$${\mathfrak {sl}}_3$$ sl 3 -Web bases, intermediate crystal bases and categorification

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 A2, or sl(3). One basis is the web basis, defined from a skein theory called the combinatorial A2 spider. The other basis is the dual canonical basis, the dual of the basis defined by Lusztig and Kashiwara. For sl(2) or A1, the web bases have been discovered many times and were recently sh...

Crystal Bases for U Q (sl(2, 1))

A construction of the crystal bases for the quantized enveloping algebra of sl(2, 1) is discussed.

G-Frames, g-orthonormal bases and g-Riesz bases

G-Frames in Hilbert spaces are a redundant set of operators which yield a representation for each vector in the space. In this paper we investigate the connection between g-frames, g-orthonormal bases and g-Riesz bases. We show that a family of bounded operators is a g-Bessel sequences if and only if the Gram matrix associated to its denes a bounded operator.

Lecture 2: Crystal Bases

Today I’ll define crystal bases, and discuss their basic properties. This will include the tensor product rule and the relationship between the crystals B(λ) of highest weight modules and the infinity crystal B(∞). I’ll then discuss an intrinsic characterization of crystal lattices and crystal bases in terms of a natural bilinear form (sometimes called “polarization”). Unless otherwise stated, ...

Crystal Bases and Young Tableaux