The exponential nature and positivity

Positivity of Dunkl's intertwining operator

For a finite reflection group on $\b R^N,$ the associated Dunkl operators are parametrized first-order differential-difference operators which generalize the usual partial derivatives. They generate a commutative algebra which is - under weak assumptions - intertwined with the algebra of partial differential operators by a unique linear and homogeneous isomorphism on polynomials. In this paper it is shown that for non-negative parameter values, this intertwining operator is positivity-preserving on polynomials and allows a positive integral representation on certain algebras of analytic functi…

拟指数空间的正定性与普适Plücker坐标

Positivity and universal Plücker coordinates for spaces of quasi-exponentials

本文研究拟指数空间的完全非负性，通过证明高阶Gaudin Hamilton量是Wronski映射的普适Plücker坐标并利用偏迹证明其半正定性，确立了Grassmann-Plücker坐标非负的结论。

Laurent positivity of quantized canonical bases for quantum cluster varieties from surfaces

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichmüller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an…