weyl -- construct a WeylFilling or apply the Weyl functor to a map

Usage:

weyl L

weyl(lambda, f)
Inputs:
- L, a list, a list of rows, each row a list of integers (weakly increasing); interpreted as the rows of a WeylFilling
- lambda, a list, a partition
- f, a matrix, a map $f : E \to F$ between free modules
Outputs:
- in the one-argument form, a WeylFilling; in the two-argument form, a Matrix giving the induced map $W_\lambda(f) : W_\lambda(E) \to W_\lambda(F)$

Description

The one-argument form is a constructor for WeylFilling:

i1 : T = weyl {{0,0,1}, {1,2}}

     +-+-+-+
o1 = |0|0|1|
     |1|2| |
     +-+-+-+

o1 : WeylFilling

i2 : class T

o2 = WeylFilling

o2 : Type

i3 : isWeylStandard T

The two-argument form is the Weyl functor applied to a morphism of free modules. Functoriality: $W_\lambda(f \circ g) = W_\lambda(f) \circ W_\lambda(g)$ and $W_\lambda(\mathrm{id}) = \mathrm{id}$:

i4 : F = matrix{{1_QQ,2},{3,4}};

              2       2
o4 : Matrix QQ  <-- QQ

i5 : weyl({2}, F)

o5 = | 1 4  4  |
     | 3 10 8  |
     | 9 24 16 |

              3       3
o5 : Matrix QQ  <-- QQ

i6 : A = random(QQ^3, QQ^3);

              3       3
o6 : Matrix QQ  <-- QQ

i7 : B = random(QQ^3, QQ^3);

              3       3
o7 : Matrix QQ  <-- QQ

i8 : weyl({2,1}, A * B) == weyl({2,1}, A) * weyl({2,1}, B)

o8 = true

For special shapes $W_\lambda$ coincides with familiar functors: $W_{(d)} = D^d$ (divided power), $W_{(1^d)} = \wedge^d$ (exterior power, also equal to $S_{(1^d)}$).

i9 : weyl({2}, F) == dividedPower(2, F)

o9 = true

Ways to use weyl:

weyl(List)
weyl(List,Matrix)

For the programmer

The object weyl is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2266:0.

weyl -- construct a WeylFilling or apply the Weyl functor to a map

Description

See also

Ways to use weyl:

For the programmer