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MASDOC - Mathematics and Statistics Doctoral Training Centre

LLT Polynomials

A useful object when dealing with domino tableau are the LLT Polynomials. They are symmetric functions and are defined as follows. For a skew shape \lambda \setminus \gamma and parameter q \in [0,1], the corresponding LLT polynomial is:

\[</p> <p>G_{\lambda \setminus \gamma}(x, q^{\frac{1}{2}}) = \sum_{P:sh(P) = \lambda \setminus \gamma} q^{spin(P)} x^{weight(P)}</p> <p>\]

If the content of P has c_1 1s, c_2 2s, ... , c_n ns, then {\textbf{x}}^{weight(P)} := \Pi_{i=1}^n x_i^{c_i} and we define spin(P) to be half the number of vertical dominoes in P

Calculating an LLT Polynomial


We wish to calculate the LLT polynomial associated with the shape [3,3], given variables x=(x_1,x_2). Because there are two variables, we look at all legal domino tableaux that have shape [3,3] that do not contain any numbers other than one or two. These can be seen below.

lltcalc.png

Considering the definition, and the diagram above, we can see that

\[</p> <p>G_{[3,3]}(x_1,x_2,q^{\frac{1}{2}})=q^{\frac{3}{2}}(x_1^3+x_1^2x_2^1+x_1^1x_2^2+x_2^3)+q^{\frac{1}{2}}(x_1^2x_2^1+x_1^1x_2^2)</p> <p>\]