对称函数的变形与有理Steenrod代数
Deformation of symmetric functions and the rational Steenrod algebra
摘要：
在1999年，Reg Wood证明了Q[x_1，.，x_n]是对称群S_n的分次正则表示.正如雷格·伍德所指出的那样，当有理斯廷罗德代数被对称函数环所取代时，这种说法的类似物是一个众所周知的结果;实际上，在这种情况下，对商的结构有更多的了解。我们引入了对称函数环的一个非交换q-变形，它在q=1时专门用于有理Steenrod代数。我们使用这种形式主义，得到一些部分结果。最后，我们描述了几个基于广泛的计算机探索。特别地，我们将Reg Wood的猜想推广到q形式和任何不具有-a/b形式的q复形，其中a在{1，.，n}，b为正自然数。
英文摘要：
In 1999, Reg Wood conjectured that the quotient of Q[x_1,...,x_n] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group S_n. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the quotient in this case. We introduce a non-commutative q-deformation of the ring of symmetric functions, which specializes at q=1 to the rational Steenrod algebra. We use this formalism to obtain some partial results. Finally, we describe several conjectures based on an extensive computer exploration. In particular, we extend Reg Wood's conjecture to q formal and to any q complex not of the form -a/b, with a in {1,...,n} and b a positive natural number.
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