论文标题
通用符号/正交功能和一般分支规则
Universal symplectic/orthogonal functions and general branching rules
论文作者
论文摘要
在本文中，我们首先介绍一个通用符号函数的家族$sp_λ（\ mathbf {x}^{\ pm}; \ \ m athbf {z}）$，其中包括symplectic schur函数$sp_λ（\ mathbf {\ mathbf {x}} $sp_λ（\ Mathbf {x}^{\ pm}; z）$，通用符号字符$sp_λ（\ Mathbf {z}）$和InterMediate Symplectic Carture作为亚家族。然后，我们通过顶点运算符实现了通用的符号函数，该功能自然会导致其偏斜版本，并表明$sp_λ（\ mathbf {x}^{\ pm}; \ m athbf {z}）$遵守一般分支规则。这还提供了奇数符号字符的Gelfand-Tsetlin表示，以及奇数符号字符和符号schur函数之间的过渡公式。 其次，我们介绍了一个通用正交功能的家族$o_λ（\ mathbf {x}^{\ pm}; \ m athbf {z}）$及其偏斜版本，我们以类似的方式提供了他们的顶点操作员实现并获得过渡公式和分支规则。通用正交功能$o_λ（\ MathBf {x}^{\ pm}; \ m athbf {z}）$ greene premalize orthogonal schur函数$o_λ（\ mathbf {x}}^{x}^{\ pm}^{\ pm}^{\ pm}）通用正交字符$O_λ（\ Mathbf {Z}）$以及中间正交字符。 第三，我们给出了$ cb $ - 间接schur函数的顶点操作员实现$ s^{cb}_λ（x;β）$由Bisi和Zygouras引入（adv。Math。，2022）和$ db $ - db $ -db $ - 中间schur schur function $ s^{db $ s $ s $ db; $ b $。作为应用程序，我们显示$ s^{cb}_λ（x;β）$等于正式成立性schur多项式$spo_λ（x/β）$，因此简短证明了brent-krent-krattenthaler-warnaar识别性获得的概括，kumari（arxiv：arxiv：arxiv：24001.01.01-23）。
In this paper, we first introduce a family of universal symplectic functions $sp_λ(\mathbf{x}^{\pm};\mathbf{z})$ that include symplectic Schur functions $sp_λ(\mathbf{x}^{\pm})$, odd symplectic characters $sp_λ(\mathbf{x}^{\pm};z)$, universal symplectic characters $sp_λ(\mathbf{z})$ and intermediate symplectic characters as subfamilies. We then realize the universal symplectic functions by vertex operators, which naturally lead to their skew versions, and show that $sp_λ(\mathbf{x}^{\pm};\mathbf{z})$ obey the general branching rules. This also gives the Gelfand-Tsetlin representations of odd symplectic characters and a transition formula between odd symplectic characters and symplectic Schur functions. Secondly we introduce a family of universal orthogonal functions $o_λ(\mathbf{x}^{\pm};\mathbf{z})$ and their skew versions in a similar manner, and we provide their vertex operator realizations and obtain transition formulas and the branching rule. The universal orthogonal functions $o_λ(\mathbf{x}^{\pm};\mathbf{z})$ generalize orthogonal Schur functions $o_λ(\mathbf{x}^{\pm})$, odd orthogonal Schur functions $so_λ(\mathbf{x}^{\pm})$, universal orthogonal characters $o_λ(\mathbf{z})$ as well as intermediate orthogonal characters. Thirdly, we give vertex operator realizations for the $CB$-interpolating Schur functions $s^{CB}_λ(x;β)$ introduced by Bisi and Zygouras (Adv. Math., 2022) and the $DB$-interpolating Schur functions $s^{DB}_λ(x;β)$ interpolating between characters of type $D$ and $B$. As an application, we show $s^{CB}_λ(x;β)$ are equal to the orthosymplectic Schur polynomials $spo_λ(x/β)$, thus give a short proof of the generalization of the Brent-Krattenthaler-Warnaar identity obtained by Kumari (arXiv:2401.01723).
