208 research outputs found
Web bases for sl(3) are not dual canonical
We compare two natural bases for the invariant space of a tensor product of
irreducible representations of A_2, or sl(3). One basis is the web basis,
defined from a skein theory called the combinatorial A_2 spider. The other
basis is the dual canonical basis, the dual of the basis defined by Lusztig and
Kashiwara. For sl(2) or A_1, the web bases have been discovered many times and
were recently shown to be dual canonical by Frenkel and Khovanov.
We prove that for sl(3), the two bases eventually diverge even though they
agree in many small cases. The first disagreement comes in the invariant space
Inv((V^+ tensor V^+ tensor V^- tensor V^-)^{tensor 3}), where V^+ and V^- are
the two 3-dimensional representations of sl(3). If the tensor factors are
listed in the indicated order, only 511 of the 512 invariant basis vectors
coincide.Comment: 18 pages. This version has very minor correction
The still-Life density problem and its generalizations
A "still Life" is a subset S of the square lattice Z^2 fixed under the
transition rule of Conway's Game of Life, i.e. a subset satisfying the
following three conditions:
1. No element of Z^2-S has exactly three neighbors in S;
2. Every element of S has at least two neighbors in S;
3. Every element of S has at most three neighbors in S.
Here a ``neighbor'' of any x \in Z^2 is one of the eight lattice points
closest to x other than x itself. The "still-Life conjecture" is the assertion
that a still Life cannot have density greater than 1/2 (a bound easily
attained, for instance by {(x,y): x is even}). We prove this conjecture,
showing that in fact condition 3 alone ensures that S has density at most 1/2.
We then consider variations of the problem such as changing the number of
allowed neighbors or the definition of neighborhoods; using a variety of
methods we find some partial results and many new open problems and
conjectures.Comment: 29 pages, including many figures drawn as LaTeX "pictures
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