500 research outputs found
Twisted Witt Groups of Flag Varieties
Calm\`es and Fasel have shown that the twisted Witt groups of split flag
varieties vanish in a large number of cases. For flag varieties over
algebraically closed fields, we sharpen their result to an if-and-only-if
statement. In particular, we show that the twisted Witt groups vanish in many
previously unknown cases. In the non-zero cases, we find that the twisted total
Witt group forms a free module of rank one over the untwisted total Witt group,
up to a difference in grading.
Our proof relies on an identification of the Witt groups of flag varieties
with the Tate cohomology groups of their K-groups, whereby the verification of
all assertions is eventually reduced to the computation of the (twisted) Tate
cohomology of the representation ring of a parabolic subgroup.Comment: inverse Cartan coefficients for E_7 in Figures 1, 2 and 3 corrected;
related mistake in the marking scheme for diagrams of type E_n corrected;
many minor corrections and clarifications; more example
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Smith-Knight/Rayleigh-Knight Essay
On a complex variety X, two different approaches to K-theory are available: the
algebraic K-theory of the variety, and the topological K-theory of the underlying
topological space. In this context, the algebraic variant known as Hermitian
K-theory corresponds to topological KO-theory. Our aim is to compare the two
approaches.
We start by constructing a comparison map from certain Hermitian K-groups
of X to the KO-groups of X. It is clear what this map must be on groups in
degree zero, but the definitions of relative and higher groups differ widely in
the algebraic and the topological setting. This difficulty can be overcome by
viewing relative and higher groups as subgroups of degree zero groups of certain
auxiliary spaces.
Once the definition of our comparison map is in place, we prove a number
of fundamental properties, in particular compatibility with pushforwards along
closed embeddings. We also show how we can use it to compare an exact
sequence relating usual algebraic K-theory to Hermitian K-theory with a portion
of the Bott sequence in topology. This finally allows us to deduce that the
map is an isomorphism on smooth cellular varieties. We conclude with some
details concerning projective spaces, for which independent computations of the
algebraic and the topological groups exist.This work was supported by the Engineering and Physical Sciences Research Council [LFAG/067]
Witt groups of complex cellular varieties
We show that the Grothendieck-Witt and Witt groups of smooth complex cellular
varieties can be identified with their topological KO-groups. As an
application, we deduce the values of the Witt groups of all irreducible
hermitian symmetric spaces, including smooth complex quadrics, spinor varieties
and symplectic Grassmannians.Comment: 43 pages. This is the final version, identical in content to the
paper published in Documenta Mathematic
KO-Rings of Full Flag Varieties
We present type-independent computations of the KO-groups of full flag
varieties, i.e. of quotient spaces G/T of compact Lie groups by their maximal
tori. Our main tool is the identification of the Witt ring, a quotient of the
KO-ring, of these varieties with the Tate cohomology of their complex K-ring.
The computations show that the Witt ring is an exterior algebra whose
generators are determined by representations of G.Comment: 22 pages; updated references and other minor change
On the preparation and NMR spectroscopic characterization of potassium aluminium tetrahydride KAlH4
Potassium aluminium tetrahydride KAlH4 of high phase purity (space group Pnma
(62)) was synthesized via a mechanochemical route. The thus obtained material
was studied by 27Al and 39K MAS NMR spectroscopy. For both nuclei precise data
for the isotropic chemical shift and the quadrupole coupling at T=295 K were
derived (27Al: delta_iso=(107.6+-0.2) ppm, C_Q = (1.29+-0.02) MHz and eta =
0.64+-0.02; 39K: delta_iso=(6.1+-0.2) ppm, C_Q = (0.562+-0.005) MHz and eta =
0.74+-0.02). The straightforward NMR spectroscopic approach applied here should
also work for other complex aluminium hydrides and for many other materials
containing half-integer nuclei experiencing small to medium-sized quadrupole
couplings.Comment: 27 pages, 5 figures, final version as published in PCCP, with
reviewer reports attached to the ES
Vector bundles of non-negative curvature over cohomogeneity one manifolds
We provide several results on the existence of metrics of non-negative
sectional curvature on vector bundles over certain cohomogeneity one manifolds
and homogeneous spaces up to suitable stabilization.
Beside explicit constructions of the metrics, this is achieved by identifying
equivariant structures upon these vector bundles via a comparison of their
equivariant and non-equivariant K-theory. For this, in particular, we
transcribe equivariant K-theory to equivariant rational cohomology and
investigate surjectivity properties of induced maps in the Borel fibration via
rational homotopy theory.Comment: some upgrades and modification
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