Computations Field with one element

1 computations

1.1 sets projective spaces
1.2 permutations flags
1.3 subsets subspaces

computations

various structures on set analogous structures on projective space, , can computed in same way:

sets projective spaces

the number of elements of p(fn

q) = p(fq), (n − 1)-dimensional projective space on finite field fq, q-integer

[
n

]

q


:=




q

n


−
1


q
−
1



=
1
+
q
+

q

2


+
⋯
+

q

n
−
1


.


{\displaystyle [n]_{q}:={\frac {q^{n}-1}{q-1}}=1+q+q^{2}+\dots +q^{n-1}.}

taking q = 1 yields [n]q = n.

the expansion of q-integer sum of powers of q corresponds schubert cell decomposition of projective space.

permutations flags

there n! permutations of set n elements, , [n]q! maximal flags in fn

q, where

[
n

]

q


!
:=
[
1

]

q


[
2

]

q


…
[
n

]

q




{\displaystyle [n]_{q}!:=[1]_{q}[2]_{q}\dots [n]_{q}}

is q-factorial. indeed, permutation of set can considered filtered set, flag filtered vector space: instance, ordering (0, 1, 2) of set {0,1,2} corresponds filtration {0} ⊂ {0,1} ⊂ {0,1,2}.

subsets subspaces

the binomial coefficient

n
!


m
!
(
n
−
m
)
!





{\displaystyle {\frac {n!}{m!(n-m)!}}}

gives number of m-element subsets of n-element set, , q-binomial coefficient

[
n

]

q


!


[
m

]

q


!
[
n
−
m

]

q


!





{\displaystyle {\frac {[n]_{q}!}{[m]_{q}![n-m]_{q}!}}}

gives number of m-dimensional subspaces of n-dimensional vector space on fq.

the expansion of q-binomial coefficient sum of powers of q corresponds schubert cell decomposition of grassmannian.