On linear shift representations.

*(English)*Zbl 1346.20071From the text: We introduce and develop the concept of (linear) shift representation. This derives from a certain action on 2-cocycle groups that preserves both cohomological equivalence and orthogonality for cocyclic designs, discovered by K. J. Horadam. Detailed information about fixed point spaces and reducibility is given. We also discuss results of computational experiments, including the calculation of shift orbit structure and searching for orthogonal cocycles.

We now summarize the content of the paper. In Section 2 we prove elementary facts about shift representations. The main results of Section 3 are a determination of fixed points under shift action in the full cocycle space, and a bound on the dimension of the fixed coboundary space. We thereby solve most of Research Problem 55 (1) in [K. J. Horadam, Hadamard matrices and their applications. Princeton: Princeton University Press (2007; Zbl 1145.05014)]. Some relevant linear group theory is then given in Section 4. This serves as background for Section 5, where we establish that a shift representation is hardly ever completely reducible. In fact, we provide criteria for deciding irreducibility and complete reducibility. As an illustration of the practical nature of our approach, in the final section we describe new results obtained from our Magma [W. Bosma et al., J. Symb. Comput. 24, No. 3-4, 235-265 (1997; Zbl 0898.68039)] implementation of procedures to compute with shift representations. Open questions arising from the computational work are posed.

We now summarize the content of the paper. In Section 2 we prove elementary facts about shift representations. The main results of Section 3 are a determination of fixed points under shift action in the full cocycle space, and a bound on the dimension of the fixed coboundary space. We thereby solve most of Research Problem 55 (1) in [K. J. Horadam, Hadamard matrices and their applications. Princeton: Princeton University Press (2007; Zbl 1145.05014)]. Some relevant linear group theory is then given in Section 4. This serves as background for Section 5, where we establish that a shift representation is hardly ever completely reducible. In fact, we provide criteria for deciding irreducibility and complete reducibility. As an illustration of the practical nature of our approach, in the final section we describe new results obtained from our Magma [W. Bosma et al., J. Symb. Comput. 24, No. 3-4, 235-265 (1997; Zbl 0898.68039)] implementation of procedures to compute with shift representations. Open questions arising from the computational work are posed.

##### MSC:

20J05 | Homological methods in group theory |

05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |

20C15 | Ordinary representations and characters |

20H30 | Other matrix groups over finite fields |

##### Keywords:

shift representations; 2-cocycle groups; cohomological equivalence; orthogonal cocyclic designs; fixed point spaces; orthogonal cocycles; linear groups; irreducibility; complete reducibility##### Software:

Magma
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\textit{D. L. Flannery} and \textit{R. Egan}, J. Pure Appl. Algebra 219, No. 8, 3482--3494 (2015; Zbl 1346.20071)

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##### References:

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