normalize -- sort each column of a Filling, with Koszul sign

Usage:

(c, T') = normalize T
Inputs:
- T, an instance of the type Filling,
Outputs:
- a sequence, a pair $(c, T')$ with $c \in \{-1, 0, 1\}$ and $T'$ either a Filling or null

Description

Each column of a Filling represents an element of an exterior power, so repeated entries in a column collapse the corresponding wedge to zero, and reordering a column changes the sign by the parity of the permutation. normalize T returns

$(0, \texttt{null})$ if any column has a repeated entry;
$(1, T')$ if every column sorts into strictly increasing order under an even permutation;
$(-1, T')$ if the total permutation is odd.

This is the first step of the straightening algorithm; after it, $T'$ lives in the standard ambient $\wedge^{\mu'_1}(E) \otimes \cdots \otimes \wedge^{\mu'_k}(E)$ with a known sign.

i1 : normalize new Filling from {{1,0}, {2,1}}

         +-+-+
o1 = (1, |0|1|)
         |1|2|
         +-+-+

o1 : Sequence

i2 : normalize new Filling from {{0,0}, {1,2}}

o2 = (0, )

o2 : Sequence

i3 : normalize new Filling from {{2,1,0}, {1,0}}

         +-+-+
o3 = (1, |0|0|)
         |1|1|
         |2| |
         +-+-+

o3 : Sequence

Ways to use normalize:

normalize(Filling)

For the programmer

The object normalize is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:1403:0.

normalize -- sort each column of a Filling, with Koszul sign

Description

See also

Ways to use normalize:

For the programmer