1,201 research outputs found
On singularities of lattice varieties
Toric varieties associated with distributive lattices arise as a fibre of a
flat degeneration of a Schubert variety in a minuscule. The singular locus of
these varieties has been studied by various authors. In this article we prove
that the number of diamonds incident on a lattice point \a in a product of
chain lattices is more than or equal to the codimension of the lattice. Using
this we also show that the lattice varieties associated with product of chain
lattices is smooth
Beck's Conjecture for Power Graphs
Beck's conjecture on coloring of graphs associated to various algebraic
objects has generated considerable interest in the community of discrete
mathematics and combinatorics since its inception in the year 1988. The version
of this conjecture for power-graphs of finite groups has been addressed and
partially settled by previous authors. In this paper we answer it in the
affirmative in complete generality, and, in effect, we establish a "nicer"
statement on a larger class of graphs. We also clear up certain ambiguities
present in the way the previous versions of the conjecture were posed
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