conjugate(Filling) -- transpose of a Young tableau

Function: conjugate
Usage:

conjugate T
Inputs:
- T, an instance of the type Filling,
Outputs:
- an instance of the type Filling, the tableau of conjugate (transposed) shape whose columns are the rows of T

Description

The conjugate partition $\mu'$ of $\mu$ has $\mu'_j = \#\{i : \mu_i \ge j\}$. As a Young diagram, $\mu'$ is the reflection of $\mu$ across its main diagonal. Since Schur tableaux are stored column-wise (each T#j is a column), conjugation swaps rows and columns.

i1 : T = new Filling from {{0,1}, {1,2}}

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

o1 : Filling

i2 : T

     +-+-+
o2 = |0|1|
     |1|2|
     +-+-+

o2 : Filling

i3 : conjugate T

     +-+-+
o3 = |0|1|
     |1|2|
     +-+-+

o3 : Filling

Conjugation is an involution and intertwines the two tableau-storage conventions:

i4 : new WeylFilling from toList conjugate T

     +-+-+
o4 = |0|1|
     |1|2|
     +-+-+

o4 : WeylFilling

Ways to use this method:

conjugate(Filling) -- transpose of a Young tableau

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurFunctors.m2:1360:0.

conjugate(Filling) -- transpose of a Young tableau

Description

See also

Ways to use this method: