从Jack多项式到双Jack多项式的超对称桥
From Jack to Double Jack Polynomials via the Supersymmetric Bridge
摘要：
Calogero-Sutherland模型出现在大量的物理环境中，直接或通过其本征函数，杰克多项式。这个模型的超对称对应物，虽然不那么普遍，但具有同样丰富的结构。特别是，它的本征函数，杰克超多项式，似乎共享非常显着的组合和结构特性作为其非超对称版本。这些超函数由具有固定玻色子和费米子度的超分区参数化。现在，当费米子次数足够大时，一个真正令人惊讶的特征就会出现：杰克超多项式稳定和因式分解。它们的稳定性是相对于它们在基本基方面的膨胀而言的，在稳定的扇区中，膨胀系数变得与费米子度无关。当费米子变量以适当的方式剥离时，可以看到它们的因式分解，这会导致两个普通杰克多项式的乘积（通过体积变换进行了一些修改），被称为双杰克多项式。在这里，除了拼写出这些结果，这是第一次获得的Macdonal超多项式的上下文中，我们提供了一个启发式推导的杰克超多项式的情况下，通过执行简单的操作上的超对称本征算子，使它们独立的粒子数和费米子度。此外，我们还导出了表征双插孔的哈密顿量的表达式。这个哈密顿量，它定义了一个新的可积系统，不仅涉及到预期的Calogero-Sutherland片，但也组合的一个潜在的仿射${\widehat{\mathfrak {sl}}_2}$代数的生成元。
英文摘要：
The Calogero-Sutherland model occurs in a large number of physical contexts, either directly or via its eigenfunctions, the Jack polynomials. The supersymmetric counterpart of this model, although much less ubiquitous, has an equally rich structure. In particular, its eigenfunctions, the Jack superpolynomials, appear to share the very same remarkable combinatorial and structural properties as their non-supersymmetric version. These super-functions are parametrized by superpartitions with fixed bosonic and fermionic degrees. Now, a truly amazing feature pops out when the fermionic degree is sufficiently large: the Jack superpolynomials stabilize and factorize. Their stability is with respect to their expansion in terms of an elementary basis where, in the stable sector, the expansion coefficients become independent of the fermionic degree. Their factorization is seen when the fermionic variables are stripped off in a suitable way which results in a product of two ordinary Jack polynomials (somewhat modified by plethystic transformations), dubbed the double Jack polynomials. Here, in addition to spelling out these results, which were first obtained in the context of Macdonal superpolynomials, we provide a heuristic derivation of the Jack superpolynomial case by performing simple manipulations on the supersymmetric eigen-operators, rendering them independent of the number of particles and of the fermionic degree. In addition, we work out the expression of the Hamiltonian which characterizes the double Jacks. This Hamiltonian, which defines a new integrable system, involves not only the expected Calogero-Sutherland pieces but also combinations of the generators of an underlying affine ${\widehat{\mathfrak {sl}}_2}$ algebra.
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