dualPieriColumn -- GL-equivariant projection wedge^d V ⊗ S_lambda V --> S_mu V

• Usage:

  dualPieriColumn(mu, cols, P)

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

• Inputs:
  • mu, a list, the target shape: $\mu = \lambda + (\text{vertical $d$-strip})$ with $d = \#\mathrm{cols}$
  • cols, a list, a list of column indices recording the strip $\mu/\lambda$
  • P, a polynomial ring, a SkewCommutative ring ($\wedge^d V$ is encoded there)
• Optional inputs:
  • Convention => a string, default value "Filling"
• Outputs:
  • a matrix, a $K$-matrix of size $\dim S_\mu V \times \bigl(\dim S_\lambda V \cdot \binom{n}{d}\bigr)$. Columns indexed by $(T_\lambda, \alpha)$ pairs ($\alpha$ an exterior monomial).

i1 : P = QQ[e_0..e_2, SkewCommutative => true];

i2 : M = pieriColumn({2,1}, {1}, P);

             6      8
o2 : Matrix P  <-- P

i3 : N = dualPieriColumn({2,1}, {1}, P);

              8       18
o3 : Matrix QQ  <-- QQ

i4 : numRows N, numColumns N

o4 = (8, 18)

o4 : Sequence