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Fundam. Prikl. Mat., 2001, Volume 7, Issue 2, Pages 387–421 (Mi fpm580)  

This article is cited in 7 scientific papers (total in 7 papers)

The Procesi–Razmyslov theorem for quiver representations

A. N. Zubkov

Omsk State Pedagogical University

Abstract: We find the generators and the defining relations of any quiver representation invariant algebra. To be precise, let $R(Q,\bar k)$ be a quiver representation space with respect to the natural action of the group consisting of all isomorphisms of the quiver representations. Denote this group by $\operatorname{GL}(\bar k)$, where $\bar k$ is a dimensional vector of the quiver representation space $R(Q,\bar k)$. For example, when our quiver $Q$ has only one vertex and several loops are incidental to this vertex we have the well-known case of the adjoint action of the general linear group on the space of several $n\times n$-matrices. In the characteristic zero case Artin and Procesi described the quotient of the last variety under this action in their classic works. In the case of arbitrary infinite ground field this result can be deduced from some results by Procesi and Donkin. In a similar manner we can define the quotient of the quiver representation space $R(Q,\bar k)$ by the action of the group $\operatorname{GL}(\bar k)$. By the definition we have that $K[R(Q,\bar k)/\operatorname{GL}(\bar k)]\cong K[R(Q,\bar k)]^{\operatorname{GL}(\bar k)}$. Donkin has recently found the generators of that algebra. In this article we define a free quiver representation invariant algebra. Then we prove that the kernel of its canonical epimorphism onto $K[R(Q,\bar k)]^{\operatorname{GL}(\bar k)}$ is generated as a T-ideal by the values of the coefficients of the characteristic polynomial with sufficiently large number. This result generalizes the well-known Procesi–Razmyslov theorem about trace matrix identities. Besides, by an alternative way we can deduce Donkin's result about the generators of $K[R(Q,\bar k)]^{\operatorname{GL}(\bar k)}$.

Full text: PDF file (1739 kB)

Bibliographic databases:
UDC: 512.64
Received: 01.01.1998

Citation: A. N. Zubkov, “The Procesi–Razmyslov theorem for quiver representations”, Fundam. Prikl. Mat., 7:2 (2001), 387–421

Citation in format AMSBIB
\Bibitem{Zub01}
\by A.~N.~Zubkov
\paper The Procesi--Razmyslov theorem for quiver representations
\jour Fundam. Prikl. Mat.
\yr 2001
\vol 7
\issue 2
\pages 387--421
\mathnet{http://mi.mathnet.ru/fpm580}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1866464}
\zmath{https://zbmath.org/?q=an:1014.16016}


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    This publication is cited in the following articles:
    1. Lopatin A.A., “The algebra of invariants of 3 x 3 matrices over a field of arbitrary characteristic”, Communications in Algebra, 32:7 (2004), 2863–2883  crossref  mathscinet  zmath  isi
    2. Lopatin A.A., Zubkov A.N., “Semi-invariants of mixed representations of quivers”, Transformation Groups, 12:2 (2007), 341–369  crossref  mathscinet  zmath  isi  elib
    3. Lopatin A.A., “Invariants of quivers under the action of classical groups”, Journal of Algebra, 321:4 (2009), 1079–1106  crossref  mathscinet  zmath  isi
    4. A. A. Lopatin, “Indecomposable invariants of quivers for dimension $(2,…,2)$ and maximal paths, II”, Sib. elektron. matem. izv., 7 (2010), 350–371  mathnet  elib
    5. Lopatin A.A., “Indecomposable Invariants of Quivers for Dimension (2,...,2) and Maximal Paths”, Comm Algebra, 38:10 (2010), 3539–3555  crossref  mathscinet  zmath  isi  elib
    6. Lopatin A.A., “Minimal generating set for semi-invariants of quivers of dimension two”, Linear Algebra Appl, 434:8 (2011), 1920–1944  crossref  mathscinet  zmath  isi  elib
    7. Lopatin A.A., “Relations Between O(N)-Invariants of Several Matrices”, Algebr. Represent. Theory, 15:5 (2012), 855–882  crossref  mathscinet  zmath  isi  elib
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