Field With One Element - Computations

Computations

Various structures on a set are analogous to structures on a projective space, and can be computed in the same way:

Sets are projective spaces

The number of elements of, the (n − 1)-dimensional projective space over the finite field F_q, is the q-integer

Taking q = 1 yields _q = n.

The expansion of the q-integer into a sum of powers of q corresponds to the Schubert cell decomposition of projective space.

Permutations are flags

There are n! permutations of a set with n elements, and _q! maximal flags in Fn
q, where

is the q-factorial. Indeed, a permutation of a set can be considered a filtered set, as a flag is a filtered vector space: for instance, the permutation (0, 1, 2) corresponds to the filtration .

Subsets are subspaces

The binomial coefficient gives the number of m-element subsets of an n-element set, and the q-binomial coefficient gives the number of m-dimensional subspaces of an n-dimensional vector space over F_q.

The expansion of the q-binomial coefficient into a sum of powers of q corresponds to the Schubert cell decomposition of the Grassmannian.