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Tensor product of representations of quivers

Pradeep Das, Umesh V. Dubey, N. Raghavendra

Abstract

In this article, we define the tensor product $V\otimes W$ of a representation $V$ of a quiver $Q$ with a representation $W$ of an another quiver $Q'$, and show that the representation $V\otimes W$ is semistable if $V$ and $W$ are semistable. Over the field of complex numbers, we also describe a relation between the natural line bundles, and between the universal representations on the fine moduli spaces $N_1, N_2$ and $N_3$ of representations of $Q, Q'$ and $Q\otimes Q'$ respectively. We then prove that the internal product $\tilde{Q}\otimes \tilde{Q'}$ of covering quivers is a sub-quiver of the covering quiver $\widetilde{Q\otimes Q'}$. We deduce the relation between stability of the representations $\widetilde{V\otimes W}$ and $\tilde{V} \otimes \tilde{W}$. We also lift the relation between natural line bundles on the product of moduli spaces $\tilde{N_1} \times \tilde{N_2}$.

Tensor product of representations of quivers

Abstract

In this article, we define the tensor product of a representation of a quiver with a representation of an another quiver , and show that the representation is semistable if and are semistable. Over the field of complex numbers, we also describe a relation between the natural line bundles, and between the universal representations on the fine moduli spaces and of representations of and respectively. We then prove that the internal product of covering quivers is a sub-quiver of the covering quiver . We deduce the relation between stability of the representations and . We also lift the relation between natural line bundles on the product of moduli spaces .
Paper Structure (14 sections, 17 theorems, 87 equations)

This paper contains 14 sections, 17 theorems, 87 equations.

Table of Contents

  1. The moduli space of quiver representations
  2. Quivers and their representations
  3. Stability and family of representations
  4. The moduli of $\theta$-semistable representations of type $d$
  5. A description of the moduli space
  6. Moduli space over the field of complex numbers
  7. Tensor products of representations and their semistability
  8. Semistability of tensor product of representations over arbitrary fields
  9. Semistability of tensor product of representations over the field of complex numbers
  10. Relation between the bundles
  11. Relation between the natural line bundles over $\mathbf{C}$
  12. Relation between the universal representations on fine moduli spaces over arbitrary fields
  13. Covering quiver
  14. Relation between natural line bundles on the moduli of covering quivers over $\mathbf{C}$

Key Result

Proposition 1.1

Kin There exists a connected normal quasi-projective $k$-scheme $M(\theta,d)$, which universally corepresents the functor $F(\theta,d)$.

Theorems & Definitions (30)

  • Proposition 1.1
  • Proposition 1.2: Kin
  • Proposition 1.3
  • Remark 1.4
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • ...and 20 more