pieriColumn -- native column-form (vertical-strip) Pieri inclusion

Usage:

pieriColumn(mu, cols, P)
Inputs:
- mu, a list, a partition
- cols, a list, a list of 1-indexed column indices. One box is removed from the bottom of each listed column; the cumulative removal must form a vertical strip (no two boxes in the same row).
- P, a polynomial ring, a SkewCommutative $\mathbb{Q}$-algebra in $n$ variables. Its generators carry the wedge factor $\wedge^{|\mathrm{cols}|} V$ on the output side.
Optional inputs:
- Convention => ..., default value "Filling"
Outputs:
- a matrix, a matrix from $S_\mu V$ to $\wedge^{|\mathrm{cols}|} V \otimes S_\lambda V$, with rows indexed by standard Fillings of $\lambda$ and columns indexed by standard Fillings of $\mu$.

Description

The dual of pieri. Where pieri(mu, rows, P) removes a horizontal strip and lands in $\mathrm{Sym}^k V$, pieriColumn(mu, cols, P) removes a vertical strip and lands in $\wedge^k V$. Both maps are unique (up to scalar) $GL(V)$-equivariant inclusions $S_\mu V \to (\,?\text{-tensor}\,) \otimes S_\lambda V$; pieri uses Olver's formula with $c_J = \prod (\mu_{J_q} - \mu_k + k - J_q)$ on the symmetric ($\mathrm{Sym}$) side, while pieriColumn uses the dual formula with $c'_J = \prod (\mu'_k - \mu'_{J_q} + J_q - k)$ on the wedge side, where $\mu'$ is the conjugate partition. The extra sign in $c'_J$ relative to a literal transpose accounts for the wedge anticommutativity.

i1 : needsPackage "SchurFunctors";

i2 : P = QQ[e_0..e_3, SkewCommutative => true];

i3 : M = pieriColumn({3,2,1}, {1}, P);

             60      64
o3 : Matrix P   <-- P

i4 : numRows M, numColumns M

o4 = (60, 64)

o4 : Sequence

i5 : -- Multi-strip: remove a vertical strip of length 2.
     P = QQ[e_0..e_3, SkewCommutative => true];

i6 : M = pieriColumn({3,2,1}, {1,2}, P);

             45      64
o6 : Matrix P   <-- P

i7 : numRows M, numColumns M

o7 = (45, 64)

o7 : Sequence

i8 :

Ways to use pieriColumn:

pieriColumn(List,List,PolynomialRing)

For the programmer

The object pieriColumn 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.m2:899:0.

pieriColumn -- native column-form (vertical-strip) Pieri inclusion

Description

See also

Ways to use pieriColumn:

For the programmer