融合代数、对称多项式、N群的轨道和秩-级对偶
Fusion algebras, symmetric polynomials, orbits of N-groups, and rank-level duality
摘要：
最近，A. M. M.提出了一种在$k$水平上计算$A_{n-1}$型李代数融合系数的方法。Feingold和M. Weiner \cite{FW}使用$\mathbb{Z}_n^k$在$S_k$对$k$ -元组的置换作用下的轨道。他们只得到了n = 2和3的融合系数。我们将把这种方法推广到所有的$n \geq 2$和所有的$k \geq 1$。首先，我们展示了杨图和$S_k$ -轨道的$\mathbb{Z}_n ^k$之间的连接，并使用Pieri规则，我们证明了这种方法的某些特定的权重，产生的融合代数。然后我们证明了轨道方法在一般情况下不起作用，但借助于Jacobi-Trudi行列式，我们给出了一种再现所有A型融合产物的迭代法。
英文摘要：
A method of computing fusion coefficients for Lie algebras of type $A_{n-1}$ on level $k$ was recently developed by A. Feingold and M. Weiner \cite{FW} using orbits of $\mathbb{Z}_n^k$ under the permutation action of $S_k$ on $k$-tuples. They got the fusion coefficients only for n = 2 and 3. We will extend this method to all $n \geq 2$ and all $k \geq 1$. First we show a connection between Young diagrams and $S_k$-orbits of $\mathbb{Z}_n ^k$, and using Pieri rules we prove that this method works for certain specific weights that generate the fusion algebra. Then we show that the orbit method does not work in general, but with the help of the Jacobi-Trudi determinant, we give an iterative method to reproduce all type A fusion products.
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