Andersen's conjecture on modular and quantum tilting characters

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Let GG be a semisimple, simply connected algebraic group with root system Φf\Phi_{\rm f} over an algebraically closed field of characteristic pp, let A0A_0 be the dominant weights in the interior of the fundamental pp-alcove, and let \prescriptpW\prescript{p}{}{W} denote the relevant set of dominant elements for the pp-dilated dot action. For a dominant weight λ\lambda, write T(λ)T(\lambda) for the indecomposable modular tilting module and Tp(λ)T_p(\lambda) for the corresponding quantum tilting module.

Andersen's conjecture. Suppose Lusztig's conjecture holds for GG in characteristic pp. For λ∈A0\lambda \in A_0 and dominant x∈\prescriptpWx \in \prescript{p}{}{W}, one has

ch⁡T(x⋅pλ)=ch⁡Tp(x⋅pλ).\operatorname{ch} T(x \cdot_p \lambda)=\operatorname{ch} T_p(x \cdot_p \lambda).

This conjecture predicts equality between modular and quantum tilting characters in the region governed by Lusztig's conjecture, strengthening Andersen's general lower-bound observation. The supplied text does not state whether it has been resolved.

References

Primary source

Amit Hazi, “Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups”, arXiv:1708.07072 (2025).

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