Pascal's Identities for $q$-binomials

Question

The recursive identity for building Pascal's triangle is

$$ \binom{n}{k}=\binom{n-1}{k}+\binom{n-1}{k-1}. \tag{$\circ$}$$

This has a simple combinatorial interpretation: every $k$-subset of $\{1,\cdots,n\}$ is either a $k$-subset of $\{1,\cdots,n-1\}$ or it's a $(k-1)$-subset of $\{1,\cdots,n-1\}$ with $\{n\}$ adjoined.

From Wikipedia there are two analogous identities for $q$-binomials:

$$ \left[\begin{array}{c} n \\ k \end{array}\right]_q = q^k\left[\begin{array}{c} n-1 \\ k \end{array}\right]_q + \left[\begin{array}{c} n-1 \\ k-1 \end{array}\right]_q, \tag{I}$$

$$ \left[\begin{array}{c} n \\ k \end{array}\right]_q=\left[\begin{array}{c} n-1 \\ k \end{array}\right]_q + q^{n-k} \left[\begin{array}{l} n-1 \\ k-1 \end{array}\right]_q. \tag{II}$$

(We interpret the $q$-binomial as the number of $k$-dimensional subspaces of $\mathbb{F}_q^n$.)

The first identity $(\mathrm{I})$ has a $\mathbb{F}_q$-interpretation as follows: we have two cases,

1. The subspace $V$'s projection into $\mathbb{F}_q^{n-1}$ is still $k$-dimensional, in which case it is determined by its image $W$ under the projection and the corresponding section $W\to V$, and that section is the sum of the identity map on $W$ and an arbitrary linear map $W\to\mathbb{F}_qe_n$.

2. The subspace $V$ is a direct sum of a $(k-1)$-dim subspace of $\mathbb{F}_q^{n-1}$ and $\mathbb{F}_qe_n$.

However I can't figure out a $\mathbb{F}_q$-interpretation of the second identity $(\mathrm{II})$. What is it? Surely I'm missing something obvious. Presumably it involves taking a complementary subspace to a $k$-dim subspace and then using another linear map from/to it.

Answer 1

Found a way. For subspaces $V$ of $\mathbb{F}_q^n$ there are two cases:

1. $V$ is a subspace of $\mathbb{F}_q^{n-1}$
2. Otherwise, $V$ is determined by picking a $(k-1)$-dim subspace $K$ of $\mathbb{F}_q^{n-1}$ to be the kernel of the projection $\pi:\mathbb{F}_q^n\to\mathbb{F}_q$ onto the $n$th coordinate, then picking any element $\alpha$ of the quotient space $\mathbb{F}_q^{n-1}/K$ to determine the fiber $\pi^{-1}(1)=\alpha$.

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Edit: I generalized this argument in this answer to prove the $q$-analog of Chu-Vandermonde convolution. This yields both versions of Pascal's rule as a special case (as the convolution $q$-identity is "asymmetric," we can apply it for $(n-1)+1$ and $1+(n-1)$ to get the two versions of the rule).