FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 2, PAGES 387-421

**The Procesi--Razmyslov theorem for quiver representations**

A. N. Zubkov

Abstract

View as HTML
View as gif image
View as LaTeX source

We find the generators and the defining relations of any
quiver representation invariant algebra.
To be precise, let $R(Q,$**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 $GL($**k**),
where $$**k**
is a dimensional vector of
the quiver representation space $R(Q,$**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 $n$´ 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 $R(Q,$**k**)
by the action of the group
$GL($**k**).
By the definition we have that $K[R(Q,$**k**) / GL(**k**)]
´
K[R(Q,**k**)]^{GL(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
$K[R(Q,$**k**)]^{GL(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 $K[R(Q,$**k**)]^{GL(k)}.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k01/k012/k01206h.htm.

Last modified: October 31, 2001.