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Publications and source records attributed to A. A. Lopatin.
The algebra of invariants of several 3 x 3 matrices under the action of the orthogonal group by simultaneous conjugation is considered over a field of characteristic different from two. The maximal degree of elements of minimal system of generators is described with deviation 3.
A linear group G<GL(n) acts on d-tuples of n x n matrices by simultaneous conjugation. In [Adv. Math. 19 (1976), 306-381] Procesi established generators and relations between them for G-invariants, where G is GL(n), O(n), and Sp(n) and the characteristic of base field is zero. We continue generalization of the mentioned results to the case of positive characteristic originated by Donkin in [Invent. Math. 110 (1992), 389-401]. We investigate relations between generators for O(n)-invariants.
The notion of mixed representations of quivers can be derived from ordinary quiver representations by considering the dual action of groups on "vertex" vector spaces together with the usual action. A generating system for the algebra of semi-invariants of mixed representations of a quiver is determined. This is done by reducing the problem to the case of bipartite quivers of the special form and by introducing a function DP on three matrices, which is a mixture of the determinant and two pfaffians.
A classical linear group $G<GL(n)$ acts on $d$-tuples of $n\times n$ matrices by simultaneous conjugation. Working over an infinite field of characteristic different from two we establish that the ideal of free relations, i.e. relations valid for matrices of any order, between generators for matrix O(n)- and $\Sp(n)$-invariants is zero. We also prove similar result for invariants of mixed representations of quivers. These results can be considered as a generalization of the characteristic isomorphism ${\rm ch}:\Sym\to J$ between the graded ring $\Sym=\otimes_{d=0}^{\infty} \Sym_d$, where $\Sym_d$ is the character group of the symmetric group $S_d$, and the inverse limit $J$ with respect to $n$ of rings of symmetric polynomials in $n$ variables. As a consequence, we complete the description of relations between generators for O(n)-invariants as well as the description of relations for invariants of mixed representations of quivers. We also obtain an independent proof of the result that the ideal of free relations for $GL(n)$-invariants is zero, which was proved by Donkin in [Math. Proc. Cambridge Philos. Soc. 113 (1993), 23--43].
The orthogonal group acts on the space of several $n\times n$ matrices by simultaneous conjugation. For an infinite field of characteristic different from two, relations between generators for the algebra of invariants are described. As an application, the maximal degree of elements of a minimal system of generators is described with deviation $3$. This note contains concise but precise description of the results. All proofs can be found in arXiv: 0902.4266 and arXiv: 1011.5201.
An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the reduction to the problem of description of maximal paths satisfying certain condition.
A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,...,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees, where in the case of characteristic different from two we take only admissible paths. As a consequence, we describe relations modulo decomposable semi-invariants.
An upper bound on degrees of elements of a minimal generating system for invariants of quivers of dimension (2,...,2) is established over a field of arbitrary characteristic and its precision is estimated. The proof is based on the reduction to the problem of description of maximal paths satisfying certain condition.
The algebra of invariants of d-tuples of n x n skew-symmetric matrices under the action of the orthogonal group by simultaneous conjugation is considered over an infinite field of characteristic different from two. For n=3 and d>0 a minimal set of generators is established. A homogeneous system of parameters (i.e., an algebraically independent set such that the algebra of invariants is a finitely generated free module over subalgebra generated by this set) is described for n=3 and d>0, for n=4 and d=2,3, for n=5 and d=2.
This paper is a survey on invariants of representations of quivers and their generalizations. We present the description of generating systems for invariants and relations between generators.
Amitsur's formula, which expresses det(A+B) as a polynomial in coefficients of the characteristic polynomial of a matrix, is generalized for partial linearizations of the pfaffian of block matrices. As applications, in upcoming papers we determine generators for the SO(n)-invariants of several matrices and relations for the O(n)-invariants of several matrices over a field of arbitrary characteristic.
Let R_{n,d} be the ring of invariants of d-tuples of n x n matrices under the simultaneous conjugation action of the general linear group. A minimal generating system and a homogeneous system of parameters for R_{3,3} are determined. Homogeneous systems of parameters for R_{3,2}, R_{4,2} are also pointed out.
We consider a generalization of representations of quivers that can be derived from the ordinary representations of quivers by considering a product of arbitrary classical groups instead of a product of the general linear groups and by considering the dual action of groups on "vertex" vector spaces together with the usual action. A generating system for the corresponding algebra of invariants is found. In particular, a generating system for the algebra of SO(n)-invariants of several matrices is constructed over a field of characteristic different from 2. The proof uses the reduction to semi-invariants of mixed representations of a quiver and the decomposition formula that generalizes Amitsur's formula for the determinant.
A base of a relatively free associative algebra with the identity x^3=0 over a field of arbitrary characteristic is found. As an application a minimal generating system of the 3x3 matrix invariant algebra is determined.
The least upper bound on degrees of elements of a minimal system of generators of the algebra of invariants of 3x3 matrices is found, and the nilpotency degree of a relatively free finitely generated algebra with the identity x^3=0 is established.