coho-incidence

This project contains the source code for replicating the numerical evidence showcased in the author's paper "Even-carry polynomials and cohomology of the incidence correspondence in positive characteristic" (see arxiv:2404.04166). It contains the following files:

• print.m2: Helper code for formatting pictures of symmetric polynomials.
• S.m2: Basic methods for symmetric polynomials.
• h0.m2: Computes $K(d,e)$ by the method given in Proposition 2.2.
• prim.m2: Computes primitive cohomology $K(d,e)^{\mathrm{prim}}$ as described in Section 4.1.
• tiles.m2: Explores tiles and produces the tables in Section 4.2.
• general.m2: Explores the questions asked in Section 4.3.

Each of these files automatically loads all the preceding ones, except that the last two are independent. So you can launch M2 in the directory where you downloaded this project and run:

load("tiles.m2")
load("general.m2")

Some examples of usage follow.

h0.m2

To display a particular $\kappa(d,e)$:

graph kappa(2,5)

To display all $\kappa(d,e)$ up to a given maximum value of $e$ ($e \leq 30$ by default):

allKappa(emax => 10)

To compute an individual coefficient of $\kappa(d,e)$ (faster than computing the whole character), or a basis for an individual eigenspace of $K(d,e)$:

h0b(2,5,{1,2,3})
H0b(2,5,{1,2,3})

Note: The values of $n$ and $p$ are coded in S.m2. By default, $n = p = 3$. To change these values, modify and reload the file S.m2, or use the updatenp routine, e.g.

n = 4
p = 2
updatenp()

prim.m2

To separate $K(d,e)$ into primitive and nonprimitive components:

graph kappa(3,4)
graph nonprim(3,4)
graph primChar(3,4)

To print a table of primitive characters (like Table 1):

primTable(dif => 1, mmax => 10)

To print a table of conjectured primitive characters, assuming certain structural hypotheses that are weakened versions of Conjectures 4.7 and 4.9:

primGuessTable(dif => 1, mmax => 10)

To check that these guesses are correct:

primTestAll(dif => 1, mmax => 10)

tiles.m2

To find tiles (as in Tables 2, 3, 4):

mineTiles(nmin => 3, nmax => 4, pmax => 7, mmax => 15)

general.m2

The contents of this file are more more speculative. To guess $\kappa(d,e)$, based on a formula similar to Conjecture 4.10 but missing the unknown terms $P_i(m)$ and trying to infer them instead by truncation:

testAll(emax => 15)
testEdge(emax => 30)

Final note

If you use jEdit to edit code, please check out my syntax highlighting modes for MAGMA and Macaulay2.

Written with StackEdit.