# $q$ as a prime power and a root of unity

The number of points on the $$(n-1)$$-dimensional projective space $$P^{n-1}(\mathbb{F}_q)$$ over a finite field $$\mathbb{F}_q$$ is the $$q$$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of points on the $$(n-1)$$-dimensional projective space $$P^{n-1}(\mathbb{F}_{un})$$ over the field with one element $$\mathbb{F}_{un}$$ is, simply by taking $$q=1$$, $$[n]_1=n$$. Similarly, whenever the number of $$\mathbb{F}_q$$ points on a variety is a polynomial in $$q$$, we can take the $$q=1$$ limit just as well.

On the flip side of the coin, the $$q$$-integers appear in many $$q$$-analogs, including representation theory of quantum groups. In this context, $$q$$ is often a root of unity $$q=e^{\frac{2\pi i}{k}}$$ or more generally a complex number, and $$q\rightarrow 1$$ limit is understood as a "classical limit".

The connection is probably not superficial, because there are other instances where $$q$$ is interpreted in two different ways. For instance, according to this slide, it's a theorem of Katz that for a smooth quasi-projective variety $$X$$ defined over $$\mathbb{Z}$$, if the number of $$\mathbb{F}_q$$ points is a polynomial in $$q$$, then it is the E-polynomial of $$X$$, which is a specialization of the weight polynomial $$WH(X;q,t)$$. On the other hand, Chuang-Diaconescu-Pan related the weight polynomial to the refined Gopakumar-Vafa expansion.

So, my question is, is there any simple explanation of the apperance of $$q$$ in two different guises, a power of a prime and a root of unity?

• Somehow, prime powers and roots of unity feel "dual" to each other. Perhaps this can be made precise through the fact that as locally compact abelian groups, $\mathbb{Z}$ and $\mathbb{S}^1$ are dual to each other, but don't ask me how to relate this fact to the examples that you mention... Jan 22, 2019 at 9:27

This is very not rigorous, but it's a way of thinking about this topic which I find personally helpful. Several $$\mathbb F_1$$ papers contains remarks about the idea going back to Weil and Iwasawa that adjoining roots of unity is analogous to base field extensions. This means that $$\mathbb F_1,\mathbb F_{1^2},\dots$$ shouldnt be thought of as the same "$$F_{un}$$", rather you should think of $$\mathbb F_{1^n}$$ as $$\mathbb F_1$$ together with $$n$$-th roots of unity. Notice that the number of $$\mathbb F_1$$ points will be the same as the number of $$\mathbb F_{1^n}$$ points since plugging in $$q=1$$ or $$1^n$$ gives the same thing, however things get more interesting when considering the relation between the fields rather than individually.

Just like $$\mathbb F_q$$ points are the Frobenius fixed points of $$\mathbb F_{q^n}$$, in combinatorics land $$\mathbb F_1$$ points should be the points of a $$\mathbb F_{1^n}$$ scheme fixed by a cyclic group of order $$n$$. Just like how you expect to count $$\mathbb F_1$$ points by plugging in $$q=1$$ formally when the point count is uniform over all finite fields, counting $$\mathbb F_1$$ points in a variety over $$\mathbb F_{1^n}$$, should have something to do with plugging in formally $$q^n=1$$, or in other words some primitive $$n$$-th root of unity. This perspective makes cyclic sieving natural in many combinatorial contexts.

One explicit connection is to look in non-describing characteristic, i.e. over an algebraically closed field $$k$$ of characteristic $$p$$ not dividing q, so $$q$$ can be simultaneously a root of unity and a prime power.

Let's say we are in type $$A$$, then the Iwahori-Hecke algebra $$\mathcal{H}_q(n)$$ and the $$q$$-Schur algebra $$S(q,n)$$ over $$k$$ arise as endomorphism algebras of certain unipotent representations of $$GL_n(\mathbb{F}_q)$$ and control many aspects of the unipotent representations, in fact there is an equivalence of categories between $$S(q,n)$$-modules and unipotent representations (for $$\mathcal{H}_q(n)$$-modules there are biadjoint functors, but not equivalences in general).

For example we can see that if the order of $$q$$ (mod p) is bigger than $$n$$ then the order of $$GL_n(\mathbb{F}_q)$$ is prime to $$p$$ and hence every representation is completely reducible. This corresponds to the fact in characteristic zero that the algebras $$\mathcal{H}_q(n)$$ and $$S(q,n)$$ fail to be semisimple exactly when $$q$$ is a root of unity of order less than $$n.$$

One can relate the cases of positive characteristic and characteristic zero using model theory. In particular if you let the characteristic $$p$$ tend to infinity while always choosing $$q$$ to be a primitive $$d$$th root of unity then the representation theory of $$S(q,n)$$ (or $$\mathcal{H}_q(n)$$) in characteristic $$p$$ "converges" (in a model theoretic sense) to that of $$S(\zeta,n)$$ in characteristic zero (where $$\zeta$$ is a primitive $$d$$th root of unity).

Hence you can prove things about Schur algebras and Iwahori-Hecke algebras at roots of unity in characteristic zero (and therefore about quantum groups) by looking at unipotent representations of $$GL_n(\mathbb{F}_q)$$ in non-describing characteristic. For example it's not too hard to turn the above paragraph into an actual proof of for which values of $$q$$ these algebras are semisimple in characteristic zero.