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From brick manifolds to Grassmannians of bimodules

Evgeny Feigin, Markus Reineke

TL;DR

The paper generalizes brick manifolds to Grassmannians of sub-bimodules over path algebras of acyclic quivers without parallel paths. It constructs $M(V_*)$ and the Grassmannian $X(V_*)={\rm Gr}^{\mathbf f(V_*)}_{A\otimes A^{\rm op}}(M(V_*))$, proves smoothness and a cellular decomposition, and provides a framed moduli interpretation that yields a recursive formula for motives. It develops explicit examples including linearly oriented type $A$, alternating orientation, and type $D$ quivers, showing how the construction recovers known brick manifolds and related quiver Grassmannians, and it extends the framework to quivers with parallel paths, where irreducibility can fail and singular examples arise. The work connects representations of quivers and bimodules with compactifications of representation spaces and motivic invariants, and it furnishes a concrete toolkit for computing motives via framed moduli spaces and fixed-point combinatorics.

Abstract

We study a class of Grassmannians of sub-bimodules over the path algebras of quivers. Our quiver Grassmannians include Escobar's brick manifolds as well as Labelle's generalizations. We give an explicit construction of the varieties in question, provide examples and clarify connection with the quiver representation spaces. We also prove smoothness of our Grassmannians, construct cellular decompositions and derive a realization as framed moduli spaces. The framed moduli realization leads to a recursive formula for the motives.

From brick manifolds to Grassmannians of bimodules

TL;DR

The paper generalizes brick manifolds to Grassmannians of sub-bimodules over path algebras of acyclic quivers without parallel paths. It constructs and the Grassmannian , proves smoothness and a cellular decomposition, and provides a framed moduli interpretation that yields a recursive formula for motives. It develops explicit examples including linearly oriented type , alternating orientation, and type quivers, showing how the construction recovers known brick manifolds and related quiver Grassmannians, and it extends the framework to quivers with parallel paths, where irreducibility can fail and singular examples arise. The work connects representations of quivers and bimodules with compactifications of representation spaces and motivic invariants, and it furnishes a concrete toolkit for computing motives via framed moduli spaces and fixed-point combinatorics.

Abstract

We study a class of Grassmannians of sub-bimodules over the path algebras of quivers. Our quiver Grassmannians include Escobar's brick manifolds as well as Labelle's generalizations. We give an explicit construction of the varieties in question, provide examples and clarify connection with the quiver representation spaces. We also prove smoothness of our Grassmannians, construct cellular decompositions and derive a realization as framed moduli spaces. The framed moduli realization leads to a recursive formula for the motives.
Paper Structure (13 sections, 11 theorems, 63 equations)

This paper contains 13 sections, 11 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Quiver representations and bimodules
  3. A class of Grassmannians of sub-bimodules
  4. Examples
  5. Linearly oriented type A
  6. Alternating orientation quivers
  7. Type D quiver
  8. Compactification of representation spaces
  9. Smoothness
  10. Framed moduli spaces and motivic generating series
  11. Torus fixed points
  12. Framed moduli spaces
  13. Quivers with parallel paths

Key Result

Theorem A

Assume that $Q$ is acyclic and has no parallel paths. Then $X(V_*)$ is a smooth $G_{\bf d}$-equivariant compactification of the representation space $R_{\bf d}(Q)$.

Theorems & Definitions (21)

  • Theorem A
  • Theorem B
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 5.1
  • proof
  • Theorem 6.1
  • proof
  • ...and 11 more