Equivariant Schubert Calculator

The Equivariant Schubert Calculator is a Maple package able to perform computations in the equivariant K-theory ring KT(X) of any generalized flag variety X=G/P. For cominuscule varieties it can also do calculations in the (small) equivariant quantum K-theory ring QKT(X). In addition, the package can present its results in (and perform conversions between) several common indexing conventions for Schubert classes.

At its core, this software package implements general functions for doing calculations with Weyl groups and root systems. These functions are used to implement calculations in equivariant (quantum) K-theory, as well as computations of curve neighborhoods of Schubert varieties.

To install the Equivariant Schubert Calculator, download the file
and unpack it using the (unix) command:
    % tar zxvf EquivCalc-1.0.1.tar.gz

This will create a directory called EquivCalc-1.0.1 that contains the Maple program file equivcalc, a (very preliminary) user manual, and several examples. The user manual will be expanded later.

For a quick demonstration of the capabilities, the following Maple commands will compute the multiplication table of the quantum K-theory ring of the Grassmannian Gr(2,4):

   % cd EquivCalc-1.0.1
   % maple
   > read equivcalc;
   > with(equivcalc):
   > Gr(2,4);
   > for u in weyl_group() do for v in weyl_group() do
   >   lprint(weyl_comin(X[op(u)] * X[op(v)] = comin_qkmult(u, v)));
   > od; od;

More examples can be found in the files ex_twostep. ex_qkgrass, ex_qktP2, ex_kminu, ex_curvenbhd, ex_kpieri, and ex_qkcayley. For example, the first example can be executed with the (unix) command:

    % maple -q < ex_twostep
Notice that the examples ex_kpieri and ex_qkcayley take a while to execute, and some versions of Maple require a large amount of memory to complete these examples. All of the examples can be executed using Maple 14 on my own computer with 4GB of RAM. However, I have not been able to execute the last two examples with Maple 15 or Maple 16. I expect that Maple 17 will perform better.

I will be grateful for any comments or bug reports for this package. Enjoy!

Anders Buch

During the development of this software, the author was partially supported by NSF grants DMS-0603822, DMS-0906148, and DMS-1205351.