rt.representation theory – Interpolated simple integral fusion categories of Lie type


$DeclareMathOperatorPSL{PSL} DeclareMathOperatorRep{Rep}$The idea motivating this post is that there should exist a global understanding of the unitary fusion categories $Rep(G(q))$, with $G(q)$ a finite group of Lie type (on the finite field $mathbb{F}_q$, with $q$ a prime power), allowing to interpolate an extended family denoted “$Rep(G(n))$“, for all (odd, at least) integer $n>1$. The integer $n$ corresponds to a virtual “finite field of order $n$“, in the same flavor than the noncommutative geometry or the field with one element. In particular, for $G$ classical, it is NOT given by $G(mathbb{Z}/nmathbb{Z})$, because $G(q)$ is already different from $G(mathbb{Z}/qmathbb{Z})$ when $q=p^r$, with $p$ prime and $r>1$.

Such constructions would provide infinite families of non-group-like simple integral fusion categories, and so a lot of examples of non weakly-group-theoretical integral fusion categories, answering Question 2 in Etingof-Nikshych-Ostrik (2011), thanks to their Proposition 9.11.

A global understanding of the type of $Rep(PSL(2,q))$ is known:

  • if $q equiv 0 pmod2$, it is of type $((1,1),(q-1,frac{q}{2}),(q,1),(q+1,frac{q-2}{2}))$,
  • if $q equiv 1 pmod4$, it is of type $((1,1),(frac{q+1}{2},2),(q-1,frac{q-1}{4}),(q,1),(q+1,frac{q-5}{4}))$,
  • if $q equiv 3 pmod4$, it is of type $((1,1),(frac{q-1}{2},2),(q-1,frac{q-3}{4}),(q,1),(q+1,frac{q-3}{4})),$

and so is for the character table, see for example the webpage Character table of $PSL(2,mathbb{F}_q)$ by J. Adams (up to typos).

Question: Is there a global understanding of the F-symbols (also called 6j-symbols) for $Rep(PSL(2,q))$?
If not (yet), how to compute them for $q$ small? (it could be enough to guess for some small $n$).

Note that the knowledge of the F-symbols is exactly what is lost when we consider the Grothendieck ring of a fusion category.

The types and character tables mentionned above can immediately be interpolated, by replacing the prime power $q$ by an integer $n$. The problem here is the existence of a global understanding of the unitary fusion category $Rep(PSL(2,q))$, in order to interpolate the extended family “$Rep(PSL(2,n))$“. Note that by the Schur orthogonality relations we can already compute what would be their Grothendieck rings. For example, the Grothendieck ring of “$Rep(PSL(2,6))$” would be exactly the (first) fusion ring mentionned in this post, and the one of “$Rep(PSL(2,15))$” would be the following of rank $10$:

$$small{begin{smallmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1end{smallmatrix} , begin{smallmatrix}0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \ 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1end{smallmatrix} , begin{smallmatrix}0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1end{smallmatrix} , begin{smallmatrix}0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \ 1 & 0 & 0 & 2 & 2 & 2 & 1 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2end{smallmatrix} , begin{smallmatrix}0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 2 \ 1 & 1 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 \ 0 & 0 & 0 & 1 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2end{smallmatrix}},$$ $$small{begin{smallmatrix}0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 2 & 1 & 1 & 2 & 2 & 2 & 2 \ 0 & 0 & 0 & 1 & 2 & 2 & 2 & 2 & 2 & 2 \ 1 & 1 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2end{smallmatrix} , begin{smallmatrix}0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 1 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 2 \ 1 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 3end{smallmatrix} , begin{smallmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 2 & 2 \ 1 & 1 & 1 & 2 & 2 & 2 & 3 & 2 & 3 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 2 & 3 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 3end{smallmatrix} , begin{smallmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 2 & 3 \ 1 & 1 & 1 & 2 & 2 & 2 & 3 & 2 & 2 & 3 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 & 2end{smallmatrix} , begin{smallmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \ 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 2 & 3 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 2 & 3 & 3 \ 0 & 1 & 1 & 2 & 2 & 2 & 2 & 3 & 3 & 2 \ 1 & 1 & 1 & 2 & 2 & 2 & 3 & 3 & 2 & 2end{smallmatrix}} $$

Here is its character table:
$$scriptsize{begin{array}{c|c}
text{class}&C_1&C_2&C_3&C_4&C_5&C_6&C_7&C_8&C_9&C_{10} newline
text{dim}&1&240 & 240& 240& 112& 112& 105& 210& 210& 210 newline hline
chi_1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 newline
chi_2 & 7 & 0 & 0 & 0 & frac{-1+isqrt{15}}{2} & frac{-1-isqrt{15}}{2} & -1 & 1 & 1 & -1 newline
chi_3 & 7 & 0 & 0 & 0 & frac{-1-isqrt{15}}{2} & frac{-1+isqrt{15}}{2} & -1 & 1 & 1 & -1 newline
chi_4 & 14 & 0 & 0 & 0 & -1 & -1 & -2 & 0 & 0 & 2 newline
chi_5 & 14 & 0 & 0 & 0 & -1 & -1 & 2 & sqrt{2} & -sqrt{2} & 0 newline
chi_6 & 14 & 0 & 0 & 0 & -1 & -1 & 2 & -sqrt{2} & sqrt{2} & 0 newline
chi_7 & 15 & 1 & 1 & 1 & 0 & 0 & -1 & -1 & -1 & -1 newline
chi_8 & 16 & 2cos(frac{2pi}{7}) & -2cos(frac{3pi}{7})& -2cos(frac{pi}{7}) & 1 & 1 & 0 & 0 & 0 & 0 newline
chi_9 & 16 & -2cos(frac{3pi}{7})& -2cos(frac{pi}{7}) & 2cos(frac{2pi}{7}) & 1 & 1 & 0 & 0 & 0 & 0 newline
chi_{10}& 16 & -2cos(frac{pi}{7}) & 2cos(frac{2pi}{7}) & -2cos(frac{3pi}{7}) & 1 & 1 & 0 & 0 & 0 & 0 newline
end{array}}$$