The weak order is the partial order on the symmetric group S_n (or other Coxeter group) whose cover relations correspond to simple transpositions; this talk will describe results from two projects about the weak order, both are joint with Yibo Gao.
Maximal chains in the weak order on S_n are the same thing as sorting networks. We study maximal collections of antichains, showing that the weak order has the Sperner property: no antichain is larger than the largest level set. This answers open problems of Björner and of Stanley. We prove this theorem by exhibiting a representation of the Lie algebra sl_2 which respects the weak order and the Bruhat order, leading to new formulas for principal specializations of Schubert polynomials.
Next, we study length-additive multiplicative decompositions of Weyl groups (such as a parabolic quotient times a parabolic subgroup). In the case of S_n, we show that such decompositions always come from weak order intervals and that they are in natural bijection with faces of the associahedron, answering an open problem of Björner-Wachs. We also show that multiplying permutations from certain pairs of weak order intervals always surjects onto S_n, resolving an open problem of Morales-Pak-Panova about linear extensions of 2-dimensional posets.
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