The fattened Davis complex and weighted L2–(co)homology of Coxeter groups

Abstract We show that, under weak assumptions, the automorphism group of a $\textrm{CAT(0)}$ cube complex $X$ coincides with the automorphism group of Hagen’s contact graph $\mathcal{C}(X)$. The result holds, in particular, for universal covers of Salvetti complexes, where it provides an analogue of Ivanov’s theorem on curve graphs of non-sporadic surfaces. This highlights a contrast between contact graphs and Kim–Koberda extension graphs, which have much larger automorphism group. We also study contact graphs associated with Davis complexes of right-angled Coxeter groups. We show that these contact graphs are less well behaved and describe exactly when they have more automorphisms than the universal cover of the Davis complex.

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Automorphisms of odd Coxeter groups

Monatshefte für Mathematik ◽

10.1007/s00605-020-01496-3 ◽

2021 ◽

Author(s):

Tushar Kanta Naik ◽

Mahender Singh

Keyword(s):

Coxeter Groups

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The 300 “correlators” suggests 4D, $$ \mathcal{N} $$ = 1 SUSY is a solution to a set of Sudoku puzzles

Journal of High Energy Physics ◽

10.1007/jhep05(2021)077 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Aleksander J. Cianciara ◽

S. James Gates ◽

Yangrui Hu ◽

Renée Kirk

Keyword(s):

Coxeter Groups ◽

Auxiliary Field ◽

Permutation Groups ◽

Weight Space ◽

Field Problem ◽

Truncated Octahedron ◽

Color Problem ◽

Minimal Representations

Abstract A conjecture is made that the weight space for 4D, $$ \mathcal{N} $$ N -extended supersymmetrical representations is embedded within the permutahedra associated with permutation groups 𝕊d. Adinkras and Coxeter Groups associated with minimal representations of 4D, $$ \mathcal{N} $$ N = 1 supersymmetry provide evidence supporting this conjecture. It is shown that the appearance of the mathematics of 4D, $$ \mathcal{N} $$ N = 1 minimal off-shell supersymmetry representations is equivalent to solving a four color problem on the truncated octahedron. This observation suggest an entirely new way to approach the off-shell SUSY auxiliary field problem based on IT algorithms probing the properties of 𝕊d.

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