Orthogonal webs and semisimplification

We define a diagrammatic category that is equivalent to tilting representations for the orthogonal group. Our construction works in characteristic not equal to two. We also describe the semisimplification of this category.

💡 Research Summary

The paper “Orthogonal webs and semisimplification” develops a diagrammatic framework for the tilting representation theory of the orthogonal group O(N) over an infinite field F of characteristic p≠2 (including characteristic 0). The authors introduce a new monoidal category of “orthogonal webs”, defined as planar graphs whose edges are labelled by integers 1,…,N and whose vertices satisfy the condition that the three incident labels are k, l and k + l. Crossings are allowed in the usual sense, and cutting a closed web yields morphisms between tensor products of exterior powers Λ^k(V) of the standard O(N)-module V. By imposing a collection of local relations (including >N=0, kℓ=k+ℓ, and the usual associativity/coassociativity relations) they obtain a symmetric ribbon category 𝑊𝑒𝑏_F(O(N)).

The first main result (Theorem 1.3) is that there is a fully faithful symmetric ribbon functor  𝑊𝑒𝑏_F(O(N)) → Rep_F(O(N)), sending the object labelled k to the k‑th exterior power Λ^k(V). After taking the additive idempotent completion of the web category, this functor becomes an equivalence with the category Tilt_F(O(N)) of all tilting O(N)-modules. In other words, orthogonal webs provide a complete diagrammatic presentation of the tilting category. The proof adapts the strategy of Cautis‑Kamnitzer‑Morrison for type‑A webs: one constructs a homomorphism Ψ_m from the integral form ˙U_N^ℤ(so_{2m}) of the quantum group to the endomorphism algebra of the web category, shows that Chevalley generators and their divided powers are represented by ladder‑shaped webs, and uses a ladder‑ization argument to prove surjectivity. The authors rely on Elias’s integral version of the Cautis‑Kamnitzer‑Morrison theorem to guarantee that Ψ_m respects the higher Serre relations, thereby establishing full faithfulness of the functor.

The second main result (Theorem 1.4) describes the semisimplification of the tilting category. Writing the integer N in its p‑adic expansion N = Σ_i N_i p^i, the authors prove an equivalence of symmetric ribbon categories  Tilt_F(O(N)) ≅ ⊠_{i≥0} Tilt_F(O(N_i)), where ⊠ denotes Deligne’s tensor product. Each factor Tilt_F(O(N_i)) is a Verlinde (or “Verschiebung”) category, which is already well understood. Consequently, the semisimplified orthogonal tilting category decomposes as a Deligne tensor product of these familiar categories, and this decomposition holds for any characteristic p≠2, without any restriction that p be larger than N.

The paper also contains a detailed discussion of the background material: Schur–Weyl duality, Brauer algebras, Howe duality for GL(N)–GL(m), and their integral and modular versions; the construction of the integral forms ˙U_N^ℤ(gl_m) and ˙U_N^ℤ(so_{2m}); and the relationship between type‑A webs and orthogonal webs via the inclusion GL(N)⊂O(N). The authors explain how the orthogonal web category over ℤ reduces to the known characteristic‑zero construction after base change to ℚ, and how the integral version allows them to work uniformly in positive characteristic.

Technical novelties include:

1. An explicit integral version of Howe’s orthogonal duality (Andersen–Riche) suitable for diagrammatic use.
2. A verification that ladder‑shaped webs satisfy the higher Serre relations in the integral quantum group, either directly or by invoking Elias’s result.
3. The construction of the idempotent completion of the web category and the identification of its objects with all tilting modules.
4. The application of Deligne’s tensor product to express the semisimplification in terms of p‑adic digits of N, extending previous results for GL(N) to the orthogonal case.

The authors conclude by suggesting future directions: extending the framework to symplectic groups (where exterior powers are not simple even in characteristic 0), exploring 2‑categorical enhancements of orthogonal webs, and investigating connections with categorified quantum invariants. Overall, the paper provides a comprehensive diagrammatic description of orthogonal tilting representations and a clear structural picture of their semisimplification, bridging characteristic‑zero and modular representation theory through the language of webs.