Orthogonal representations
η
n
:
S
n
↷
R
N
\eta _n\colon S_n\curvearrowright \mathbb {R}^N
of the symmetric groups
S
n
S_n
,
n
≥
4
n\ge 4
, with
N
=
n
!
/
8
N=n!/8
, emerging from symmetries of double ratios are treated. For
n
=
5
n=5
, the representation
η
5
\eta _5
is decomposed into irreducible components and it is shown that a certain component yields a solution of the equations that describe the Möbius structures in the class of sub-Möbius structures. In this sense, a condition determining the Möbius structures is implicit already in symmetries of double ratios.
The hippocampal formation has long been suggested to underlie both memory formation and spatial navigation. We discuss how neural mechanisms identified in spatial navigation research operate across information domains to support a wide spectrum of cognitive functions. In our framework, place and grid cell population codes provide a representational format to map variable dimensions of cognitive spaces. This highly dynamic mapping system enables rapid reorganization of codes through remapping between orthogonal representations across behavioral contexts, yielding a multitude of stable cognitive spaces at different resolutions and hierarchical levels. Action sequences result in trajectories through cognitive space, which can be simulated via sequential coding in the hippocampus. In this way, the spatial representational format of the hippocampal formation has the capacity to support flexible cognition and behavior.
Systematization of orthogonal representations of probability density functions of random processes
