论文标题
带有一个列图和矩阵倒置的Koornwinder多项式的分支规则
Branching Rules for Koornwinder Polynomials with One Column Diagrams and Matrix Inversions
论文作者
论文摘要
我们为过渡矩阵$ \ MATHCAL {C} $提供了一个明确的公式$ m _ {（1^{r}）}（x）$。矩阵$ \ mathcal {c} $的条目享受四个术语递归关系。这些递归为我们提供了带有一个列图的Koornwinder多项式的分支规则，即限制规则从$ bc_n $到$ bc_ {n-1} $。要对所涉及的过渡矩阵有很好的描述，我们介绍了Koornwinder多项式的以下变性方案：$ p _ {（1^r）}（x | a，b，b，c，c，d | q，d | q，t）\ longftrightArrow p _ {（1^r） p _ {（1^r）}（x | a，-a，c，-c | q，t）\ longleftrightarrow p _ {（1^r）} \ big（x | t^{1/2} c，-t}，-t^{1/2} p _ {（1^r）} \ big（x | t^{1/2}， - t^{1/2}，1，-1 | q，t \ big）$。我们证明，与这些变性步骤相关的每个变性步骤相关的过渡矩阵都是根据Bressoud的矩阵反转公式给出的。作为一个应用程序，我们给出了$ b_n $的Kostka多项式的明确公式，即Schur多项式的过渡矩阵$ p^{（b_n，b_n，b_n）} _ {（1^r）}（x | q，q）$ $ p^{（b_n，b_n）} _ {（1^r）}（x | t; 0，t）$。我们还提出了$ b_n $ $ $ q $ -toda操作员的渐近免费本征函数的猜想，可以将其视为$ b_n $ $ q $ q $ -toDA eigenfunction的分支公式，仅限于$ a_ {n-1} $ q $ q $ q $ -toda eigen eigenfunctions。
We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $BC_n$ Koornwinder polynomials $P_{(1^r)}(x|a,b,c,d|q,t)$ with one column diagrams, to the type $BC_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from $BC_n$ to $BC_{n-1}$. To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: $P_{(1^r)}(x|a,b,c,d|q,t) \longleftrightarrow P_{(1^r)}(x|a,-a,c,d|q,t)\longleftrightarrow P_{(1^r)}(x|a,-a,c,-c|q,t) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2}c,-t^{1/2}c,c,-c|q,t\big) \longleftrightarrow P_{(1^r)}\big(x|t^{1/2},-t^{1/2},1,-1|q,t\big)$. We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type $B_n$, namely the transition matrix from the Schur polynomials $P^{(B_n,B_n)}_{(1^r)}(x|q;q,q)$ to the Hall-Littlewood polynomials $P^{(B_n,B_n)}_{(1^r)}(x|t;0,t)$. We also present a conjecture for the asymptotically free eigenfunctions of the $B_n$ $q$-Toda operator, which can be regarded as a branching formula from the $B_n$ $q$-Toda eigenfunction restricted to the $A_{n-1}$ $q$-Toda eigenfunctions.
