Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator
Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator
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DOI：
10.1007/s00220-013-1876-x
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发表时间：
2012-09
影响因子：
2.4
通讯作者：
N. Tsilevich;A. Vershik
中科院分区：
文献类型：
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作者：
N. Tsilevich;A. Vershik
我们将经典的Schur-Weyl对偶推广到群和无限对称群的表示之间。我们的结构是基于舒尔-韦尔对偶的“动态”或归纳的方案。它引出了无限对称群的一类新的表示，这是以前没有出现过的。我们描述了这些表示，特别是找到了它们关于Gelfand-Tsetlin代数的谱类型。这种表示的主要例子作用于不完全无限张量积。作为一个重要的应用，我们在这些表示中考虑所谓的Coxeter-Laplace算子的弱极限，它本质上是XXX Heisenberg模型的哈密顿量。
We extend the classical Schur–Weyl duality between representations of the groupsandto the case ofand the infinite symmetric group. Our construction is based on a “dynamic,” or inductive, scheme of Schur–Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which has not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand–Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important application, we consider the weak limit of the so-called Coxeter–Laplace operator, which is essentially the Hamiltonian of the XXX Heisenberg model, in these representations.