schubmult 1.0.20

Computing Littlewood-Richardson coefficients of Schubert polynomials

schubmult

Program and package for computing Littlewood-Richardson coefficients of Schubert polynomials

This is a set of python scripts written by Matt Samuel for computing Littlewood-Richardson coefficients of (ordinary or double) Schubert polynomials. It has the same command line syntax as the program "schubmult" in lrcalc by Anders Buch. Example:

schubmult_py 1 2 4 9 11 6 8 12 3 5 7 10 - 6 8 1 2 3 4 7 10 12 14 5 9 11 13  
schubmult_double 1 3 4 6 2 5 - 2 1 5 7 3 4 6  
schubmult_yz 1 3 4 6 2 5 - 2 1 5 7 3 4 6  

Runtime will vary tremendously by case. The general problem is #P-hard. Though the result is always nonnegative and the problem is in GapP, it is not known to be in #P at this time.

schubmult_py is for multiplying ordinary Schubert polynomials. schubmult_yz is for multiplying double Schubert polynomials in different sets of coefficient variables (labeled y and z), and schubmult_double is for multiplying double Schubert polynomials in the same set of coefficient variables. Both have the same command line syntax as schubmult. schubmult_double displays the result with nonnegative coefficients in terms of the negative simple roots. Both are of course slower than schubmult_py, and expressing the result positively for schubmult_double slows it down even more.

When imported as a python package, the relevant packages are schubmult.perm_lib, which has various permutation manipulation functions, and three modules that have functions of the same name (function name is "schubmult"): schubmult.schubmult_py, schubmult.schubmult_yz, schubmult.schubmult_double. Function takes a permutation dictionary (keys are tuples of ints, which must be trimmed permutations, and values are either integers or symengine values, which can also be integers) as well as a permutation as its second argument, which is the (double) Schubert polynomial to multiply by. Returns a dictionary of the same form with the coefficients.

from schubmult.schubmult_yz import schubmult  
  
coeff_dict = schubmult({(1,3,4,6,2,5): 1},(2,1,5,7,3,4,6)) # outputs dictionary with results  

from schubmult.schubmult_py import schubmult  
  
coeff_dict = schubmult({(1,3,4,6,2,5): 1},(2,1,5,7,3,4,6))

Note versions 1.0.15 and prior had a bug that failed to upgrade the executable. The coefficients it computed were correct, but it was not using the updated version.

Version 1.0.18 adds the command line argument --display-positive to schubmult_yz, which displays the result positively (if possible, this is still only always possible conjecturally). This is highly processor intensive.

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