applyDualPieri -- apply dualPieri to a single basis pair (poly, T_lambda)

• Usage:

  applyDualPieri((mu, boxes), poly, T, n)

  applyDualPieri(..., Convention => "Row" | "Filling" | "Weyl")

• Inputs:
  • (mu, boxes), a sequence, the same data passed to dualPieri
  • poly, a ring element, a homogeneous polynomial of degree $d = \#\mathrm{boxes}$ in the polynomial ring P (an element of $\mathrm{Sym}^d V$)
  • T, a basic list, a tableau of shape $\lambda$. Accepted as either a List or a Filling. In "Row"/"Weyl" convention a List is the PM row form; in "Filling" convention a List is column form (auto-wrapped). A Filling is always treated as column form. Non-standard inputs are straightened first using the convention's relations.
  • n, an integer, $\dim V$
• Optional inputs:
  • Convention => a string, default value "Row", , basis for source $S_\lambda$ and target $S_\mu$.
• Outputs:
  • a list, a list of pairs {coefficient, T_mu} representing the image as a sum $\sum c_i \cdot T_\mu^{(i)}$ in $S_\mu V$. Zero coefficients are dropped.

i1 : P = QQ[a,b,c];

i2 : Tlam = {{0,0},{1}};   -- standard PM tableau of {2,1}

i3 : applyDualPieri(({3,1}, {1}), a, Tlam, 3)

       1
o3 = {{-, {{0, 0, 0}, {1}}}}
       3

o3 : List

i4 : -- Non-standard input is straightened first (row-Garnir).
     P = QQ[a,b,c,d];

i5 : applyDualPieri(({3,2,1}, {3}), a, {{1,0},{2}}, 4)

o5 = {}

o5 : List

i6 :