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Artem A. Lopatin

Publications and source records attributed to Artem A. Lopatin.

5 recordsLinked to original sources
arXiv2023

Minimal generating and separating sets for O(3)-invariants of several matrices

Given an algebra $F[H]^G$ of polynomial invariants of an action of the group $G$ over the vector space $H$, a subset $S$ of $F[H]^G$ is called separating if $S$ separates all orbits that can be separated by $F[H]^G$. A minimal separating set is found for some algebras of matrix invariants of several matrices over an infinite field of arbitrary characteristic different from two in case of the orthogonal group. Namely, we consider the following cases: 1) $GL(3)$-invariants of two matrices; 2) $O(3)$-invariants of $d>0$ skew-symmetric matrices; 3) $O(4)$-invariants of two skew-symmetric matrices; 4) $O(3)$-invariants of two symmetric matrices. A minimal generating set is also given for the algebra of orthogonal invariants of three $3\times 3$ symmetric matrices.

math.RA↗
arXiv2014

Identities for matrix invariants of the symplectic group

The general linear group acts on the space of several linear maps on the vector space as the basis change. Similarly, we have the actions of the orthogonal and symplectic groups. Generators and identities for the corresponding polynomial invariants over a characteristic zero field were described by Sibirskii, Procesi and Razmyslov in 1970s. In 1992 Donkin started to transfer these results to the case of infinite fields of arbitrary characteristic. We completed this transference for fields of odd characteristic by establishing identities for the symplectic matrix invariants over infinite fields of odd characteristic.

math.RA↗
arXiv2012

Matrix identities with forms

Consider the algebra M(n,F) of n x n matrices over an infinite field F of arbitrary characteristic. An identity for M(n,F) with forms is such a polynomial in n x n generic matrices and in σ_k(x), 0 n he introduced partial linearizations of σ_t and proved that they together with the well-known free relations and the Cayley--Hamilton polynomial generate T(n) as a T-ideal. We show that it is enough to take partial linearizations of σ_t for n<t\leq 2n. In particular, the T-ideal T(n) is finitely based. Working over a field of characteristic different from two, we obtain a similar result for the ideal of identities with forms for the F-algebra generated by n x n generic and transpose generic matrices. These results imply that ideals of identities for the algebras of matrix GL(n)- and O(n)-invariants are generated by the well-known free relations together with partial linearizations of σ_t for n<t\leq 2n and partial linearizations of σ_{t,r} for n<t+2r\leq 2n, respectively.

math.RA↗
arXiv2012

On the nilpotency degree of the algebra with identity x^n=0

Denote by C_{n,d} the nilpotency degree of a relatively free algebra generated by d elements and satisfying the identity x^n=0. Under assumption that the characteristic p of the base field is greater than n/2, it is shown that C_{n,d} n/2. For p\neq2 the nilpotency degree C_{4,d} is described with deviation 4 for all d. As an application, a finite generating set for the algebra R^{GL(n)} of GL(n)-invariants of d matrices is established in terms of C_{n,d}. Several conjectures are formulated.

math.RA↗