论文标题
纳扎罗夫 - 斯基兰氏素洛杉矶国际裔运算符的光谱理论
Spectral Theory of the Nazarov-Sklyanin Lax Operator
论文作者
论文摘要
在对杰克多项式的研究中，纳扎罗夫 - 斯基兰素引入了一个了不起的新级线性算子$ {\ Mathcal l} \ colon f [w] \ rightarrow f [w]，其中$ f $是符号函数的环，$ w $是可变的。在本文中，我们（1）将环状分解$ f [w] \bigoplus_λz（j_λ，{\ nathcal l}）$纳入有限二维$ {\ natercal l} $ - j polynomials $j_λ$ cycit $ cycit cyc vercription和2）每个$ z（j_λ，{\ mathcal l}）$的l} $具有由$λ$的Young Young图的Addable Corners $ s $的各向异性内容$ [s] $提供的简单频谱。我们对（1）和（2）的证明依赖于与$ {\ Mathcal l} $相关的可集成层次结构的通勤和频谱定理，均由Nazarov-Sklyanin建立。最后，我们猜测$ {\ Mathcal l} $ - eigenfunctions $ψ_λ^s {\ in f [w]} $ {带有eigenvalue $ [s] $ and contand} $ cop} $ c = s | _ {w = 0}整数系数。
In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon F[w] \rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1) establish a cyclic decomposition $F[w] \cong \bigoplus_λ Z(j_λ, {\mathcal L})$ into finite-dimensional ${\mathcal L}$-cyclic subspaces in which Jack polynomials $j_λ$ may be taken as cyclic vectors and (2) prove that the restriction of ${\mathcal L}$ to each $Z(j_λ, {\mathcal L})$ has simple spectrum given by the anisotropic contents $[s]$ of the addable corners $s$ of the Young diagram of $λ$. Our proofs of (1) and (2) rely on the commutativity and spectral theorem for the integrable hierarchy associated to ${\mathcal L}$, both established by Nazarov-Sklyanin. Finally, we conjecture that the ${\mathcal L}$-eigenfunctions $ψ_λ^s {\in F[w]}$ {with eigenvalue $[s]$ and constant term} $ψ_λ^s|_{w=0} = j_λ$ are polynomials in the rescaled power sum basis $V_μ w^l$ of $F[w]$ with integer coefficients.
