Three obstructions to a naive dual BN-pair

Definitions and mathematical statement

Problem 3.4 (D. Bessis). Is there a structure theory of Lie groups or algebraic groups analogous to the dual braid monoid, and is there a dual notion of $BN$-pairs? The source does not specify axioms for the desired dual structure or a class of groups. The extraction's following section "4. Garside Structures" is neighboring exposition, not part of the numbered question; it is retained below as context.

Reported obstruction to three naive templates. Let $(W,S)$ be a finite Coxeter system, let $T$ be its reflections, and let $\ell_T$ denote reflection length. For a Coxeter element $c$, define absolute order and the noncrossing interval by \[ u\leq_Tv\quad\Longleftrightarrow\quad \ell_T(u)+\ell_T(u^{-1}v)=\ell_T(v),\qquad \operatorname{NC}(W,c)=[1,c]_T. \] Assume some pair $s,t\in S$ has $m_{st}\geq3$; irreducibility is not required. The aid reports the following three obstructions.

1. Although $c\in\operatorname{NC}(W,c)$, one has $c^2\notin\operatorname{NC}(W,c)$. Hence this set is not a group under the inherited multiplication in $W$, and cannot serve unchanged as an ordinary Weyl quotient with that multiplication. This does not forbid an unrelated group structure on a set of the same cardinality.
2. Whenever a group $G$ has an ordinary $BN$-pair with Weyl group $W=N/(B\cap N)$, and Bruhat decomposition $G=\bigsqcup_{w\in W}B\dot wB$, the union \[ \bigcup_{w\in\operatorname{NC}(W,c)}B\dot wB \] is not a subgroup. It contains a representative of $c$, while its square has Weyl label $c^2$, which is excluded. This is conditional on such a $BN$-pair existing and does not assert that every noncrystallographic $W$ occurs for a Lie group.
3. The poset $\operatorname{NC}(W,c)$ contains a rank-two dihedral interval with $m_{st}$ atoms. Every rank-two interval in Coxeter Bruhat order has exactly two atoms. Therefore $\operatorname{NC}(W,c)$ is not isomorphic to a Coxeter Bruhat interval.

The noncommuting-edge assumption matters: products of commuting $A_1$ factors are outside this obstruction. The conclusions exclude retaining ordinary group multiplication, ordinary Bruhat-cell subgroup unions or ordinary Bruhat intervals as the proposed dual structure. They do not rule out Garside categories, different cells/orders or genuinely new axioms for a dual $BN$-pair.

Spilled-over Garside context. The next section credits David Bessis [6] and Tom Brady [12] with the general noncrossing lattice in their study of Artin groups $A(W)$. It describes labeled Hasse diagrams yielding monoids $M(P)$ and groups $G(P)$, with appropriate interval-path relations. Additional labeling/Garside axioms give useful normal forms and word/conjugacy algorithms; those axioms are not supplied, so such algorithmic conclusions are not assigned to every arbitrary labeled poset. Jon McCammond's survey [35] is recommended.

The aid reports later analogues involving Garside categories, Deligne--Lusztig theory, complex reflection groups and categorical ADE constructions, but says it found no accepted $\operatorname{NC}(W,c)$-indexed dual-$BN$ structure theorem in the literature checked. No full references, date or precise intended axioms accompany that assessment. It is not a nonexistence result, and the three elementary template obstructions do not settle the open-ended source question.

Short English statement

Bessis asks for a dual-braid analogue of group structure theory and $BN$-pairs. The aid excludes three direct templates when the Coxeter system has a noncommuting edge: inherited noncrossing multiplication, a subgroup union of ordinary Bruhat cells, and an ordinary Bruhat interval. The adjacent Garside exposition and broader possible analogues are retained separately.

Sources

• Local UnsolvedMath registry, permanent ID $20002714$, catalogue entry, aim-probability-notes.json, notes[155], workshop Braid groups, clusters and free probability, Problem 3.4, D. Bessis, and appended Section 4: https://aimath.org/WWN/braidgroups/braidgroups.pdf.
• Original references by label: David Bessis [6], dual braid monoids; Tom Brady [12], noncrossing partitions and Artin groups; Jon McCammond [35], Garside survey. Full bibliography entries are not supplied.
• Supplied research summary on the three template obstructions and later Garside-category/Deligne--Lusztig/complex-reflection/categorical ADE analogues. Attempt 1: partial_result, status at run "open", rigor "proved", low novelty confidence; registry status partially_solved. The machine-generated aid is not peer reviewed and requires independent verification before citation. The supplied verification reports page 8 inspection; this draft has not visited the PDF.