FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2001, VOLUME 7, NUMBER 2, PAGES 387-421
A. N. Zubkov
Abstract
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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 $\mathrm{GL}(\bar k)$ , 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 $\mathrm{GL}(\bar k)$ .
By the definition we have that
$K[R(Q,\bar k)/\mathrm{GL}(\bar k)]\cong K[R(Q,\bar k)]^{\mathrm{GL}(\bar k)}$ .
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
$K[R(Q,\bar k)]^{\mathrm{GL}(\bar k)}$ 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 $K[R(Q,\bar k)]^{\mathrm{GL}(\bar k)}$ .
All articles are published in Russian.
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Last modified: October 31, 2001.