weylModule -- Weyl module of a partition applied to a free module

Usage:

weylModule(lambda, E)
Inputs:
- lambda, a list, a partition $\lambda = (\lambda_1 \ge \cdots \ge \lambda_r)$
- E, a module, a free module over some ring $R$
Outputs:
- a module, the Weyl module $W_\lambda(E)$, a free $R$-module whose rank equals the number of Weyl-semistandard fillings of shape $\lambda$ with entries in $\{0,\ldots,\mathrm{rank}(E) - 1\}$

Description

The Weyl functor $W_\lambda$ is the divided-power analogue of $S_\lambda$. It is realized here as the quotient $$D^{\lambda_1}(E) \otimes \cdots \otimes D^{\lambda_r}(E) \;\twoheadrightarrow\; W_\lambda(E)$$ by the divided-power Garnir relations across adjacent rows; see Akin-Buchsbaum-Weyman, Schur functors and Schur complexes, Adv. Math. 44 (1982).

In characteristic zero $W_\lambda \cong S_\lambda$ canonically, but over $\mathbb{Z}$ (or in positive characteristic) $W_\lambda$ and $S_\lambda$ are distinct functors related by duality.

The rank matches the usual Weyl-dimension count:

i1 : E = QQ^3;

i2 : weylModule({2}, E)          -- divided square: rank C(3+1, 2) = 6

       6
o2 = QQ

o2 : QQ-module, free

i3 : rank oo

o3 = 6

i4 : weylModule({1,1}, E)        -- W_{(1,1)} = ∧^2, rank 3

       3
o4 = QQ

o4 : QQ-module, free

i5 : rank oo

o5 = 3

For a genuinely mixed shape:

i6 : W = weylModule({2,1}, E);

i7 : rank W

o7 = 8

i8 : standardWeylTableaux(3, {2,1})

      +-+-+  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+
o8 = {|0|0|, |0|0|, |0|1|, |0|1|, |1|1|, |0|2|, |0|2|, |1|2|}
      |1| |  |2| |  |1| |  |2| |  |2| |  |1| |  |2| |  |2| |
      +-+-+  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+

o8 : List

Cached data. The returned module W stores in W.cache#"Weyl" the 4-tuple (f, finv, AT, ST), exactly parallel to schurModule:

i9 : W.cache#"Weyl"

                                                                          
o9 = {| 0 1 0 -1 0 0 0  0 0 0  0 0 0  0  0 0  0  0 |, | 0 0 0 0 0 0 0 0 |,
      | 0 0 1 0  0 0 0  0 0 -1 0 0 0  0  0 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  0 0  0  0 |  | 0 1 0 0 0 0 0 0 | 
      | 0 0 0 0  0 1 0  0 0 0  0 0 -1 0  0 0  0  0 |  | 0 0 0 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 1 0 0 0 0 0 | 
      | 0 0 0 0  0 0 0  0 0 0  1 0 -1 0  0 0  0  0 |  | 0 0 0 1 0 0 0 0 | 
      | 0 0 0 0  0 0 0  0 0 0  0 1 0  0  0 -1 0  0 |  | 0 0 0 0 0 0 0 0 | 
      | 0 0 0 0  0 0 0  0 0 0  0 0 0  0  1 0  -1 0 |  | 0 0 0 0 0 0 0 0 | 
                                                      | 0 0 0 0 1 0 0 0 | 
                                                      | 0 0 0 0 0 0 0 0 | 
                                                      | 0 0 0 0 0 1 0 0 | 
                                                      | 0 0 0 0 0 0 1 0 | 
                                                      | 0 0 0 0 0 0 0 0 | 
                                                      | 0 0 0 0 0 0 0 0 | 
                                                      | 0 0 0 0 0 0 0 1 | 
                                                      | 0 0 0 0 0 0 0 0 | 
                                                      | 0 0 0 0 0 0 0 0 | 
                                                      | 0 0 0 0 0 0 0 0 | 
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
                                                                          
     ------------------------------------------------------------------------
               +-+-+                   +-+-+
     HashTable{|0|0| => 0 }, HashTable{|0|0| => 0}}
               |0| |                   |1| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|0| => 1              |0|0| => 1
               |1| |                   |2| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|0| => 2              |0|1| => 2
               |2| |                   |1| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|1| => 3              |0|1| => 3
               |0| |                   |2| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|1| => 4              |0|2| => 5
               |1| |                   |1| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|1| => 5              |0|2| => 6
               |2| |                   |2| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|2| => 9              |1|1| => 4
               |0| |                   |2| |
               +-+-+                   +-+-+
               +-+-+                   +-+-+
               |0|2| => 10             |1|2| => 7
               |1| |                   |2| |
               +-+-+                   +-+-+
               +-+-+
               |0|2| => 11
               |2| |
               +-+-+
               +-+-+
               |1|1| => 6
               |0| |
               +-+-+
               +-+-+
               |1|1| => 7
               |1| |
               +-+-+
               +-+-+
               |1|1| => 8
               |2| |
               +-+-+
               +-+-+
               |1|2| => 12
               |0| |
               +-+-+
               +-+-+
               |1|2| => 13
               |1| |
               +-+-+
               +-+-+
               |1|2| => 14
               |2| |
               +-+-+
               +-+-+
               |2|2| => 15
               |0| |
               +-+-+
               +-+-+
               |2|2| => 16
               |1| |
               +-+-+
               +-+-+
               |2|2| => 17
               |2| |
               +-+-+

o9 : List

i10 : (f, finv, AT, ST) = toSequence W.cache#"Weyl";

i11 : f * finv == id_W

o11 = true

Caveat

The partition lambda should be nonempty and weakly decreasing. Entries of lambda need not bound the rank of E, but the output module may of course have small rank.

Ways to use weylModule:

weylModule(List,Module)

For the programmer

The object weylModule is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2211:0.

weylModule -- Weyl module of a partition applied to a free module

Description

Caveat

See also

Ways to use weylModule:

For the programmer