# multiplicity of a weight in the basic representation of $\hat{sl_2}$

it is known by Weyl character formula that multiplicity of the weight $\wedge_{0}- n.\delta$ in the basic representation $L(\wedge_{0})$ of $\hat{sl_2}$is equal to the number of partitions of $n$.It is known that LS(Lakshmibai seshadri) paths of shape $\wedge_{0}$ gives a parametrization for the basis of $L(\wedge_{0})$.There is a paper of Yasmine B.Sanderson on dimensions of Demazure modules for rank two affine algebras where she has found L-S paths for the basic representation of $\hat{sl_2}$ explicitly. Here is the explicit description of paths. The L-S Paths of shape $\wedge_{0}$ are those paths $\pi = (\sigma,a)$ such that $\sigma:w_{n+k} > w_{n+k-1} > w_{n+k-2} >...> w_{n}$ where $w_i$ is the product of $i$ reflections $r_{0}.r_{1}.r_{0}.r_{1}...$ and $n,k \geq 0$ and $a:0 < a_{n+k} < a_{n+k-1} < ...... < a_{n+1} < 1$ where $a_j.d_j\epsilon Z$ and $d_j = -j$ and $a_{i} \epsilon Q$ Now the question is how can one show that there are exactly $P(n)$ of L-S paths of shape $\wedge_{0}$ whose weight is $\wedge_{0} - n.\delta$ ?

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This is impossible to read/understand for me. Please edit and explain your notation. – Marc Palm Mar 27 '13 at 10:37
You can check the notations in Yasmine B.Sanderson 's paper on dimensions of demazure modules for rank two affine algebras in the journal Compositio Mathematica ,tome 101,n^0 2(1996),p.115-131 for the notations. – Rekha Biswal Mar 27 '13 at 12:19
I still suggest to edit the question. You probably should provide the background required and not give reference to a paper, where the notation is explained. – Marc Palm Mar 31 '13 at 10:45