$\widehat{s\ell}(n)$-模 $V(k\Lambda_0)$ 中若干极大支配权的重数
表示论
2020-05-01 v1
摘要
对 n ≥ 2 n \geq 2 n ≥ 2 { α i ∣ 0 ≤ i ≤ n − 1 } \{\alpha_i \mid 0 \leq i \leq n-1\} { α i ∣ 0 ≤ i ≤ n − 1 } s ℓ ^ ( n ) \widehat{s\ell}(n) s ℓ ( n ) V ( k Λ 0 ) , k ∈ Z ≥ 1 V(k\Lambda_0), \, k \in \mathbb{Z}_{\geq 1} V ( k Λ 0 ) , k ∈ Z ≥ 1 k Λ 0 k\Lambda_0 k Λ 0 s ℓ ^ ( n ) \widehat{s\ell}(n) s ℓ ( n ) V ( k Λ 0 ) V(k\Lambda_0) V ( k Λ 0 ) V ( k Λ 0 ) V(k\Lambda_0) V ( k Λ 0 ) k Λ 0 − λ a , b ℓ k\Lambda_0 - \lambda^\ell_{a,b} k Λ 0 − λ a , b ℓ λ a , b ℓ = ℓ α 0 + ( ℓ − b ) α 1 + ( ℓ − ( b + 1 ) ) α 2 + ⋯ + α ℓ − b + α n − ℓ + a + 2 α n − ℓ + a + 1 + … + ( ℓ − a ) α n − 1 \lambda^\ell_{a,b} = \ell\alpha_0 + (\ell-b)\alpha_1 + (\ell-(b+1))\alpha_2 + \cdots + \alpha_{\ell-b} + \alpha_{n-\ell+a} + 2\alpha_{n - \ell+a+1} + \ldots + (\ell-a)\alpha_{n-1} λ a , b ℓ = ℓ α 0 + ( ℓ − b ) α 1 + ( ℓ − ( b + 1 )) α 2 + ⋯ + α ℓ − b + α n − ℓ + a + 2 α n − ℓ + a + 1 + … + ( ℓ − a ) α n − 1 k ≥ a + b k \geq a+b k ≥ a + b a , b ∈ Z ≥ 1 a,b \in \mathbb{Z}_{\geq 1} a , b ∈ Z ≥ 1 max { a , b } ≤ ℓ ≤ ⌊ n + a + b 2 ⌋ − 1 \max\{a,b\} \leq \ell \leq \left \lfloor \frac{n+a+b}{2} \right \rfloor-1 max { a , b } ≤ ℓ ≤ ⌊ 2 n + a + b ⌋ − 1 { 1 , 2 , 3 , … ℓ } \{1, 2, 3, \ldots \ell\} { 1 , 2 , 3 , … ℓ }
引用
@article{arxiv.2004.14470,
title = {Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(k\Lambda_0)$},
author = {Rebecca L. Jayne and Kailash C. Misra},
journal= {arXiv preprint arXiv:2004.14470},
year = {2020}
}