On the realization of a class of $\text{SL}(2,\mathbb{Z})$ representations
Let p<q be odd primes and \rho_{1} and \rho_{2} be irreducible representations of \text{SL}(2,\mathbb{Z}_{p}) and \text{SL}(2,\mathbb{Z}_{q}) of dimensions \frac{p+1}{2} and \frac{q+1}{2} , respectively. We show that if \rho_{1}\oplus\rho_{2} can be realized as a modular representation associate...
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| Published in | Journal of noncommutative geometry Vol. 18; no. 4; pp. 1521 - 1542 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English Japanese |
| Published |
25.06.2024
|
| Online Access | Get full text |
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| Summary: | Let
p<q
be odd primes and
\rho_{1}
and
\rho_{2}
be irreducible representations of
\text{SL}(2,\mathbb{Z}_{p})
and
\text{SL}(2,\mathbb{Z}_{q})
of dimensions
\frac{p+1}{2}
and
\frac{q+1}{2}
, respectively. We show that if
\rho_{1}\oplus\rho_{2}
can be realized as a modular representation associated with a modular fusion category
\mathcal{C}
, then
q-p=4
. Moreover, if
\mathcal{C}
contains a non-trivial étale algebra, then
\mathcal{C}\boxtimes\mathcal{C}(\mathbb{Z}_{p},\eta)\cong\mathcal{Z}(\mathcal{A})
as a braided fusion category, where
\mathcal{A}
is a near-group fusion category of type
(\mathbb{Z}_{p},p)
, which gives a partial answer to the conjecture of D. Evans and T. Gannon. We also show that there exists a non-trivial
\mathbb{Z}_{2}
-extension of
\mathcal{A}
that contains simple objects of Frobenius–Perron dimension
\frac{\sqrt{p}+\sqrt{q}}{2}
. |
|---|---|
| ISSN: | 1661-6952 1661-6960 |
| DOI: | 10.4171/jncg/578 |