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.
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