论文标题
$ \ wideHat {s \ ell}（n）$ - 模块$ v（kλ_0）$的某些最大优势权重的多重性
Multiplicities of some maximal dominant weights of the $\widehat{s\ell}(n)$-modules $V(kΛ_0)$
论文作者
论文摘要
对于$ n \ geq 2 $，请考虑使用简单根$ \ {α_i\ mid 0 \ leq I \ leq i \ leq n-1 \} $的Aggine lie lie lelgebra $ \ widehat {s \ ell}（n）$。令$ v（kλ_0），\，k \ in \ mathbb {z} _ {\ geq 1} $表示可集成的最高权重$ \ wideHat {s \ ell}（n）$ - 具有最高权重$kλ_0$的模块。众所周知，$ v（kλ_0）$的最大主要优势权重有限。使用$ V（kλ_0）$和晶格路径组合的晶体基础实现，我们确定了形式的一大大最大优势权重的多样性，$kλ_0-λ_0-λ^\ ell_ {a，b} $ {a，b} $ $ ve + \ cdot +α_ {\ ell-b} +α_{n- \ ell + a} +2α_{n- \ ell + a + a + a + a + 1} + \ ldots +（\ ell-a）α___________{n-1 {n-1 {n-1} $ $ \ max \ {a，b \} \ leq \ ell \ leq \ left \ lfloor \ frac {n+a+a+b} {2} {2} \ right \ rfloor-1 $。我们表明，这些权重多重性是由某些模式的数量给出的，避免了$ \ {1、2、3，\ ldots \ ell \} $的排列。
For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{α_i \mid 0 \leq i \leq n-1\}$. Let $V(kΛ_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module with highest weight $kΛ_0$. It is known that there are finitely many maximal dominant weights of $V(kΛ_0)$. Using the crystal base realization of $V(kΛ_0)$ and lattice path combinatorics we determine the multiplicities of a large set of maximal dominant weights of the form $kΛ_0 - λ^\ell_{a,b}$ where $ λ^\ell_{a,b} = \ellα_0 + (\ell-b)α_1 + (\ell-(b+1))α_2 + \cdots + α_{\ell-b} + α_{n-\ell+a} + 2α_{n - \ell+a+1} + \ldots + (\ell-a)α_{n-1}$, and $k \geq a+b$, $a,b \in \mathbb{Z}_{\geq 1}$, $\max\{a,b\} \leq \ell \leq \left \lfloor \frac{n+a+b}{2} \right \rfloor-1 $. We show that these weight multiplicities are given by the number of certain pattern avoiding permutations of $\{1, 2, 3, \ldots \ell\}$.
