applyLR -- apply Psi_Q to a single tableau of shape lambda

Usage:

applyLR((lambda, mu, nu), Q, T, n)

applyLR(..., Convention => "Row" | "Filling" | "Weyl")
Inputs:
- (lambda, mu, nu), a sequence, three partitions specifying the LR map
- Q, a list, an LR tableau as returned by lrTableaux
- T, a basic list, a tableau of shape $\lambda$. May be supplied as a List or as a Filling. In "Row"/"Weyl" convention a List is interpreted as PM row form; in "Filling" convention a List is interpreted as column form (and wrapped automatically). A Filling is always treated as column form. T need not be standard; it is straightened internally using the convention's straightening relations.
- n, an integer, the dimension of $V$
Optional inputs:
- Convention => a string, default value "Row", , basis convention used to interpret T and label the output $T_\nu$, $T_\mu$.
Outputs:
- a list, a list of triples {c, T_nu, T_mu} representing the image as the sum $\sum c_i \cdot T_\nu^{(i)} \otimes T_\mu^{(i)}$ in $S_\nu V \otimes S_\mu V$

Description

Computes the image of a single basis vector (or, more generally, of any tableau, which is first straightened into the SSYT basis of $S_\lambda V$) under the LR inclusion $\Psi_Q$. The output is intended for human reading or interactive exploration when the full lrMap matrix is too large to display. Use displayLRImage for pretty-printing.

When Convention => "Filling" is chosen, the input T must be a SchurFunctors Filling, and the output uses Fillings on both target sides. Straightening uses SchurFunctors' column-Garnir relations. When Convention => "Row" or "Weyl", T is a list of rows and PieriMaps' row-Garnir straightening is applied.

i1 : Qs = lrTableaux({3,2,1}, {2,1}, {2,1});

i2 : T = {{0,0,0},{1,1},{2}};   -- highest-weight vector of S_(3,2,1) QQ^3

i3 : applyLR(({3,2,1}, {2,1}, {2,1}), Qs#0, T, 3)

       27                                                                
o3 = {{--, {{0, 0}, {1}}, {{0, 1}, {2}, {}}}, {9, {{0, 0}, {1}}, {{0, 2},
        4                                                                
     ------------------------------------------------------------------------
                   63                                        9          
     {1}, {}}}, {- --, {{0, 0}, {2}}, {{0, 1}, {1}, {}}}, {- -, {{0, 1},
                    4                                        2          
     ------------------------------------------------------------------------
                                27                                        9
     {1}}, {{0, 0}, {2}, {}}}, {--, {{0, 1}, {2}}, {{0, 0}, {1}, {}}}, {- -,
                                 4                                        4
     ------------------------------------------------------------------------
     {{0, 2}, {1}}, {{0, 0}, {1}, {}}}}

o3 : List

i4 : displayLRImage applyLR(({3,2,1}, {2,1}, {2,1}), Qs#0, T, 3)

o4 =   +  27/4 . [1 1 ] ⊗ [1 2 ]
                  2        3 
       +  9 . [1 1 ] ⊗ [1 3 ]
               2        2 
       -  63/4 . [1 1 ] ⊗ [1 2 ]
                  3        2 
       -  9/2 . [1 2 ] ⊗ [1 1 ]
                 2        3 
       +  27/4 . [1 2 ] ⊗ [1 1 ]
                  3        2 
       -  9/4 . [1 3 ] ⊗ [1 1 ]
                 2        2

i5 : -- Non-standard input: row-Garnir straightening before applying.
     Tnonstd = {{1,0},{2}};   -- row 0 not weakly-increasing

i6 : applyLR(({2,1}, {1}, {1,1}), (lrTableaux({2,1},{1},{1,1}))#0, Tnonstd, 3)

       3                           3
o6 = {{-, {{0}, {2}}, {{1}, {}}}, {-, {{1}, {2}}, {{0}, {}}}}
       2                           2

o6 : List

i7 :

Ways to use applyLR:

applyLR(Sequence,List,BasicList,ZZ)

For the programmer

The object applyLR is a method function with options.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/PieriMaps/doc.m2:619:0.

applyLR -- apply Psi_Q to a single tableau of shape lambda

Description

See also

Ways to use applyLR:

For the programmer