论文标题
统一根源扭曲的经典字符的分解：ii
Factorization of classical characters twisted by roots of unity: II
论文作者
论文摘要
修复自然数$ n \ geq 1 $，$ t \ geq 2 $和一个原始$ t^{\ text {th}} $ Unity $ω$的根。在与A. Ayyer的先前工作（J. alg。，2022）中，我们研究了$ \ text {gl} _ {tn} $，$ \ text {so} _ {2tn+1}的专业不可减至的不可还原字符的分解， $ 0 \ leq j \ leq t-1 $和$ 1 \ leq i \ leq n $的元素到$ω^j x_i $。在这项工作中，我们将结果扩展到$ \ text {gl} _ {tn+m} $ $（0 \ leq m \ leq t-1）$，$ \ text {so} _ {2tn+3} $，$ \ $ \ text {sp}专业：（1）对于$ \ text {gl} _ {tn+m}（\ mathbb {c}）$ case，我们将第一个$ tn $元素设置为$ω^j x_i $ for $ 0 \ leq j \ leq j \ leq j \ leq t-1 $和$ 1 \ leq i \ leq i \ leq i \ leq $ $ $ $， ω^{m-1} y $; （2）对于其他三个家庭，具有相同的专业，但$ m = 1 $。本文的主要结果是对分区的表征，这些字符消失并将非零字符分解为较小的经典群体。我们的动机是Wagh和Prasad（Manuscripta Math。，2020）的猜想，将$ \ text {spin} _ {2n+1} $和$ \ text {sl} _ {sl} _ {2n} _ {2n} $，$ \ text} $，$ \ text} AS $ \ text {spin} _ {2n+2} $和$ \ text {sp} _ {2n} $。我们的证明使用Weyl字符公式和$ t $ core分区的Beta-stet。最后，我们进行了培训，以证明这些字符无限的$ t $ core分区无限。
Fix natural numbers $n \geq 1$, $t \geq 2$ and a primitive $t^{\text{th}}$ root of unity $ω$. In previous work with A. Ayyer (J. Alg., 2022), we studied the factorization of specialized irreducible characters of $\text{GL}_{tn}$, $\text{SO}_{2tn+1},$ $\text{Sp}_{2tn}$ and $\text{O}_{2tn}$ evaluated at elements to $ω^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$. In this work, we extend the results to the groups $\text{GL}_{tn+m}$ $(0 \leq m \leq t-1)$, $\text{SO}_{2tn+3}$, $\text{Sp}_{2tn+2}$ and $\text{O}_{2tn+2}$ evaluated at similar specializations: (1) for the $\text{GL}_{tn+m}(\mathbb{C})$ case, we set the first $tn$ elements to $ω^j x_i$ for $0 \leq j \leq t-1$ and $1 \leq i \leq n$ and the remaining $m$ to $y, ωy, \dots, ω^{m-1} y$; (2) for the other three families, the same specializations but with $m=1$. The main results of this paper are a characterization of partitions for which these characters vanish and a factorization of nonzero characters into those of smaller classical groups. Our motivation is the conjectures of Wagh and Prasad (Manuscripta Math., 2020) relating the irreducible representations of $\text{Spin}_{2n+1}$ and $\text{SL}_{2n}$, $\text{SL}_{2n+1}$ and $\text{Sp}_{2n}$ as well as $\text{Spin}_{2n+2}$ and $\text{Sp}_{2n}$. Our proofs use the Weyl character formulas and the beta-sets of $t$-core partitions. Lastly, we give a bijection to prove that there are infinitely many $t$-core partitions for which these characters are nonzero.
