scholarly journals Geometric realization and coincidence for reducible non-unimodular Pisot tiling spaces with an application to \beta -shifts

2006 ◽  
Vol 56 (7) ◽  
pp. 2213-2248 ◽  
Author(s):  
Veronica Baker ◽  
Marcy Barge ◽  
Jaroslaw Kwapisz
Keyword(s):  
Geometric Realization ◽  
Tiling Spaces

scholarly journals Geometric realization of Khovanov-Lauda-Rouquier algebras associated with Borcherds-Cartan data

2013 ◽  
Vol 107 (4) ◽  
pp. 907-931 ◽  
Author(s):  
Seok-Jin Kang ◽  
Masaki Kashiwara ◽  
Euiyong Park
Keyword(s):  
Geometric Realization

scholarly journals The cohomology of semi-infinite Deligne–Lusztig varieties

2020 ◽  
Vol 2020 (768) ◽  
pp. 93-147
Author(s):  
Charlotte Chan
Keyword(s):  
Large Class ◽  
Finite Type ◽  
Cohomology Group ◽  
Geometric Realization ◽  
Local Fields ◽  
Irreducible Representations ◽  
Division Algebras ◽  
Full Generality ◽  
Homology Groups ◽  
Supercuspidal Representations

AbstractWe prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes {X_{h}}. Boyarchenko’s two conjectures are on the maximality of {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant {1/n} in the case {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of {X_{h}} attains its Weil–Deligne bound, so that the cohomology of {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

scholarly journals Dirac masses and mixings in the (geo)SM(EFT) and beyond

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Jim Talbert ◽  
Michael Trott
Keyword(s):  
Cp Violation ◽  
Effective Field Theory ◽  
Geometric Realization ◽  
Effective Field ◽  
Renormalization Group Flow ◽  
Quark Masses ◽  
The Standard Model ◽  
Mass Dimension ◽  
Symmetry Transformations ◽  
Group Flow

Abstract We report a set of exact formulae for computing Dirac masses, mixings, and CP-violation parameter(s) from 3×3 Yukawa matrices Y valid when YY† → U†YY†U under global $$ \mathrm{U}{(3)}_{Q_L} $$ U 3 Q L flavour symmetry transformations U. The results apply to the Standard Model Effective Field Theory (SMEFT) and its ‘geometric’ realization (geoSMEFT). We thereby complete, in the Dirac flavour sector, the catalogue of geoSMEFT parameters derived at all orders in the $$ \sqrt{2\left\langle {H}^{\dagger }H\right\rangle } $$ 2 H † H /Λ expansion. The formalism is basis-independent, and can be applied to models with decoupled ultraviolet flavour dynamics, as well as to models whose infrared dynamics are not minimally flavour violating. We highlight these points with explicit examples and, as a further demonstration of the formalism’s utility, we derive expressions for the renormalization group flow of quark masses, mixings, and CP-violation at all mass dimension and perturbative loop orders in the (geo)SM(EFT) and beyond.

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