Crystal

Worksheet by David Joyner, Package by David Joyner, Roland Martin, Michael Fourte

Abstract: Worksheet explains how to use 'crystal' to compute "3 tensor 3 tensor 3".

Application Areas/Subjects: Algebra

Keywords: Tensor, Lie, crystal graphs, Kashiwara, dynkin

Prerequisites: Requires coxeter and weyl packages from Share Library.

> restart:

Crystal test worksheet #1

This maple worksheet will explain how to use crystal to compute "3 tensor 3 tensor 3" - the triple tensor product of the identity representation of SU(3) with itself. "3 tensor 3" breaks up into an irreducible 6 dimensional representation plus the "3-bar" (the contragredient of 3). 3 tensor 3-bar is an irreducible 8 dimensional plus a 1.

> restart;

> with(plots):
with(share):
with(coxeter):
with(weyl):
with(crystal):

The identity representation (i.e., "3") is represented by the list L1 of its weights as follows:

> weyl[weights](A2);
L1:=crystal[weight_system](-(1/3*e2-2/3*e1+1/3*e3),A2);

We use L2 to denote another copy of the same representation as L1.

> weyl[weyl_dim](1/3*e2-2/3*e1+1/3*e3,A2);
L2:=crystal[weight_system](-(1/3*e2-2/3*e1+1/3*e3),A2);

To see the crystal graph of 3, simply type the commands:

> crystal[graphrep](L1,A2,G1);
crystal[graphrep](L2,A2,G2);
crystal[showgraph](G1,1,3);

The crystal graph product of the two is denoted G3:

> crystal[graphprodrep](G1,G2,G3);

To see this, simply type the command:

> crystal[showgraphprod](G3,1,3,1,3);

We pick out the component of this graph which contains the highest weight vector 2*e1 and call it G4:

> crystal[linsubgraphrep](2*e1,A2,G3,G4);

In other words, we denote by G4 the component of G3 which contains the highest weight vertex v1X1 (the so-called "Cartan product"):

> crystal[linsubgraphrep](v1X1,A2,G3,G4);

By the way, to see that v1X1 is the highest weight, simply type the vweight command to see it's associated weight vector:

> networks[vweight](G3);
print(networks[vertices](G3));

Though we don't need this, to see the (root) labels of the edges, type:

> networks[eweight](G3);

The graph G4 can be viewed as a linear graph using the showgraph command:

> crystal[showgraph](G4,1,6);

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