| Title:
|
Real Representations of $C_2$-Graded Groups: The Linear and Hermitian Theories (English) |
| Author:
|
Rumynin, Dmitriy |
| Author:
|
Taylor, James |
| Language:
|
English |
| Journal:
|
Higher Structures |
| ISSN:
|
2209-0606 |
| Volume:
|
6 |
| Issue:
|
1 |
| Year:
|
2022 |
| Pages:
|
359-374 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the category of hermitian representations, as an $\infty$-category, is equivalent to a category of usual representations. (English) |
| Keyword:
|
Real representation |
| Keyword:
|
finite group |
| Keyword:
|
topological category |
| Keyword:
|
$\infty$-category |
| MSC:
|
18D99 |
| MSC:
|
20C99 |
| idZBL:
|
Zbl 1547.18035 |
| idMR:
|
MR4456598 |
| DOI:
|
10.21136/HS.2022.07 |
| . |
| Date available:
|
2026-03-13T09:59:42Z |
| Last updated:
|
2026-03-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153451 |
| . |
| Reference:
|
[1] Atiyah, M., Segal, G.: Equivariant K-theory and completion..J. Differential Geometry, 3:1–18 |
| Reference:
|
[2] Borevič, Z., Devjatko, G.: On a Frobenius-Moore formula..Algebra and Number Theory (Ordzhonikidze), 3:3–8. (Russian) |
| Reference:
|
[3] Dimmock, J. O.: Representation theory for nonunitary groups..J. Mathematical Phys., 4:1307–1311 |
| Reference:
|
[4] Dyson, F.: The threefold way. Algebraic structure of symmetry groups and ensembles in quantum mechanics..J. Mathematical Phys., 3:1199–1215 |
| Reference:
|
[5] Fok, C.-K.: The Real K-theory of compact Lie groups..SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 022, 26 MR 3210613 |
| Reference:
|
[6] Hristova, K., Jones, J., Rumynin, D.: General comodule-contramodule correspondence..arxiv:2004.12953 http://arxiv.org/pdf/2004.12953 MR 4837288 |
| Reference:
|
[7] Karoubi, M.: Sur la K-théorie équivariante..In Séminaire Heidelberg-Saarbrücken-Strasbourg sur la K-théorie (1967/68), LNM, Vol. 136, pages 187–253. Springer |
| Reference:
|
[8] Lurie, J.: Higher topos theory, volume 170 of Annals of Mathematics Studies..Princeton University Press, Princeton, NJ MR 2522659 |
| Reference:
|
[9] Morel, F., Voevodsky, V.: A¹-homotopy theory of schemes..Inst. Hautes Études Sci. Publ. Math., (90):45–143, 1999 10.1007/BF02698831 |
| Reference:
|
[10] Neeb, K.-H., Ólafsson, G.: Antiunitary representations and modular theory..Banach Center Publications, Volume 113, pages 291–362 MR 3791713, 10.4064/bc113-0-16 |
| Reference:
|
[11] Noohi, B., Young, M.: Twisted loop transgression and higher jandl gerbes over finite groupoids..arxiv:1910.01422 http://arxiv.org/pdf/1910.01422, to appear in Algebraic and Geometric Topology MR 4495667 |
| Reference:
|
[12] Rozenbaum, K.: Induced symplectic modules..Izv. Vyssh. Uchebn. Zaved. Mat., 42:34–43 |
| Reference:
|
[13] Rumynin, D., Taylor, J.: Real representations of C₂-graded groups: the antilinear theory..Linear Algebra and its Applications, 610:135–168 MR 4159287 |
| Reference:
|
[14] Rumynin, D., Young, M.: Burnside rings for Real 2-representation theory: The linear theory..Communications in Contemporary Mathematics, 23(5):Paper No. 2050012 MR 4289902, 10.1142/S0219199720500121 |
| Reference:
|
[15] Schwede, S.: Global homotopy theory, volume 34 of New Mathematical Monographs..Cambridge University Press, Cambridge MR 3838307 |
| Reference:
|
[16] Wigner, E.: Group theory and its application to the quantum mechanics of atomic spectra..Academic Press, New York |
| Reference:
|
[17] Young, M.: Real representation theory of finite categorical groups..High. Struct., 5(1):18–70 MR 4367217 |
| . |
