插值多项式、条形单项式及其正性
Interpolation polynomials, bar monomials, and their positivity
摘要：
证明了Knop和Sahi关于插值多项式的一个正性结果。这些多项式最早是由Sahi在约旦代数的Capelli本征值问题中引入的，后来被Knop-Sahi和Okounkov-Olshanski证明与Jack多项式有关。本文证明的正性结果是Macdonald关于Jack多项式的正性猜想的非齐次推广。我们还制定和证明的非对称版本的诺普-萨希猜想，事实上，我们推导出一切从一个更强的积极性的结果。这最后一个结果涉及某些非齐次类似的普通单项式，我们称之为酒吧单项式。它们的积极性在本质上涉及到一种新的成分偏序，我们称之为小节序，以及一种新的操作，我们称之为滑音。
英文摘要：
We prove a positivity result for interpolation polynomials that was conjectured by Knop and Sahi. These polynomials were first introduced by Sahi in the context of the Capelli eigenvalue problem for Jordan algebras, and were later shown to be related to Jack polynomials by Knop-Sahi and Okounkov-Olshanski. The positivity result proved here is an inhomogeneous generalization of Macdonald's positivity conjecture for Jack polynomials. We also formulate and prove the non-symmetric version of the Knop-Sahi conjecture, and in fact we deduce everything from an even stronger positivity result. This last result concerns certain inhomogeneous analogues of ordinary monomials that we call bar monomials. Their positivity involves in an essential way a new partial order on compositions that we call the bar order, and a new operation that we call a glissade.
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