pmToFillingMatrix -- QQ matrix of the PM-to-Filling change of basis on a Schur module

Usage:

pmToFillingMatrix(mu, n)
Inputs:
- mu, a list, a partition (the Schur shape)
- n, an integer, the dimension of $V$
Outputs:
- a matrix, a $\mathbb{Q}$ matrix; rows indexed by Fillings of column heights $\mu^t$, columns indexed by PieriMaps' standard tableaux of shape $\mu$. Entry $(i, j)$ is the coefficient of Filling $i$ in pmToFilling(PMtab_j).

Description

This matrix realizes the equivariant change of basis between PieriMaps' row-form basis of $S_\mu V$ and SchurFunctors' column-form basis. It is invertible over $\mathbb{Q}$ (up to a shape-dependent overall scalar) and is the workhorse behind pieri(...,Convention=>...)/lrMap(...,Convention=>...) for the Filling convention.

Source/target basis labels are attached so symbolicForm can display the change-of-basis on each PM tableau:

i1 : symbolicForm pmToFillingMatrix({2,1}, 3)

     +-----+-------------+
     |+-+-+|      +-+-+  |
o1 = ||0|0||1  *  |0|0|  |
     ||1| ||      |1| |  |
     |+-+-+|      +-+-+  |
     +-----+-------------+
     |+-+-+|      +-+-+  |
     ||0|0||1  *  |0|0|  |
     ||2| ||      |2| |  |
     |+-+-+|      +-+-+  |
     +-----+-------------+
     |+-+-+|1     +-+-+  |
     ||0|1||-  *  |0|1|  |
     ||1| ||2     |1| |  |
     |+-+-+|      +-+-+  |
     +-----+-------------+
     |+-+-+|  1     +-+-+|
     ||0|1||- -  *  |0|2||
     ||2| ||  2     |1| ||
     |+-+-+|        +-+-+|
     |     |      +-+-+  |
     |     |1  *  |0|1|  |
     |     |      |2| |  |
     |     |      +-+-+  |
     +-----+-------------+
     |+-+-+|      +-+-+  |
     ||0|2||1  *  |0|2|  |
     ||1| ||      |1| |  |
     |+-+-+|      +-+-+  |
     |     |  1     +-+-+|
     |     |- -  *  |0|1||
     |     |  2     |2| ||
     |     |        +-+-+|
     +-----+-------------+
     |+-+-+|1     +-+-+  |
     ||0|2||-  *  |0|2|  |
     ||2| ||2     |2| |  |
     |+-+-+|      +-+-+  |
     +-----+-------------+
     |+-+-+|      +-+-+  |
     ||1|1||1  *  |1|1|  |
     ||2| ||      |2| |  |
     |+-+-+|      +-+-+  |
     +-----+-------------+
     |+-+-+|1     +-+-+  |
     ||1|2||-  *  |1|2|  |
     ||2| ||2     |2| |  |
     |+-+-+|      +-+-+  |
     +-----+-------------+

For the programmer

The object pmToFillingMatrix is a function closure.

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/PieriMaps/doc.m2:826:0.

pmToFillingMatrix -- QQ matrix of the PM-to-Filling change of basis on a Schur module

Description

See also

For the programmer