Triangle presentations and tilting modules for \(\operatorname{SL}_{2k+1}\) (Q2233683)

Let \(\Bbbk\) be an algebraically closed field of positive characteristic \(p\). The paper under review constructs fiber functors, i.e. monoidal \(\Bbbk\)-linear functors to the category \(\mathbf{Vec}\) of finite-dimensional vector spaces, from the category \(\operatorname{Tilt}(\operatorname{SL}_{n})\) of tilting modules for \(\operatorname{SL}_{n}(\Bbbk)\), for odd \(n\). Two diagrammatic categories, \(\operatorname{Web}(\operatorname{SL}_{n}^{+})\) and \(\operatorname{Web}(\operatorname{SL}_{n}^{-})\) are defined, using the diagrammatic presentation of the category of \(\operatorname{GL}_{n}(\Bbbk)\)-modules generated by exterior powers of the standard \(n\)-dimensional representation, constructed in [\textit{J. Brundan} et al., Adv. Math. 375, 37 p. (2020; Zbl 1494.18011)] . For all \(n\), the category \(\operatorname{Web}(\operatorname{SL}_{n}^{+})\) is monoidally equivalent to \(\operatorname{Tilt}(\operatorname{SL}_{n})\). If \(n\) is odd, we have \(\operatorname{Web}(\operatorname{SL}_{n}^{+}) = \operatorname{Web}(\operatorname{SL}_{n}^{-})\). For even \(n\), the two constructions are different. Given a triangle presentation \(\mathfrak{T}\) of type \(\widetilde{A}_{n-1}\), a fiber functor \(F_{\mathfrak{T}}\) on \(\operatorname{Web}(\operatorname{SL}_{n}^{-})\) is constructed. More precisely, the construction requires \(p \geq n-1\) and a finite projective geometry \(\Pi\) with involution, of algebraic dimension \(n\) and order \(q\), with \(q \equiv 1 \text{ mod } p\). The triangle presentation \(\mathfrak{T}\) is a combinatorial structure on \(\Pi\), and the construction of the fiber functor only uses the combinatorial properties of \(\mathfrak{T}\). For completeness, the article contains a short account of the setting of finite projective geometries and affine buildings, where the triangle presentations naturally arise. In particular, it contains the description of a group \(\Gamma\) associated to \(\mathfrak{T}\). By definition, the fiber functor \(F_\mathfrak{T}\) factors through the forgetful functor from the category \(\mathbf{Vec}(\Gamma)\) of \(\Gamma\)-graded vector spaces to \(\mathbf{Vec}\). For \(n\) odd, a fiber functor on \(\operatorname{Tilt}(\operatorname{SL}_{n})\) is obtained by combining the equivalence of \(\operatorname{Tilt}(\operatorname{SL}_{n})\) with \(\operatorname{Web}(\operatorname{SL}_{n}^{+})\), equality of \(\operatorname{Web}(\operatorname{SL}_{n}^{+})\) with \(\operatorname{Web}(\operatorname{SL}_{n}^{-})\), and the fiber functor \(F_{\mathfrak{T}}\) on \(\operatorname{Web}(\operatorname{SL}_{n}^{-})\). The category \(\operatorname{Web}(\operatorname{SL}_{n}^{-})\) is not braided, unless \(n\) is even. Despite that, it is shown that the obtained fiber functors yield involutive solutions to the Yang-Baxter equation, which are given explicitly as linear maps and further also described as ``positive characteristic deformations'' of familiar solutions in characteristic zero.