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For any groupoid, it's groupoid cardinality is the sum of the reciprocals of the automorphism groups over the isomorphism classes. Let us consider the category of vector spaces over a finite field $\mathbb F_q$ with only invertible morphisms allowed.

Then, the size of the automorphism groups are $g_n= \prod_{i=1}^n(q^n-q^{i-1}) = q^{n\choose 2}\prod_{i=1}^n(q^i-1)$ so it's groupoid cardinality is the following infinite sum: $$\sum_{n \geq 1}\frac{1}{g_n}.$$

Is there a closed form expression for it? Note that as $q\to 1$ in an appropriate sense, the general linear group is supposed to converge to the symmetric group and the groupoid of vector spaces should converge to the groupoid of finite sets.

In this sense, the above infinite sum is an analog of the usual one for $e$ and can perhaps reasonably be called a q-analog of $e$.

Alternatively, in the case of abelian, (even semi-simple) categories, is there a better replacement for the groupoid cardinality that I should be using?

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    $\begingroup$ Is the sum $q$-hypergeometric, i.e., the ratio $g_{n+1}/g_n$ is a rational function of $q^n$? If it is then you can apply the $q$-Gosper algorithm. $\endgroup$ Jun 17 '20 at 5:54
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Upon substituting $x=\frac{1}{q}$ we obtain $$\sum_{n\geq 0}\frac{1}{|\mathrm{GL}_n(\mathbb F_q)|}=\sum_{n\geq 0}\frac{x^{n^2}}{(1-x)(1-x^2)\cdots(1-x^n)}$$ and this evaluates to the product $\prod_{i\geq 1}\frac{1}{(1-x^{5i-4})(1-x^{5i-1})}$ by the first Rogers-Ramanujan identity.

You can also interpret the second Rogers-Ramanujan identity as giving a groupoid cardinality for the category of affine spaces over $\mathbb F_q$. where the sum becomes $\sum_{n\geq 0}\frac{1}{|\mathrm{AGL}_n(\mathbb F_q)|}$. (Note that I wrote these sums to contain a term for $n=0$, corresponding to the zero dimensional vector/affine space.)

I don't think that there is a good explanation for this evaluation that avoids proving the R-R identities first. In particular it is unclear how to directly relate the exponents of the infinite product side with the category of vector/affine spaces.

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    $\begingroup$ Sorry, let me repost that comment with correct links. I wonder if there's a way to do it using cylindric plane partitions as in arxiv.org/abs/1209.1807 and doi.org/10.1090/proc/13373. Cylindric plane partitions are known to be related to affine crystals (arxiv.org/abs/math/0702062). $\endgroup$ Oct 7 '20 at 2:03
  • $\begingroup$ @SamHopkins That sounds very cool. I would love to see if there is a connection there. $\endgroup$ Oct 7 '20 at 2:17
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    $\begingroup$ WOW! I know, such comments are discouraged, but - WOW! $\endgroup$ Oct 7 '20 at 4:26
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    $\begingroup$ I can't help seconding @მამუკაჯიბლაძე 's sentiments. I suppose the following is a standard question about the Rogers-Ramanujan identities (which are new to me), but: Why 5? $\endgroup$
    – Tim Campion
    Oct 7 '20 at 18:08

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