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combinatorics / Poset topology

The Hamaker-Reiner Conjecture on ASM Weak Order Intervals

Hamaker and Reiner conjectured that the order complex of an open interval $(u,w)$ in the ASM weak order is contractible unless $w$ is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on $S_n$ whose order complex is not contractible even though $w$ has no such form, detected by a nonzero Mobius function value.

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combinatoricsMay 8, 2026Significance 15/100Registry: unreviewed

The Hamaker-Reiner Conjecture on ASM Weak Order Intervals

Prior state unknown→disproved

Hamaker and Reiner conjectured that the order complex of an open interval $(u,w)$ in the ASM weak order is contractible unless $w$ is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on $S_n$ whose order complex is not contractible even though $w$ has no such form, detected by a nonzero Mobius function value.

Source authenticatedThe Hamaker-Reiner Conjecture on ASM Weak Order Intervals →
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Hamaker and Reiner conjectured that the order complex of an open interval $(u,w)$ in the ASM weak order is contractible unless $w$ is the long element of a standard parabolic subgroup, in which case it is homotopy equivalent to a sphere. False: there is an interval in the ASM weak order on $S_n$ whose order complex is not contractible even though $w$ has no such form, detected by a nonzero Mobius function value.

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