Geodesic
An algorithm for constructing irreducible decompositions of permutation representations of wreath products of finite groups
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 107-139 
V. V. Kornyak. An algorithm for constructing irreducible decompositions of permutation representations of wreath products of finite groups. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXXI, Tome 485 (2019), pp. 107-139. http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a6/
@article{ZNSL_2019_485_a6,
     author = {V. V. Kornyak},
     title = {An algorithm for constructing irreducible decompositions of permutation representations of wreath products of finite groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {107--139},
     year = {2019},
     volume = {485},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a6/}
}
TY  - JOUR
AU  - V. V. Kornyak
TI  - An algorithm for constructing irreducible decompositions of permutation representations of wreath products of finite groups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2019
SP  - 107
EP  - 139
VL  - 485
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a6/
LA  - ru
ID  - ZNSL_2019_485_a6
ER  - 
%0 Journal Article
%A V. V. Kornyak
%T An algorithm for constructing irreducible decompositions of permutation representations of wreath products of finite groups
%J Zapiski Nauchnykh Seminarov POMI
%D 2019
%P 107-139
%V 485
%U http://geodesic.mathdoc.fr/item/ZNSL_2019_485_a6/
%G ru
%F ZNSL_2019_485_a6

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We describe an algorithm for decomposing permutation representations of wreath products of finite groups into irreducible components. The algorithm is based on the construction of a complete set of mutually orthogonal projection operators into irreducible invariant subspaces of the Hilbert space of the representation under consideration. In constructive models of quantum mechanics, the invariant subspaces of representations of wreath products describe the states of multicomponent quantum systems. The proposed algorithm uses methods of computer algebra and computational group theory. The C implementation of the algorithm is capable of constructing irreducible decompositions of representations of wreath products of high dimensions and ranks. Examples of calculations are given.

[1] M. Van Raamsdonk, “Building up spacetime from quantum entanglement”, Gen. Relativ. Grav., 42 (2010), 2323–2329 | DOI | MR | Zbl

[2] C. Cao, S. M. Carroll, S. Michalakis, “Space from Hilbert space: Recovering geometry from bulk entanglement”, Phys. Rev. D, 95 (2017), 024031 | DOI | MR

[3] V. V. Kornyak, “Modeling quantum behavior in the framework of permutation groups”, EPJ Web of Conferences, 173 (2018), 01007 | DOI

[4] V. V. Kornyak, “Quantum models based on finite groups”, J. Phys.: Conf. Ser., 965 (2018), 012023 | DOI

[5] J. D. P. Meldrum, Wreath Products of Groups and Semigroups, Longman/Wiley, New York, 1995 | MR | Zbl

[6] G. D. James, A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley, Reading, 1981 | MR | Zbl

[7] M. Kholl, Teoriya grupp, Izd-vo inostr. lit., M., 1962

[8] P. J. Cameron, Permutation Groups, Cambridge University Press, Cambridge, 1999 | MR | Zbl

[9] Eiichi Bannai, Tatsuro Ito, Algebraic Combinatorics I: Association Schemes, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1984 | MR | Zbl

[10] N. H. Vasilevu, “Splitting permutation representations of finite groups by polynomial algebra methods”, CASC 2018, LNCS, 11077, eds. Gerdt V. P. et al., Springer, Cham, 2018, 304–318 | MR

[11] V. V. Kornyak, “Algoritm razlozheniya predstavlenii konechnykh grupp s pomoschyu invariantnykh proektorov”, Zap. nauchn. semin. POMI, 468, 2018

[12] N. Jacobson, Structure of Rings, Colloquium Publications, 37, Amer. Math. Soc., Providence, R.I., 1956 | DOI | MR | Zbl

[13] L. H. Rowen, Ring Theory, Academic Press, Inc., Boston, 1991 | MR | Zbl

[14] R. Wilson et al., Atlas of finite group representations, http://brauer.maths.qmul.ac.uk/Atlas/v3

[15] W.-H. Steeb, Matrix Calculus and the Kronecker Product with Applications and C++ Programs, World Scientific Publishing Co., Inc., River Edge, NJ, USA, 1997 | MR | Zbl