The su($n$) WZNW fusion ring as integrable model : a new algorithm to compute fusion coefficients (Infinite Analysis 2010 Developments in Quantum Integrable Systems)
The su($n$) WZNW fusion ring as integrable model : a new algorithm to compute fusion coefficients (Infinite Analysis 2010 Developments in Quantum Integrable Systems)
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作为可积模型的 su($n$) WZNW 融合环：计算融合系数的新算法（Infinite Analysis 2010 量子可积系统的发展）
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发表时间：
2011
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通讯作者：
C. Korff
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作者：
C. Korff
这是一篇会议论文，回顾了C. Stroppel和作者近期对\(\hat{su}(n)_k\) WZNW融合环的一种组合构造。它包含一个新颖的方面：明确推导了一种不同于卡茨 - 沃尔顿公式的融合系数计算算法。讨论是从统计力学中的顶点模型的角度展开的，其配分函数生成融合系数。通过将其转移矩阵与杨 - 巴克斯特方程的一个特解相联系，可以证明该统计模型是可积的。这个转移矩阵可被认定为在一个非交换字母表中的一组（无穷）多项式的生成函数：局部仿射普拉蒂克代数的生成元。后者是在罗宾逊 - 申斯泰特对应背景下出现的普拉蒂克代数的一种推广。人们可以在这个非交换字母表中定义舒尔多项式的类似物，当它们作为可积模型的态空间上的自同态表示时，这些类似物与融合矩阵相同。关键是本征基（即贝塞向量）的构造，它们是融合代数的幂等元。
This is a proceedings article reviewing a recent combinatorial construction of the ŝu(n)k WZNW fusion ring by C. Stroppel and the author. It contains one novel aspect: the explicit derivation of an algorithm for the computation of fusion coefficients different from the KacWalton formula. The discussion is presented from the point of view of a vertex model in statistical mechanics whose partition function generates the fusion coefficients. The statistical model can be shown to be integrable by linking its transfer matrix to a particular solution of the Yang-Baxter equation. This transfer matrix can be identified with the generating function of an (infinite) set of polynomials in a noncommutative alphabet: the generators of the local affine plactic algebra. The latter is a generalisation of the plactic algebra occurring in the context of the Robinson-Schensted correspondence. One can define analogues of Schur polynomials in this noncommutative alphabet which become identical to the fusion matrices when represented as endomorphisms over the state space of the integrable model. Crucial is the construction of an eigenbasis, the Bethe vectors, which are the idempotents of the fusion algebra.