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Compactifications of discrete quantum groups
Given a discrete quantum group A we construct a certain Hopf *-algebra AP
which is a unital *-subalgebra of the multiplier algebra of A. The structure
maps for AP are inherited from M(A) and thus the construction yields a
compactification of A which is analogous to the Bohr compactification of a
locally compact group. This algebra has the expected universal property with
respect to homomorphisms from multiplier Hopf algebras of compact type (and is
therefore unique). This provides an easy proof of the fact that for a discrete
quantum group with an infinite dimensional algebra the multiplier algebra is
never a Hopf algebra
Flag varieties and interpretations of Young tableau algorithms
The conjugacy classes of nilpotent matrices can be parametrised
by partitions of , and for a nilpotent in the class
parametrised by , the variety of -stable flags has its
irreducible components parametrised by the standard Young tableaux of shape
. We indicate how several algorithmic constructions defined for Young
tableaux have significance in this context, thus extending Steinberg's result
that the relative position of flags generically chosen in the irreducible
components of parametrised by tableaux and , is the permutation
associated to under the Robinson-Schensted correspondence. Other
constructions for which we give interpretations are Sch\"utzenberger's
involution of the set of Young tableaux, jeu de taquin (leading also to an
interpretation of Littlewood-Richardson coefficients), and the transpose
Robinson-Schensted correspondence (defined using column insertion). In each
case we use a doubly indexed family of partitions, defined in terms of the flag
(or pair of flags) determined by a point chosen in the variety under
consideration. We show that for generic choices, the family satisfies certain
combinatorial relations, whence the family describes the computation of the
algorithmic operation being interpreted, as we described in a previous
publication.Comment: 16 page
The shortcomings of semi-local and hybrid functionals: what we can learn from surface science studies
A study of the adsorption of CO on late 4d and transition metal (111)
surfaces (Ru, Rh, Pd, Ag, Os, Ir, and Pt) considering atop and hollow site
adsorption is presented. The applied functionals include the gradient corrected
PBE and BLYP functional, and the corresponding hybrid Hartree-Fock density
functionals HSE and B3LYP. We find that PBE based hybrid functionals
(specifically HSE) yield, with the exception of Pt, the correct site order on
all considered metals, but they also considerably overestimate the adsorption
energies compared to experiment. On the other hand, the semi-local BLYP
functional and the corresponding hybrid functional B3LYP yield very
satisfactory adsorption energies and the correct adsorption site for all
surfaces. We are thus faced with a Procrustean problem: the B3LYP and BLYP
functionals seem to be the overall best choice for describing adsorption on
metal surfaces, but they simultaneously fail to account well for the properties
of the metal, vastly overestimating the equilibrium volume and underestimating
the atomization energies. Setting out from these observations, general
conclusions are drawn on the relative merits and drawbacks of various
semi-local and hybrid functionals. The discussion includes a revised version of
the PBE functional specifically optimized for bulk properties and surface
energies (PBEsol), a revised version of the PBE functional specifically
optimized to predict accurate adsorption energies (rPBE), as well as the
aforementioned BLYP functional. We conclude that no semi-local functional is
capable to describe all aspects properly, and including non-local exchange also
only improves some, but worsens other properties.Comment: 12 pages, 6 figures; to be published in New Journal of Physic
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