$q$-immanants与高阶量子Capelli恒等式
The $q$-immanants and higher quantum Capelli identities
AI总结：
本文构造了以杨图为参数的多项式，其系数为量子包络代数的中心元素，并证明了高阶Capelli恒等式与Newton型恒等式的量子类似。
摘要：
我们构造了以杨图$μ$为参数的多项式$\mathbb{S}_μ(z)$，其系数是量子包络代数${\rm U}_q({\mathfrak{gl}}_n)$的中心元素。其常数项与Drinfeld和Reshetikhin一般构造提供的中心元素一致。对于$z$的另一特殊值，我们得到${\mathfrak{gl}}_n$的Okounkov量子immanants的$q$-类似。我们证明$\mathbb{S}_μ(z)$的Harish-Chandra像是量子Schur多项式。我们证明了高阶Capelli恒等式和Newton型恒等式的量子类似。
英文摘要：
We construct polynomials ${\mathbb{S}}_μ(z)$ parameterized by Young diagrams $μ$, whose coefficients are central elements of the quantized enveloping algebra ${\rm U}_q({\mathfrak{gl}}_n)$. Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of $z$, we get $q$-analogues of Okounkov's quantum immanants for ${\mathfrak{gl}}_n$. We show that the Harish-Chandra image of ${\mathbb{S}}_μ(z)$ is a factorial Schur polynomial. We also prove quantum analogues of the higher Capelli identities and derive Newton-type identities.
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