0 0 0 0 3 0 |
| 0 0 0 0 0 0 0 3/2 |
9 8
o9 : Matrix QQ <-- QQ
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All three are GL-equivariant injections of $S_\lambda V$ into $S_\nu V \otimes S_\mu V$. The Row and Weyl matrices have the same raw entries (only the source/target bases are reinterpreted on the divided-power side); the Filling matrix is obtained by post-hoc basis change. All three have the same rank.
i10 : shapes = ({2,1}, {1}, {1,1});
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i11 : Q = (lrTableaux shapes)#0;
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i12 : Mrow = lrMap(shapes, Q, 3, Convention => "Row");
9 8
o12 : Matrix QQ <-- QQ
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i13 : Mfilling = lrMap(shapes, Q, 3, Convention => "Filling");
9 8
o13 : Matrix QQ <-- QQ
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i14 : Mweyl = lrMap(shapes, Q, 3, Convention => "Weyl");
9 8
o14 : Matrix QQ <-- QQ
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i15 : (rank Mrow, rank Mfilling, rank Mweyl)
o15 = (8, 8, 8)
o15 : Sequence
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When $c^\lambda_{\mu,\nu} > 1$, distinct LR tableaux $Q$ give linearly independent inclusions; the joint matrix has rank $c^\lambda_{\mu,\nu} \cdot \dim S_\lambda V$. The canonical multiplicity-2 example is $c^{(3,2,1)}_{(2,1),(2,1)} = 2$:
i16 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1}); -- two distinct LR tableaux
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i17 : M1 = lrMap(({3,2,1}, {2,1}, {2,1}), Qs#0, 3);
64 8
o17 : Matrix QQ <-- QQ
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i18 : M2 = lrMap(({3,2,1}, {2,1}, {2,1}), Qs#1, 3);
64 8
o18 : Matrix QQ <-- QQ
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i19 : rank M1, rank M2 -- each = dim S_(3,2,1) = 8
o19 = (8, 8)
o19 : Sequence
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i20 : rank(M1 | M2) == 16 -- 2 * 8: M1 and M2 are linearly independent
o20 = true
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Use
symbolicForm to read off the action of $\Psi_Q$ on each standard tableau of $S_\lambda V$, as a sum over basis pairs of $S_\nu V \otimes S_\mu V$:
i21 : shapes = ({2,1}, {1}, {1,1});
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i22 : Q = (lrTableaux shapes)#0;
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i23 : symbolicForm lrMap(shapes, Q, 3)
+-----+-------------------+
|+-+-+| +-+ +-+ |
o23 = ||0|0||3 * |0| ⊗ |0| |
||1| || |1| +-+ |
|+-+-+| +-+ |
+-----+-------------------+
|+-+-+| +-+ +-+ |
||0|0||3 * |0| ⊗ |0| |
||2| || |2| +-+ |
|+-+-+| +-+ |
+-----+-------------------+
|+-+-+|3 +-+ +-+ |
||0|1||- * |0| ⊗ |1| |
||1| ||2 |1| +-+ |
|+-+-+| +-+ |
+-----+-------------------+
|+-+-+|3 +-+ +-+ |
||0|1||- * |0| ⊗ |1| |
||2| ||2 |2| +-+ |
|+-+-+| +-+ |
| |3 +-+ +-+ |
| |- * |1| ⊗ |0| |
| |2 |2| +-+ |
| | +-+ |
+-----+-------------------+
|+-+-+|3 +-+ +-+ |
||0|2||- * |0| ⊗ |2| |
||1| ||2 |1| +-+ |
|+-+-+| +-+ |
| | 3 +-+ +-+|
| |- - * |1| ⊗ |0||
| | 2 |2| +-+|
| | +-+ |
+-----+-------------------+
|+-+-+|3 +-+ +-+ |
||0|2||- * |0| ⊗ |2| |
||2| ||2 |2| +-+ |
|+-+-+| +-+ |
+-----+-------------------+
|+-+-+| +-+ +-+ |
||1|1||3 * |1| ⊗ |1| |
||2| || |2| +-+ |
|+-+-+| +-+ |
+-----+-------------------+
|+-+-+|3 +-+ +-+ |
||1|2||- * |1| ⊗ |2| |
||2| ||2 |2| +-+ |
|+-+-+| +-+ |
+-----+-------------------+
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