论文标题
Weyl Group Action和$ Q $ - 量子仿期代数的特征
Invariants of Weyl group action and $q$-characters of quantum affine algebras
论文作者
论文摘要
令$ w $为对应于有限维度简单的lie代数$ \ mathfrak {g} $等级$ \ ell $的weyl grout，让$ m> 1 $是整数。在[i21]中，通过应用群集突变，构建了$ \ mathcal {y} _m $上的$ w $ action。这里$ \ MATHCAL {y} _m $是$ cm \ ell $通勤变量上的有理函数字段，其中$ c \ in \ {1，2，3 \} $取决于$ \ mathfrak {g} $。这是由$ q $ -character地图$χ_q$的量子仿射代数$ u_q（\ hat {\ mathfrak {g}}）$的有限维表示的类别的动机。我们在[i21]中表明，当$ q $是团结的根源时，$ \ mathrm {im}χ_q$是$ W $ -INVARIANT subfield $ \ Mathcal {y} _mm^w $ of $ \ natcal {y} _mathcal {y} {y} _mm $。在本文中，我们对$ \ Mathcal {y} _m^w $进行了更详细的研究；对于与$ i $ th简单根相关的每个反射$ r_i \ in w $，我们描述$ r_i $ -invariant子field $ \ mathcal {y} _m^{r_i} $ of $ \ nathcal {y} _m $。
Let $W$ be the Weyl group corresponding to a finite dimensional simple Lie algebra $\mathfrak{g}$ of rank $\ell$ and let $m>1$ be an integer. In [I21], by applying cluster mutations, a $W$-action on $\mathcal{Y}_m$ was constructed. Here $\mathcal{Y}_m$ is the rational function field on $cm\ell$ commuting variables, where $c \in \{ 1, 2, 3 \}$ depends on $\mathfrak{g}$. This was motivated by the $q$-character map $χ_q$ of the category of finite dimensional representations of quantum affine algebra $U_q(\hat{\mathfrak{g}})$. We showed in [I21] that when $q$ is a root of unity, $\mathrm{Im} χ_q$ is a subring of the $W$-invariant subfield $\mathcal{Y}_m^W$ of $\mathcal{Y}_m$. In this paper, we give more detailed study on $\mathcal{Y}_m^W$; for each reflection $r_i \in W$ associated to the $i$th simple root, we describe the $r_i$-invariant subfield $\mathcal{Y}_m^{r_i}$ of $\mathcal{Y}_m$.
