/Fix-cones-in-weight-lattices

Primary LanguageTeX

DOI

The code in this repository complements the following research manuscript:

[HZ] Tobias Hemmert & Marcus Zibrowius, The Witt rings of many flag varieties are exterior algebras

Prerequisites

The code is written in Macaulay2 and requires the WeylGroups package, written by Baptiste Calmès and Viktor Petrov. It has been tested with Macaulay2 version 1.22, which includes WeylGroups package version 0.5.3. It should also run with Macaulay2 version 1.18 and WeylGroups package version 0.5.2. Older versions of the WeylGroups package contain a bug that will render the results incorrect. To see which version of the package you have, you can type readPackage "WeylGroups" in Macaulay2.

Executing the code

The computations necessary to complete the proof of [HZ, Proposition 3.3] can be run by executing all code in main.m2. The code there uses functions from the WeylGroups package and the two small auxiliary packages WeylGroupsExtra and Auxiliary provided here. The code is currently set up to peform computations for all Dynkin diagrams Σ of exceptional types (E6, E7, E8, G2, F4). This can easily be changed by editing the very last line of main.m2.

The proof of [HZ, Proposition 3.3] only requires the verification of the condition single cell for certain pairs (Σ, H). In addition, the code also checks whether the fix point moniod written as $\overline{\mathcal C}(H)^{[H]}$ in [HZ] is a free abelian monoid (in two different ways), and whether the pair (Σ, H) satisfies the weaker condition orbit basis. However, because of the associated computational costs, the verification of this last condition is restricted to root systems Σ of rank < 8. To change this behaviour, remove the conditional in the line

  if rank(R) < 8 then result#"orbitcondition" = checkIfLSatisfiesOrbitCondition(R,P);

in main.m2.

Reading the code

An effort has been made to make the code as self-explanatory as possible. The term FixedPointMonoid refers to the fixed point monoid written as $\overline{\mathcal C}(H)^{[H]}$ in [HZ].

Viewing the results

The results of the computations are written to tex files (results_G2.tex, results_F4.tex, ...). To view them, compile the auxiliary file ViewResults.tex also provided here. To display results for other than the exceptional types, the contents of ViewResults.tex need to be edited in an obvious way. For reference, results for exceptional types are already included in the folder results.

Interpreting the results

The notation in ViewResults.tex follows [HZ]. The numbering of simple roots follows the conventions of Nicolas Bourbaki, Lie groups and Lie algebras 4-6 (see plates at the end of the book). The conditions single cell and orbit basis are explained in [HZ, §1: Overview]. The condition free signifies whether the fixed point monoid $\overline{\mathcal C}(H)^{[H]}$ is free abelian.