28 research outputs found
The Ammann-Beenker tilings revisited
This paper introduces two tiles whose tilings form a one-parameter family of
tilings which can all be seen as digitization of two-dimensional planes in the
four-dimensional Euclidean space. This family contains the Ammann-Beenker
tilings as the solution of a simple optimization problem.Comment: 7 pages, 4 figure
Standard monomial theory for wonderful varieties
A general setting for a standard monomial theory on a multiset is introduced
and applied to the Cox ring of a wonderful variety. This gives a degeneration
result of the Cox ring to a multicone over a partial flag variety. Further, we
deduce that the Cox ring has rational singularities.Comment: v3: 20 pages, final version to appear on Algebras and Representation
Theory. The final publication is available at Springer via
http://dx.doi.org/10.1007/s10468-015-9586-z. v2: 20 pages, examples added in
Section 3 and in Section
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Approximate decomposability in and the canonical decomposition of 3-vectors
Given a (Figure presented.)3-vector (Formula presented.) the least distance problem from the Grassmann variety (Formula presented.) is considered. The solution of this problem is related to a decomposition of (Formula presented.) into a sum of at most five decomposable orthogonal 3-vectors in (Formula presented.). This decomposition implies a certain canonical structure for the Grassmann matrix which is a special matrix related to the decomposability properties of (Formula presented.). This special structure implies the reduction of the problem to a considerably lower dimension tensor space ⊗3R2 where the reduced least distance problem can be solved efficiently
Standard Monomial Theory for desingularized Richardson varieties in the flag variety GL(n)/B
We consider a desingularization Gamma of a Richardson variety X_w^v=X_w \cap
X^v in the flag variety Fl(n)=GL(n)/B, obtained as a fibre of a projection from
a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous
coordinate ring of Gamma inside Z, indexed by combinatorial objects which we
call w_0-standard tableaux