Rotation-Invariant Web Basis for SLₙ

• The paper introduces a combinatorial construction for SLₙ using triangular diagrams and minuscule Littelmann paths to form a rotation-invariant web basis.
• It employs the geometric Satake correspondence to relate perverse sheaves on the affine Grassmannian to invariant space bases, achieving an upper unitriangular expansion.
• Rotation invariance under cyclic permutation of boundary data is key, linking diagrammatics with cyclic sieving and applications in quantum topology.

A rotation-invariant web basis is a diagrammatically defined, combinatorial basis for spaces of invariants in tensor products of representations (notably of $\SL_n$), with the essential property that the basis is compatible with cyclic (rotational) symmetry on the boundary data. The formalism draws on the interplay between combinatorics of special graphs termed "webs," geometric representation theory (especially the geometric Satake correspondence), and crystal- and tableau-theoretic techniques. This article focuses on the explicit geometric and combinatorial realization for $\SL_n$, as constructed via triangular diagrams and their relationship to the Satake basis through the geometric Satake equivalence (Fontaine, 2011).

1. Webs and the Invariant Space for $\SL_n$

For a simple algebraic group $G$, a "web" is a planar, directed, trivalent graph with edges labeled by dominant minuscule weights, and whose boundary data matches a fixed sequence $(\lambda_1, \ldots, \lambda_k)$. Webs provide combinatorial diagrams representing specific contraction patterns for tensor invariants in the representation space $$\mathrm{Hom}_G(\mathbb{C}, V_{\lambda_1} \otimes \cdots \otimes V_{\lambda_k})$$.

One associates to this invariant space the set of minuscule Littelmann paths of given type (or alternatively, certain lattice paths determined by the sequence of weights). The main task is to construct a basis for the invariant space indexed canonically by such combinatorial data, with the added property of rotation invariance—i.e., the basis elements transform into each other under a natural cyclic permutation of the boundary.

2. Geometric Satake Correspondence and Homological Realization

The foundation for the construction is the Mirković–Vilonen geometric Satake theorem, which asserts an equivalence of tensor categories

$\mathrm{Rep}(G) \simeq \mathcal{P}(\mathrm{Gr}),$

with $\mathcal{P}(\mathrm{Gr})$ the category of perverse sheaves on the affine Grassmannian of the Langlands dual $G^\vee$, stratified by orbits $\mathrm{Gr}_\lambda$. Under this correspondence, irreducible $V_\lambda$ goes to the intersection cohomology complex $\mathrm{IC}(\overline{\mathrm{Gr}_\lambda})$.

A tensor product $V_{\lambda_1}\otimes\cdots\otimes V_{\lambda_k}$ corresponds to convolution of perverse sheaves, and the invariant space is realized as the top homology of the Satake fiber:

$$\mathrm{Hom}_G(\mathbb{C}, V_{\lambda_1}\otimes\cdots\otimes V_{\lambda_k}) \simeq H_{\mathrm{top}}(F())$$

where $F()$ is the fiber above the base "trivial lattice" in the convolution morphism of affine Grassmannians. Web combinatorics is thus directly related to the homology classes of certain Lagrangian subvarieties in $\mathrm{Gr}$.

3. Triangular Diagrams, Web Vectors, and Their Expansion

The key construction in (Fontaine, 2011) produces webs from triangular diagrams parametrized by minuscule Littelmann paths $\gamma$, with each path corresponding to a coherent web $T_\gamma$. Through a mapping $\Psi$ (from combinations of webs to invariants), each $T_\gamma$ is mapped (via the geometric Satake correspondence and pushforward of fundamental classes) to a specific invariant vector:

$$\Psi_\nu(T_\gamma) = [\overline{Q(A(\cdots))}] + \sum_{\gamma' < \gamma} c(\gamma')[\overline{Q(A(\cdots))}]$$

with respect to a natural partial order on paths. This expansion is upper unitriangular—the leading term corresponds precisely to the Satake basis vector, and all lower terms correspond to "smaller" paths. This unitriangular relationship ensures the set of web vectors from triangular diagrams forms a basis.

Notably, for $\SL_2$ and $\SL_3$ (or for weights $\omega_1, \omega_{n-1}$ in $\SL_n$), this web basis recovers earlier models—Kuperberg’s non-elliptic webs and cup diagrams—thus generalizing classical rotation-invariant constructions.

4. Diagram Chasing and the Satake Change of Basis

A central technical ingredient is a lemma stating that, after mapping through the geometric Satake correspondence and corresponding adjunctions and pushforward operations, the web vectors expand via a unitriangular transition matrix with respect to the Satake basis. The proof leverages:

• Canonical inclusions in the category of perverse sheaves yielding maps on homology.
• Projection maps corresponding to forgetting steps in a path, inducing pushforwards in homology.
• Commutativity of diagrams involving adjoint functors, guaranteeing that top homology classes of configuration spaces map in a manner coherent with the Satake identification.

Thus, the combinatorial construction is justified and precisely tracked through the perverse sheaf framework.

5. Rotation Invariance and Combinatorial Symmetry

The web basis is constructed so as to respect cyclic (rotational) symmetry; rotation of the arrangement of the boundary vertices yields, via the correspondence of webs to minuscule paths and tableaux, another basis web of the same type. The underlying symmetry manifests as rotation invariance of the web basis: webs transform amongst themselves under boundary rotation in accordance with cyclic sieving phenomena, in complete compatibility with promotion on tableaux and geometric symmetries in the affine Grassmannian.

This rotation invariance is crucial for applications to cyclic actions in combinatorics (such as cyclic sieving), categorical representation theory (modular tensor categories), and in explicit calculations of link invariants.

6. Algorithmic and Structural Properties

The paper provides effective rules for generating the web basis:

• Enumeration of minuscule Littelmann paths for given boundary data,
• Construction of triangular diagram webs for each path,
• Application of diagrammatic relations (skein-theoretic reductions and moves) to ensure coherence.

The transition matrix to the Satake basis is upper unitriangular, and the basis is compatible with all known rotation-invariant constructions in low rank and their generalizations. This aligns with the geometric realization: every web corresponds to a unique, up-to-scale, Lagrangian cycle in the affine Grassmannian.

7. Summary and Significance

The construction synthesizes combinatorial, geometric, and representation-theoretic principles to produce, for the first time in general rank, a basis of webs for the $\SL_n$ invariant space with explicit rotation invariance and geometric meaning. The interplay between diagrammatics, homology of Satake fibers, and the geometric Satake correspondence is central. The results unify prior models, generalize to higher complexity, and lay the foundation for further development in categorical representation theory and its applications in quantum topology and algebraic combinatorics.