Orthogonal representations: The smallest cone

2019 ◽  
pp. 183-210
Keyword(s):  
Orthogonal Representations

Symmetries of double ratios and an equation for Möbius structures

2021 ◽  
Vol 33 (1) ◽  
pp. 47-56
Author(s):  
S. Buyalo
Keyword(s):  
Symmetric Groups ◽  
Content Type ◽  
Irreducible Components ◽  
Orthogonal Representations

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.

scholarly journals Navigating cognition: Spatial codes for human thinking

Science ◽  
2018 ◽  
Vol 362 (6415) ◽  
pp. eaat6766 ◽  
Author(s):  
Jacob L. S. Bellmund ◽  
Peter Gärdenfors ◽  
Edvard I. Moser ◽  
Christian F. Doeller
Keyword(s):  
Hippocampal Formation ◽  
Wide Spectrum ◽  
Spatial Navigation ◽  
Grid Cell ◽  
Dynamic Mapping ◽  
Mapping System ◽  
Representational Format ◽  
Orthogonal Representations ◽  
Human Thinking ◽  
And Behavior

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.

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