Hidden symmetric group actions

该项目涉及代数拓扑，引入了等变同伦理论的思想，以及与对称群表示理论相关的代数和组合学。在拓扑学、代数和组合学中，第n个对称群的明显作用实际上是(n+1)-st个对称群的“隐藏”作用的限制。这种情况出现在树的空间和某些构型空间中。在树的第n个空间和n个分区格的神经之间存在一个已知的等变同伦等价。我们将明确和研究分区格上隐藏的额外作用。我们将探讨对称群表示的结果。

The project is in algebraic topology, bringing in ideas from equivariant homotopy theory, together with algebra and combinatorics related to representation theory of the symmetric group. It will study some related situations in topology, algebra and combinatorics where the obvious action of the n-th symmetric group is in fact the restriction of a "hidden" action of the (n+1)-st symmetric group. This situation arises for spaces of trees and certain configuration spaces. There is a known equivariant homotopy equivalence between the n-th space of trees and the nerve of the lattice of partitions of n. The hidden extra action on the partition lattice will be made explicit and studied. Consequences for representations of the symmetric group will be explored.