arXiv subjects

M. Domokos

Publications and source records attributed to M. Domokos.

43 records · Page 3Linked to original sources

arXiv2005

Typical separating invariants

It is shown that a trivial version of polarization is sufficient to produce separating systems of polynomial invariants: if two points in the direct sum of the $G$--modules $W$ and $m$ copies of $V$ can be separated by polynomial invariants, then they can be separated by invariants depending only on at most $2\dim(V)$ variables of type $V$; when $G$ is reductive, invariants depending only on at most $\dim(V)+1$ variables suffice. Similar result is valid for rational invariants. Explicit bounds on the number of type $V$ variables in a typical system of separating invariants are given for the binary polyhedral groups, and this is applied to the invariant theory of binary forms.

math.AG↗

arXiv2004

A quantum homogeneous space of nilpotent matrices

A quantum deformation of the adjoint action of the special linear group on the variety of nilpotent matrices is introduced. New non-embedded quantum homogeneous spaces are obtained related to certain maximal coadjoint orbits, and known quantum homogeneous spaces are revisited.

math.QA↗

arXiv2004

Mod 2 indecomposable orthogonal invariants

Over an algebraically closed base field $k$ of characteristic 2, the ring $R^G$ of invariants is studied, $G$ being the orthogonal group O(n) or the special orthogonal group SO(n) and acting naturally on the coordinate ring $R$ of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(n) $(n\geq 2)$ and for SO(n) $(n\geq 3)$ that there exist $m$--linear invariants with $m$ arbitrarily large that are indecomposable (i. e., not expressible as polynomials in invariants of lower degree). In fact, they are explicitly constructed for all possible values of $m$. Indecomposability of corresponding invariants over $\mathbb Z$ immediately follows. The constructions rely on analysing the Pfaffian of the skew-symmetric matrix whose entries above the diagonal are the scalar products of the vector variables.

math.RA↗

arXiv2004

Quantized trace rings

The general linear group acts on $m$-tuples of $N\times N$ matrices by simultaneous conjugation. Quantum deformations of the corresponding rings of invariants and the so-called trace rings are investigated.

math.QA↗

arXiv2003

On orthogonal invariants in characteristic 2

Working over an algebraically closed base field $k$ of characteristic 2, the ring of invariants $R^G$ is studied, where $G$ is the orthogonal group O(n) or the special orthogonal group SO(n), acting naturally on the coordinate ring $R$ of the $m$-fold direct sum $k^n \oplus...\oplus k^n$ of the standard vector representation. It is proved for O(2), $O(3)=SO(3)$, SO(4), and O(4), that there exists an $m$-linear invariant with $m$ arbitrarily large, which is not expressible as a polynomial of invariants of lower degree. This is in sharp contrast with the uniform description of the ring of invariants valid in all other characteristics, and supports the conjecture that the same phenomena occur for all $n$. For general even $n$, new O(n)-invariants are constructed, which are not expressible as polynomials of the quadratic invariants. In contrast with these results, it is shown that rational invariants have a uniform description valid in all characteristics. Similarly, if $m \leq n$, then $R^O(n)$ is generated by the obvious invariants. For all $n$, the algebra $R^G$ is a finitely generated module over the subalgebra generated by the quadratic invariants, and for odd $n$, the square of any SO(n)-invariant is a polynomial of the quadratic invariants. Finally we mention that for even $n$, an $n$-linear SO(n)-invariant is given, which distinguishes between SO(n) and O(n) (just like the determinant in all characteristics different from 2).

math.RA↗

arXiv2003

Representation rings of quantum groups

Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well known fact that the representation theory of the quantized group is completely analogous to its classical counterpart. The subalgebras of cocommutative elements in the corresponding FRT-bialgebras (defined by Faddeev, Reshetikhin, and Takhtadzhyan) are explicitly determined, using a bialgebra embedding of the FRT-bialgebra into the tensor product of the quantized coordinate ring and the one-variable polynomial ring. A parallel analysis of the subalgebras of adjoint coinvariants is carried out as well, yielding similar results with similar proofs. The basic adjoint coinvariants are interpreted as quantum traces of representations of the corresponding quantized universal enveloping algebra.

math.QA↗

arXiv2003

Orbits for the adjoint coaction on quantum matrices

Conjugation coactions of the quantum general linear group on the algebra of quantum matrices have been introduced in an earlier paper and the coinvariants have been determined. In this paper the notion of orbit is considered via co-orbit maps for these coactions. The co-orbit maps are calculated explicitly for 2 by 2 quantum matrices. In the NxN case we consider the case of a generic diagonal matrix when the deformation parameter is transcendental over the field generated by the diagonal entries.

math.QA↗