### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 011, 25 pages      arXiv:1406.4389      https://doi.org/10.3842/SIGMA.2019.011

### Decomposition of some Witten-Reshetikhin-Turaev Representations into Irreducible Factors

Julien Korinman
Fundação Universidade Federal de São Carlos, Departamento de Matemática, Rod. Washington Luís, Km 235, C.P. 676, 13565-905 São Carlos, SP, Brasil

Received October 29, 2017, in final form January 30, 2019; Published online February 12, 2019

Abstract
We decompose into irreducible factors the ${\rm SU}(2)$ Witten-Reshetikhin-Turaev representations of the mapping class group of a genus $2$ surface when the level is $p=4r$ and $p=2r^2$ with $r$ an odd prime and when $p=2r_1r_2$ with $r_1$, $r_2$ two distinct odd primes. Some partial generalizations in higher genus are also presented.

Key words: Witten-Reshetikhin-Turaev representations; mapping class group; topological quantum field theory.

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References

1. Andersen J.E., Fjelstad J., Reducibility of quantum representations of mapping class groups, Lett. Math. Phys. 91 (2010), 215-239, arXiv:0806.2539.
2. Blanchet C., Habegger N., Masbaum G., Vogel P., Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), 883-927.
3. Freedman M., Krushkal V., On the asymptotics of quantum ${\rm SU}(2)$ representations of mapping class groups, Forum Math. 18 (2006), 293-304, arXiv:math.QA/0409503.
4. Gilmer P.M., Masbaum G., Maslov index, lagrangians, mapping class groups and TQFT, Forum Math. 25 (2013), 1067-1106, arXiv:0912.4706.
5. Jones V.F.R., Index for subfactors, Invent. Math. 72 (1983), 1-25.
6. Koberda T., Santharoubane R., Irreducibility of quantum representations of mapping class groups with boundary, Quantum Topol. 9 (2018), 633-641, arXiv:1701.08901.
7. Korinman J., Irreducible factors of Weil representations and TQFT, Math. Rep., to appear, arXiv:1310.0390.
8. Lickorish W.B.R., Invariants for 3-manifolds from the combinatorics of the Jones polynomial, Pacific J. Math. 149 (1991), 337-347.
9. Masbaum G., Roberts J.D., On central extensions of mapping class groups, Math. Ann. 302 (1995), 131-150, arXiv:math.QA/9909128.
10. Masbaum G., Vogel P., $3$-valent graphs and the Kauffman bracket, Pacific J. Math. 164 (1994), 361-381.
11. Reshetikhin N., Turaev V.G., Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547-597.
12. Roberts J., Irreducibility of some quantum representations of mapping class groups, J. Knot Theory Ramifications 10 (2001), 763-767, arXiv:math.QA/9909128.
13. Wenzl H., On sequences of projections, C. R. Math. Rep. Acad. Sci. Canada 9 (1987), 5-9.
14. Witten E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351-399.