论文标题
对角线谐波交替体的基础
A basis for the Diagonal Harmonic Alternants
论文作者
论文摘要
这里将显示每个$ n \ ge 1 $，在对角线谐波上有差异操作员$ e，f $和$ h = [e，f] $，产生$ dh_n $是$ sl [2] $的表示（请参阅[3]第3章）。我们在这里的主要工作是使用$ sl [2] $理论来预测对角线谐波交替体的基础，$ dha_n $。可以证明，不可约的表示$ sl [2] $都是$ p，ep，e^2p，\ cdots，e^kp $的所有表单，带有$ fp = 0 $和$ e^{k+1} p = 0 $。已知多项式$ P $被称为“弦乐启动器”。从$ sl [2] $理论中，$ dha_n $是直接的字符串总和。到目前为止，我们的主要结果是用于弦乐启动器数量的公式。卡尔森（Carlsson）和大莫洛科夫（Oblomkov）最近的一篇论文（参见[2]）构建了代数几何工具的对角线共同变体空间的基础。有趣的是，我们是否可以从他们的结果中得出任何结果。
It will be shown here that there are differential operators $E,F$ and $H=[E,F]$ for each $n\ge 1$, acting on Diagonal Harmonics, yielding that $DH_n$ is a representation of $sl[2]$ (see [3] Chapter 3). Our main effort here is to use $sl[2]$ theory to predict a basis for the Diagonal Harmonic Alternants, $DHA_n$. It can be shown that the irreducible representations $sl[2]$ are all of the form $P,EP,E^2P,\cdots,E^kP$, with $FP=0$ and $E^{k+1}P=0$. The polynomial $P$ is known to be called a "String Starter". From $sl[2]$ theory it follows that $DHA_n$ is a direct sum of strings. Our main result so far is a formula for the number of string starters. A recent paper by Carlsson and Oblomkov (see [2]) constructs a basis for the space of Diagonal Coinvariants by Algebraic Geometrical tools. It would be interesting to see if any our results can be derived from theirs.
