Math.RootLoci.Geometry.Forget

Geometry of the degree n! finite map pi, which just forgets the order points:

pi : Q^n = P^1 x P^1 x ... x P^1  ->  P^n = P(Sym^n C^2)

It's clear that the degree of pi restricted to an open stratum corresponding to a partition mu is the multinomial coefficient corresponding to n choose mu.

It is also not hard to see that the degree of pi restricted to the intersection of the open stratum corresponding to mu with the image of the diagonal map corresponding to nu equals the number of "automorphisms" aut(mu) = prod (e_i!) where mu = (1^e1 2^e2 ... k^ek) and the number of ways nu is refinement of mu.

Note that for nu=(1,1...1) the multinomial agrees with the number of refinements.

This module contains functions to compute these numbers.

Documentation

countCoarsenings :: Partition -> Map Partition Integer

Given a partition, we list all coarser partitions together with the number of ways the input is a refinement of the coarser partition.

TODO: at the moment this is just a synonym for countCoarseningsNaive ...

countDirectCoarsenings :: Partition -> Map Partition Integer

Count coarsenings (with multiplicities) which are shorter by just 1.

countCoarseningsNaive :: Partition -> Map Partition Integer

Naive (very slow) implementation of countCoarsenings.

countPreimage :: Partition -> Map Partition Integer

Given a partition nu, we stratify the image of the corresponding diagonal Delta_nu as usual, and list the degree of pi restricted to these strata

This is just counting the coarsenings, multiplied by the number of "automorphisms" of the partition.

preimageView :: Partition -> (Integer, Map Partition Integer)

The preimage counts, but the partition itself is separated out.

countFullPreimage :: Partition -> Integer

The preimage pi^-1(x) of a point under the map pi : Q^n -> P^n is just a multinomial coefficient