[Paper Review] Modulated semi-invariants

Kiyoshi Igusa, Kent E. Orr+2 more|arXiv (Cornell University)|Jul 11, 2015

Algebraic structures and combinatorial models25 references19 citations

TL;DR

This paper establishes foundational properties of determinantal semi-invariants for finite-dimensional hereditary algebras over arbitrary fields, extending quiver representation theory to modulated quivers and virtual representation spaces. The key contribution is the $c$-vector theorem, proving that $c$-vectors of cluster tilting objects coincide with determinantal weights of semi-invariants up to sign, unifying cluster algebra theory with semi-invariant theory in a broader algebraic setting.

ABSTRACT

We prove the basic properties of determinantal semi-invariants for presentation spaces over any finite dimensional hereditary algebra over any field. These include the virtual generic decomposition theorem, stability theorem and the c-vector theorem which says that the c-vectors of a cluster tilting object are, up to sign, the determinantal weights of the determinantal semi-invariants defined on the cluster tilting objects. Applications of these theorems are given in several concurrently written papers.

Motivation & Objective

• To generalize determinantal semi-invariants from quivers to arbitrary finite-dimensional hereditary algebras over any field.
• To define virtual representation spaces via colimits of presentation spaces for integer dimension vectors.
• To establish the $c$-vector theorem, linking $c$-vectors of cluster tilting objects to determinantal weights of semi-invariants.
• To prove the virtual generic decomposition theorem for generic representations in virtual representation spaces.
• To extend stability and generic decomposition theorems to non-positive dimension vectors using presentation spaces.

Proposed method

• Define the representation space $Rep(\Lambda,\alpha)$ as a subspace of $\operatorname{Hom}_\Lambda(\operatorname{rad}P(\alpha), P(\alpha))$ for $\alpha \in \mathbb{N}^n$, generalizing quiver representations.
• Construct presentation spaces $Pres_\Lambda(\gamma_1, \gamma_0)$ for $\gamma_0, \gamma_1 \in \mathbb{N}^n$ with $\underline{\dim}P(\gamma_0) - \underline{\dim}P(\gamma_1) = \alpha$.
• Define the virtual representation space $V\!rep(\Lambda,\alpha)$ as the colimit of all such presentation spaces over all valid $\gamma_0, \gamma_1$ pairs.
• Define determinantal semi-invariants $\sigma_\beta$ on $V\!rep(\Lambda,\alpha)$ via determinants of induced maps between Hom-spaces of projective modules.
• Use the reduced norm and determinant to define reduced and full semi-invariants, respectively, with associated weights in $\mathbb{Z}^n$.
• Prove that the $c$-vectors of cluster tilting objects are, up to sign, the determinantal weights of these semi-invariants, establishing the $c$-vector theorem.

Experimental results

Research questions

• RQ1 How can the theory of semi-invariants be extended from quivers to arbitrary finite-dimensional hereditary algebras over any field, including non-algebraically closed fields?
• RQ2 What is the correct generalization of the representation space to integer dimension vectors, and how can a consistent virtual representation space be constructed?
• RQ3 How do $c$-vectors in cluster algebra theory relate to the weights of determinantal semi-invariants in the generalized setting?
• RQ4 Can the generic decomposition theorem be extended to virtual representation spaces for integer dimension vectors?
• RQ5 What is the role of the reduced norm in defining semi-invariants over division algebras, and how does it relate to the full determinant?

Key findings

• The virtual generic decomposition theorem holds: for a nonnegative integer linear combination $\alpha = \sum n_i \beta_i$ of real Schur roots $\beta_i$ with non-extending generic representations, the generic representation in $V\!rep(\Lambda,\alpha)$ is isomorphic to $\bigoplus M_{\beta_i}^{n_i}$.
• The $c$-vector theorem is proven: the $c$-vectors of a cluster tilting object are, up to sign, the determinantal weights of the semi-invariants defined on the cluster tilting object.
• For the ${\mathbb{R}}$-modulated quiver ${\mathbb{H}} \leftarrow {\mathbb{C}} \leftarrow {\mathbb{C}}$, the determinantal semi-invariant $\sigma_\beta$ for $\beta = (1,2,2)$ has full weight $(1,2,2)$, while the reduced norm semi-invariant $\overline{\sigma}_\beta$ has reduced weight $(1,1,1)$.
• The stability theorem is established: if a rigid $\Lambda$-module $M$ of dimension $\alpha$ exists, then the set of isomorphic representations in $Rep(\Lambda,\alpha)$ forms an open dense subset.
• The determinantal weight of a semi-invariant is computed via the action of automorphisms on Hom-spaces: for $z \in \operatorname{Aut}(P_3)$, the determinant scales by $|z|^4$, yielding weight $2$ for $z$'s norm, and similarly for other components.
• The reduced norm semi-invariant $\overline{\sigma}_\beta$ is defined by taking the reduced norm over $\mathbb{H}$, which gives a weight of $1$ for $h \in \mathbb{H}^*$ and $2$ for $z \in \mathbb{C}^*$, resulting in a reduced weight of $(1,1,1)$.