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Lusztig–Dyer combinatorial invariance conjecture for Kazhdan–Lusztig polynomials

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Let (W1,S1)(W_1,S_1) and (W2,S2)(W_2,S_2) be Coxeter systems, and let σ1,ω1easympW1\sigma_1,\omega_1 easymp W_1 and σ2,ω2easympW2\sigma_2,\omega_2 easymp W_2. For each pair, let G[σi,ωi]G[\sigma_i,\omega_i] denote the directed Bruhat interval, whose vertices are the elements between σi\sigma_i and ωi\omega_i in Bruhat order and whose edges are the directed reflection edges. Lusztig–Dyer's combinatorial invariance conjecture. If G[σ1,ω1]G[\sigma_1,\omega_1] and G[σ2,ω2]G[\sigma_2,\omega_2] are isomorphic as directed graphs, then

Pσ1,ω1=Pσ2,ω2.P_{\sigma_1,\omega_1}=P_{\sigma_2,\omega_2}.

The conjecture asserts that Kazhdan–Lusztig polynomials depend only on the directed-graph structure of the Bruhat interval, and is attributed in the source to Lusztig and Dyer. Its status is not specified in the supplied text.

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Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Lusztig–Dyer combinatorial invariance conjecture for Kazhdan–Lusztig polynomials

    Let WW and W′W' be Coxeter groups, with u,v∈Wu,v\in W and u′,v′∈W′u',v'\in W'. Write [u,v][u,v] and [u′,v′][u',v'] for the corresponding Bruhat intervals, and let Pu,v(q)P_{u,v}(q) and R~u,v(q)\widetilde R_{u,v}(q) denote the Kazhdan–Lusztig and modified Kazhdan–Lusztig RR-polynomials. Combinatorial invariance conjecture. If [u,v]≅[u′,v′][u,v]\cong [u',v'] as posets, then

    Pu,v(q)=Pu′,v′(q)P_{u,v}(q)=P_{u',v'}(q)

    or, equivalently,

    R~u,v(q)=R~u′,v′(q).\widetilde R_{u,v}(q)=\widetilde R_{u',v'}(q).

    The conjecture is known in several partial cases, including intervals of length at most 66, intervals in type A~2\widetilde A_2, lattice intervals, intervals beginning at the identity, and further bounded-length cases. It remains open in general and would follow from sufficiently broad hypercube-decomposition results.

    source: Margherita Zannoni, “Double shortcuts of standard hypercube decompositions”, arXiv:2605.13304 (2026).

References

Primary source

Maxim Gurevich and Chuijia Wang, “Parabolic recursions for Kazhdan-Lusztig polynomials and the hypercube decomposition”, arXiv:2303.09251 (2023).

bruhat-ordercoxeter-groupskazhdan-lusztig-polynomials
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