Does Sage support computing Gr_{F_q}(k,r), the space of k-hyperplanes
through the origin in GF(q**r)? I looked around for things like Schubert
cells and didn't see anything immediately relevant. I've started to write
some stuff for k=2. I may do a PR if it's not already available.
Thanks,
Jackson
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from itertools import combinations, product
from sage.all import GF, matrix, vector
def schubert_cells_2(q,r):
all_cells = {} # Store all Schubert cells
F = GF(q) # Finite field of order q
indices = list(combinations(range(r), 2)) # Possible pivot column indices
for p0, p1 in indices: # Iterate over pivot column pairs
all_cells[(p0, p1)] = []
M = matrix(F, 2, r) # Initialize a 2×4 zero matrix
M[:, p0] = vector(F, [1, 0]) # Place first pivot column
M[:, p1] = vector(F, [0, 1]) # Place second pivot column
# Determine free parameter positions
free_param_positions = [j for j in range(r) if j not in (p0, p1)]
# Rules for free parameters:
# - If j is between p0 and p1, assign to the top row
# - If j is after p1, assign freely to both rows
# - If j is before p0, do not assign any parameter
def num_free_params(j):
if p0 < j < p1:
return 1
if j > p1:
return 2
if j < p0:
return 0
free_param_counts = [num_free_params(j) for j in free_param_positions]
# Generate all possible parameter assignments
for params in product(*[list(F)] * sum(free_param_counts)):
M_filled = matrix(M) # Copy base matrix
param_iter = iter(params) # Iterator over parameter values
for j in free_param_positions:
if p0 < j < p1: # Free parameter on top row
# Directly assign the parameter to the top row
M_filled[0, j] = next(param_iter)
if j > p1: # Free parameters on both rows
# Directly assign the parameters to both rows, no coercion
M_filled[:, j] = vector([next(param_iter),next(param_iter)])
all_cells[(p0,p1)].append(M_filled)
return all_cells
# Example usage:
cells = schubert_cells_2(5,4) # Example with F_5
sum(len(cells[(p0, p1)]) for p0, p1 in list(combinations(range(4), 2)))
