Dyck Paths, Configuration Spaces and Polytopes For Linear Nakayama algebras

We present a combinatorial model of configuration spaces and polytopes associated to the quotients of $\mathbb{C} A_n$, the path algebra of the linearly oriented $A_n$ quiver, i.e. the algebra of upper triangular matrices. These quotient algebras are known as linear Nakayama algebras. Such configuration spaces were recently introduced for more general algebras by the second author and collaborators. In this special setting, we provide elementary proofs and explicit combinatorial constructions. From a Dyck path we define three related objects: a finite-dimensional algebra, an affine algebraic variety, and a polytope. Moreover, our constructions are natural: each relation in the poset of Dyck paths gives a morphism between the corresponding objects.

💡 Research Summary

The paper develops a completely elementary and combinatorial description of three interrelated objects—finite‑dimensional algebras, affine configuration spaces, and convex polytopes—associated to the quotients of the path algebra C Aₙ of the linearly oriented Aₙ quiver (equivalently the algebra of upper‑triangular n × n matrices). These quotients are precisely the linear Nakayama algebras. The authors show that every two‑sided ideal I⊂C Aₙ without idempotents, and therefore every linear Nakayama algebra A = C Aₙ/I, is uniquely encoded by a Dyck path D of length 2n. The correspondence D↦I_D is a poset isomorphism: D≤D′ (i.e. D lies on or below D′) iff I_D⊇I_{D′}.

From a Dyck path D the authors construct:

1. Index set I_A – the set of steps of D together with the unit squares (“diamonds”) lying strictly below D in the triangular grid of indecomposable modules M_{i,j}. Each element X∈I_A is assigned a coordinate u_X.

2. Compatibility rule – for any pair X,Y∈I_A a non‑negative integer c(X,Y) is defined. The configuration space U_A⊂ℂ^{I_A} is cut out by the family of equations
   u_X + u_Y = 1 if c(X,Y)=0,
   u_X + u_Y·u_Z = 1 in the general case (the precise form is given in Definition 3.3). These are exactly the “u‑equations” that appear in the physics literature on stringy integrals.

3. Stratification of the non‑negative part – The set (U_A){\ge0} is stratified by compatible subsets S⊆I_A; each stratum is defined by setting u_X=0 for X∈S. Inclusion of strata corresponds to inclusion of compatible subsets, so the face lattice of (U_A){\ge0} is isomorphic to the poset of compatible subsets of I_A.

4. Polytope P_A – For each grid point (i,j) lying on or below D the authors attach the F‑polynomial F_{i,j}=1+y_i+ y_i y_{i+1}+…+y_i…y_j. The Newton polytope Newt(F_{i,j}) is a simplex, and P_A is defined as the Minkowski sum of all these simplices (equation (8)). Consequently P_A is the moment polytope of a projective toric variety X_{P_A}.

The main geometric result (Theorem 4.6) states that U_A is an affine open subset of the toric variety X_{P_A}. Moreover, the positive part (U_A){\ge0} coincides with the positive part of X{P_A}, and thus the face lattice of P_A is exactly the poset of strata of (U_A)_{\ge0} (Corollary 4.7).

When two Dyck paths satisfy e D ≤ D, the corresponding ideals obey I_{e D}⊇I_D, and the authors define a monomial map ϕ: ℂ