The n-Category Café

[…] polynomials in 2-variables […]

I am a little worried that what you mean are polynomials in two variables, i.e. elements in some $K[x,y]$.

Is that so? In that case I’d unfortunately have to disappoint you: when we say “2-vectors” we don’t just mean two vectors (as in: “a pair of vectors”), but a notion of vector that is to an ordinary vector like a category is to a set.

On the other hand, if you really do mean polynomials in categorified variables, then I’d love to know more details on what exactly it is you are considering!

I should maybe remark that, after all, there is often a simple special case of a 2-something which does appear more or less as nothing but two somethings.

For instance, the 2-vector spaces $\mathbf{V}$ described and studied by John Baez and Alissa Crans in HDA IV are , in particular, two vector spaces $V_0$ and $V_1$ $$\mathbf{V} = \{V_0, V_1\} \,,$$ namely a vector space, $V_0$, of objects and a vector space, $V_0 \oplus V_1$ of morphisms.

What makes the two vector spaces a 2-vector space is extra structure on this, given by maps $$s,t : V_0 \oplus V_1 \to V_0$$ and $$i : V_0 \to V_0 \oplus V_1$$ and $$\circ : (V_0\oplus V_1)^{[2]} \to V_0 \oplus V_1$$satisfying the usual laws which say that $\circ$ is the composition of morphism which have source $s$ and target $t$.

Similar statements hold for 2-groups. A (strict) 2-group $\mathbf{G}$ is, in particular, two groups $$\mathbf{G} = (G_0, G_1) \,.$$

Again, the reason for this is that a strict 2-group, like a Baez-Crans 2-vector space, is an internal category: BC 2-vector spaces are categories internal to ordinary vector spaces, while strict 2-groups are categories internal to ordinary groups.

Since a category consists, in particular, of two objects, namely an object of objects and an object of morphisms, this gives rise to the phenomenon that 2-things often look like two things .

Sometimes (rarely, but sometimes), it indeed happens that people first study a theory of two things (like of two variables) with extra structure and properties around, and only later realize (or somebody else does, usually ;-) that what they are really studying is a 2-thing.

A nice example of this are 2-class functions and their categorical interpretation.

Like a class function on a group is a function $g \mapsto f(g)$ of a single variable that is invariant under conjugation, an $n$-class function $$(g_1,g_2,\cdots, g_n) \mapsto f(g_1,g_2,\cdots, g_n)$$ is a function of $n$-tuples of (pairwise commuting) group elements, that is invariant under simultaneous conjugation of these group elements.

So an $n$-class function is, in particular, a funcion of $n$-variables. People found that to be of use in certain contexts.

Then along came Ganter and Kaparanov and showed that like class functions come from traces of representations of ordinary groups, 2-class functions can be seen as coming from 2-traces of (2-)groups!

I think there are more examples of this kind. In fact, here we enjoy going through standard literature and trying to spot conglomerates of structures that secretly arrange themselves into single, unified $n$-categorical structures.

A more intricate example of this, for instance, is the local data for connections on gerbes. This comes in large parts of the literature as a vast array of various $p$-forms with values in various groups and Lie algebras. But staring at this structure for a while reveals that all this mess are nothing but the components of one single morphism between higher-groupoids.

There are more examples like that, but I think I’ll stop here.

If you knew all this and really do mean that you are working with categorified variables then I apologize for wasting your time with this lengthy reply here, hoping that maybe others find it helpful.

Ok, so I’ll make this more concrete. The hermite polynomials can be expressed via hypergeometric 2F0 functions and multiplicatively renormalized.

Nobuhiro Asai, Izumi Kubo, Hui-Hsiung Kuo, Multiplicative Renormalization And Generating Functions: II, Taiwanese Journal of Mathematics 8 (4) (2004) 593–628.

$\hat{\Psi}_n (x) = \frac{\Psi_n (x)}{\sqrt{\lambda_n}} = \frac{_2 F_0 \left( \frac{[- \frac{n}{2}, \frac{1}{2} - \frac{n}{2}]}{\emptyset}, - \frac{2 t^2}{x^2} \right) x^n}{\sqrt{t^{2 n} n!}}$

Which is orthnormal with respect to the measure on x, e.g., the inner products are given by the Kronecker delta

$\left\langle \hat{\Psi}_n (x,t), \hat{\Psi}_m (x,t) \right\rangle_{\mathcal{H}_{\mu_{\sigma^2}}} = \int_{- \infty}^{\infty} \hat{\Psi}_n (x,t) \hat{\Psi}_m (x,t) d \mu_{\sigma^2 (x,t)} = \delta_{n, m}$

Now we apply the pointwise Fourier transform to each element in this real orthonormal set (a discrete infinite vector of continuous vectors or distribution functions).

$T_n^{\Psi} (x, t) = \int_{- \infty}^{\infty} \hat{\Psi}_n (x, t) e^{- i x y} \d x$

The generating function for this transform can be found.. derived it heuristically and its unkown to me whether there is a systematic way to do it.

In any case, after the fourier transform the set is no longer orthogonal or normal, so it can be renormalzed again, and then the Gram-Schmidt orthogonalization process can be applied by taking the complex inner products using the complex conjugates and sucessively projecting the 0th vector onto the 1st, then the 1st projection onto the 2nd, 2nd onto 3rd, etc.

This process is burdensome but I wrote some maple code to do it fairly quickly and then work on deriviving generating functions again heuristically. I noticed that the imaginary component of the even fourier transforms =0 forall n, and the real part of the odd dimensional transforms =0 forall n, Based on this I decmoposed the transform into even and odd componets, and apply the Gram-schmidt process to each piece and then divide the dimension by 2 and sum to get an infinite seqeuence of complex orthonormal distribution functions which rapidly approach 0, and t always stays real. It’s the space variable that is complexified.

I am not aware of any concrete progress in these questions. I also haven’t thought about it much since then. One day I should again.