Type B web divided-power exchange relation conjecture

Let Xi(k)\mathsf{X}_i^{(k)} denote the nonclassical divided-power web at position ii in Web(so2n+1)\mathbf{Web}(\mathfrak{so}_{2n+1}), with 1≤k≤n1\leq k\leq n. Divided-power exchange relation conjecture. For 1≤a,b,c≤n1\leq a,b,c\leq n, there exist ξ∈C(q)\xi\in\mathbb{C}(q) and integers 1≤a′,b′,c′≤n1\leq a',b',c'\leq n such that

Xi(a)Xi±1(b)Xi(c)=ξ Xi±1(a′)Xi(b′)Xi±1(c′)+LOTa,b,c,\mathsf{X}_i^{(a)}\mathsf{X}_{i\pm1}^{(b)}\mathsf{X}_i^{(c)}=\xi\,\mathsf{X}_{i\pm1}^{(a')}\mathsf{X}_i^{(b')}\mathsf{X}_{i\pm1}^{(c')}+\mathrm{LOT}_{a,b,c},

where LOTa,b,c\mathrm{LOT}_{a,b,c} is a linear combination of terms of the forms Xi(k)Xi±1(ℓ)\mathsf{X}_i^{(k)}\mathsf{X}_{i\pm1}^{(\ell)} and Xi±1(ℓ)Xi(k)\mathsf{X}_{i\pm1}^{(\ell)}\mathsf{X}_i^{(k)}. This conjectures further relations among nonclassical divided powers in type B webs, generalizing the cited iota-Serre relation; the coefficients and indices are not specified more precisely here, so deriving an explicit complete relation remains open.

References

Primary source

Elijah Bodish, Ben Elias and David E. V. Rose, “Type B Webs”, arXiv:2607.13252 (2026).

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