Description
PieriMaps' row-form representation $S_\lambda V \subset \mathrm{Sym}^{\lambda_1} V \otimes \cdots$ and SchurFunctors' column-form representation $S_\lambda V \subset \wedge^{\lambda'_1} V \otimes \cdots$ are equivariant isomorphisms (over $\mathbb{Q}$). This function realizes that iso explicitly: each PM monomial $T = (m_1, \ldots, m_r)$ is symmetrized into $V^{\otimes |\lambda|}$ via the row-symmetrizer (with normalization $1/\prod \lambda_i!$), reordered by columns, and then projected to wedges (sort $+$ sign) per column. The result is then straightened into the standard
Filling basis via SchurFunctors'
straighten.
Round-tripping
pmToFilling then
fillingToPM gives a uniform shape-dependent scalar $c_\lambda \cdot \mathrm{id}$ (the Young symmetrizer normalization).
i1 : needsPackage "SchurFunctors";
|
i2 : pmToFilling {{0,1},{2}}
+-+-+ 1
o2 = HashTable{|0|2| => - -}
|1| | 2
+-+-+
+-+-+
|0|1| => 1
|2| |
+-+-+
o2 : HashTable
|
i3 : pmToFilling {{0,0,0},{1,1},{2}} -- highest-weight vector of S_(3,2,1) Q^3
+-+-+-+
o3 = HashTable{|0|0|0| => 1}
|1|1| |
|2| | |
+-+-+-+
o3 : HashTable
|