8 March 2001
Kristin and Nick
Jim requested that someone pick up an article that is in the Physics Library: Baxter, Kelland and Wu: J. Phys A_9, 397 (1976)
Jim also requested that we work on determining how to mutate a grid diagram by local moves in order to get all other possibilities.
Geir explained how to use q-analogs.
He explained this in relation to lattice paths. Start with a grid and by moves of only down or to the left, go from upper left vertex to lower right vertex. How many paths are there? A way to represent the answer is to consider the area underneath the lattice path and encode the info with a polynomial in q. The coefficient of q^i equals the number of lattice paths with an area of i. So, in the example of a 2x2 square grid, the number of lattice paths can be given by: 1+q+2q^2+q^3+q^4.
Next Geir defined [n]=(1-q^n)/(1-q)=1+q+q^2+...+q^(n-1) He went on to define [n]!=[n][n-1][n-2]... and you can substitute for [n] in terms of the q notation previously defined. [n]! is called a q-factorial. This allows us to define q-binomial coefficients, which are analogous to the standard coefficents, but with the bracket notation. So, the q-binomial coefficient looks like this: [n]!/([n-k]![k]!)
This can be applied to the example of the 2x2 grid, where the number of lattice paths is given by !/(!!) And Geir worked this out and verified it.
So, in general, on the kx(n-k) grid, you use the q binomial coefficient to determine the number of lattice paths. Why is this useful? The q-analog gives more useful information. For example, it allows you determine how many paths there are for a given area under a path.
This can also be applied to the lozenge tilings of a semiregular hexagon with sides (a,b,c). We have a formula that tells us the number of tililings of such a region, which is the triple product of the quantity (i+j+k-1)/(i+j+k-2).
If we think of the three dimensional picture of cubes in a box that the hexagon tiling represents and take the triple product of [i+j+k-1]/[i+j+k-2] we get the number of cubes.
Jim went on to explain how q-analog can also be applied to the number of tilings of an aztec diamond. I am a little fuzzy on this though and Nick did not have much in his notes about this either.
Next we went next door and Hal showed us his really beautiful applet and Jim showed us another program which I do not know the name of and could barely see. There was also a 'tripped-out' program that was run for us.