论文标题
国际象棋tableaux，两个的力量和仿射谎言代数
Chess tableaux, powers of two and affine Lie algebras
论文作者
论文摘要
国际象棋Tableaux是一种特殊的标准年轻tableaux，在年轻图的棋盘着色中，甚至数字总是出现在白色细胞中，而黑细胞中的奇数。 If, for $λ$ a partition of $n$, $\text{Chess}(λ)$ denotes the number of chess tableaux of shape $λ$, then Chow, Eriksson and Fan observed that $\displaystyle\sum_{λ\vdash n} \text{Chess}(λ)^2$ is divisible by unusually large powers of $2$.在本文中，我们给出了这种现象的解释，证明了$ n-o（\ sqrt {n}）$的下限，以$ 2 $ - 添加此总和的估值及其概括。我们通过利用Aggine Lie代数$ \ wideHat {\ Mathfrak {\ Mathfrak {sl} _2} $在矢量空间上以分区索引的基础索引的连接来实现这一目标。然后，我们关于国际象棋棋子的结果遵循了对$ \ wideHat {\ mathfrak {sl} _2} $的基本表示的研究，并从具有奇数分母的理性数字中获取的系数。
Chess tableaux are a special kind of standard Young tableaux where, in the chessboard coloring of the Young diagram, even numbers always appear in white cells and odd numbers in black cells. If, for $λ$ a partition of $n$, $\text{Chess}(λ)$ denotes the number of chess tableaux of shape $λ$, then Chow, Eriksson and Fan observed that $\displaystyle\sum_{λ\vdash n} \text{Chess}(λ)^2$ is divisible by unusually large powers of $2$. In this paper, we give an explanation for this phenomenon, proving a lower bound of $n-O(\sqrt{n})$ for the $2$-adic valuation of this sum and a generalization of it. We do this by exploiting a connection with a certain representation of the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ on the vector space with basis indexed by partitions. Our result about chess tableaux then follows from a study of the basic representation of $\widehat{\mathfrak{sl}_2}$ with coefficients taken from the ring of rational numbers with odd denominators.
