dividedPower -- divided-power functor on free modules and matrices

Usage:

dividedPower(d, E)

dividedPower(lambda, E)

dividedPower(d, f)

dividedPower(lambda, f)
Inputs:
- d, an integer,
- lambda, a list, a list of nonnegative integers; the output is the tensor product of divided powers $D^{\lambda_1}(\cdot) \otimes \cdots \otimes D^{\lambda_k}(\cdot)$
- E, a module, a free module
- f, a matrix, a map between free modules
Outputs:
- the divided power $D^d(E)$ or the induced map $D^d(f)$, and for list input the tensor product of divided powers

Description

The $d$-th divided power $D^d(E)$ of a free module $E$ of rank $n$ is the degree-$d$ part of the divided-power algebra on $E$. It is free of rank $\binom{n+d-1}{d}$, with a basis indexed by weakly-increasing tuples of length $d$ in $\{0,\ldots,n-1\}$ (equivalently, compositions of $d$ into $n$ parts, or multisets of size $d$ from $\{0,\ldots,n-1\}$).

Over a field of characteristic zero $D^d(E) \cong \mathrm{Sym}^d(E)$ canonically (the isomorphism rescales basis vectors by factorials); over $\mathbb{Z}$ the two functors differ.

i1 : E = QQ^3;

i2 : dividedPower(2, E)                   -- rank C(4,2) = 6

       6
o2 = QQ

o2 : QQ-module, free

i3 : dividedPower({2,1,0,1}, E)           -- D^2 ⊗ D^1 ⊗ D^0 ⊗ D^1

       54
o3 = QQ

o3 : QQ-module, free

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

              2       2
o4 : Matrix QQ  <-- QQ

i5 : dividedPower(2, f)                   -- induced map on D^2

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

              3       3
o5 : Matrix QQ  <-- QQ

The single-row (non-list) form with a List partition builds the ambient tensor product used by weylModule:

i6 : dividedPower({2,1}, E) == dividedPower(2, E) ** dividedPower(1, E)

o6 = true

dividedPower is used as the ambient object in the construction of weylModule, playing the role that exteriorPower plays for schurModule.

Ways to use dividedPower:

dividedPower(List,Matrix)
dividedPower(List,Module)
dividedPower(ZZ,Matrix)
dividedPower(ZZ,Module)

For the programmer

The object dividedPower is a method function.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:2436:0.

dividedPower -- divided-power functor on free modules and matrices

Description

See also

Ways to use dividedPower:

For the programmer