§ F1 or Fun : The field with one element

• Many combinatorial phenomena can be recovered as the "limit" of geometric phenomena over the "field with one element", a mathematical mirage.

§ Cardinality ~ Lines

• Consider projective space of dimension $n$ over $F_p$. How many lines are there?
• Note that for each non-zero vector, we get a 'direction'. So there are $p^n - 1$ potential directions.
• See that for any choice of direction $d \in F_p - \vec 0$, there are $(p - 1)$ "linearly equivalent" directions, given by $1 \cdot d$, $2 \cdot d$, \dots, $(p - 1) \cdot d$ which are all distinct since field multiplication is a group.
• Thus, we have $(p^n - 1)/(p - 1)$ lines. This is equal to $1 + p + p^2 + \dots + p^{n-1}$, which is $p^0 + p^1 + \dots + p^{n-1}$
• If we plug in $p = 1$ (study the "field with one element", we recover $\sum_{i=0}^{n-1} p^i = n$.
• Thus, "cardinality of a set of size $n$" is the "number of lines of $n$-dimensional projective space over $F_1$!
• Since $[n] \equiv \{1, 2, \dots, n\}$ is the set of size $n$, it is only natural that $[n]_p$ is defined to be the lines in $F_p^n$. We will abuse notation and conflate $[n]_p$ with the cardinality, $[n]_p \equiv (p^n - 1)/(p - 1)$.

§ Permutation ~ Maximal flags

• Recall that a maximal flag is a sequence of subspaces $V_1 \subseteq V_2 \subseteq \dots \subseteq V$. At each step, the dimension increases by $1$, and we start with dimension $1$. So we pick a line $l_1$ through the origin for $V_1$. Then we pick a plane through the origin that contains the line $l_1$through the origin. Said differently, we pick a plane $p_2$ spanned by $l_1, l_2$. And so on.
• How many ways can we pick a line? That's $[n]_p$. Now we need to pick another line orthogonal to the first line. So we build the quotient space $F_p^n/L$, which is $F_p^{n-1}$. Thus picking another line here is $[n-1]_p$. On multiplying all of these, we get $[n]_p [n-1]_p \dots [1]_p$.
• In the case of finite sets, this gives us $1 \cdot 2 \cdot \dots n = n!$.

§ Combinations ~ Grassmanian

• Recall that a grassmanian consists of $k$ dimensional subspaces of an $n$ dimensional space.

• Reference: This week's finds 184 by baez