arXiv subjects
Publications and source records attributed to M. Domokos.
The present paper completes the computation of the separating Noether numbers for the groups with order strictly less than $32$. Most of the results are proved for the case of a general (possibly finite) base field containing an element whose multiplicative order equals the size of the group.
It is shown that certain transformations on quiver-dimension vector pairs induce isomorphisms on the corresponding moduli spaces of quiver representations and map a stable dimension vector to a stable dimension vector. This result combined with a combinatorial analysis of dimension vectors in the fundamental set of a wild quiver is applied to prove that in each dimension there are only finitely many projective algebraic varieties occurring as a moduli space of representations of a quiver with a dimension vector that satisfies some simple constraints.
The problem of finding generators of the $GL$-ideal of the relations between the generators of the algebra of invariants of the dihedral group acting on $m$-tuples of vectors from its defining $2$-dimensional representation is studied. It is shown that this $GL$-ideal is generated by relations depending on no more than $3$ vector variables. A minimal $GL$-ideal generating system is found for the case when $m=2$, and for the case of the dihedral group of order $8$ and arbitrary $m$.
Sets of monomials separating Zariski closed orbits under diagonalizable group actions are characterized in terms of the monoid of zero-sum sequences over the character group. This is applied to compare the degree bounds for separating invariants and generating invariants of diagonalizable group actions.
For each irreducible finite dimensional representation of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$ of $2\times 2$ traceless matrices, an explicit uniform upper bound is given for the multiplicities in the cocharacter sequence of the polynomial identities satisfied by the given representation.
To a finite dimensional representation of a complex Lie group $G$, an associative algebra of adjoint covariant polynomial maps from the direct sum of $m$ copies of the Lie algebra $\mathfrak{g}$ of $G$ into an algebra of complex matrices is associated. When the tangent representation of the given representation is irreducible, the center of this algebra of concomitants can be identified with the algebra of adjoint invariant polynomial functions on $m$-tuples of elements of $\mathfrak{g}$. For irreducible finite dimensional representations of $\mathrm{SL}_2(\mathbb{C})$ minimal generating systems of the corresponding algebras of concomitants are determined, both as an algebra and as a module over its center.
Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.
Let $so_3(K)$ be the Lie algebra of $3\times 3$ skew-symmetric matrices over a field $K$ of characteristic 0. The ideal $I(M_3(K),so_3(K))$ of the weak polynomial identities of the pair $(M_3(K),so_3(K))$ consists of the elements $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle X\rangle$ with the property that $f(a_1,\ldots,a_n)=0$ in the algebra $M_3(K)$ of all $3\times 3$ matrices for all $a_1,\ldots,a_n\in so_3(K)$. The generators of $I(M_3(K),so_3(K))$ were found by Razmyslov in the 1980's. In this paper the cocharacter sequence of $I(M_3(K),so_3(K))$ is computed. In other words, the ${\mathrm{GL}}_p(K)$-module structure of the algebra generated by $p$ generic skew-symmetric matrices is determined. Moreover, the same is done for the closely related algebra of $\mathrm{SO}_3(K)$-equivariant polynomial maps from the space of $p$-tuples of $3\times 3$ skew-symmetric matrices into $M_3(K)$ (endowed with the conjugation action). In the special case $p=3$ the latter algebra is a module over a $6$-variable polynomial subring in the algebra of $\mathrm{SO}_3(K)$-invariants of triples of $3\times 3$ skew-symmetric matrices, and a free resolution of this module is found. The proofs involve methods and results of classical invariant theory, representation theory of the general linear group and explicit computations with matrices.
A minimal homogeneous generating system of the algebra of semi-invariants of tuples of two-by-two matrices over an infinite field of characteristic two or over the ring of integers is given. In an alternative interpretation this yields a minimal system of homogeneous generators for the vector invariants of the special orthogonal group of degree four over a field of characteristic two or over the ring of integers. An irredundant separating system of semi-invariants of tuples of two-by-two matrices is also determined, it turns out to be independent of the characteristic.
The problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert-Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.
It is well known that results on zero-sum sequences over a finitely generated abelian group can be translated to statements on generators of rings of invariants of the dual group. Here the direction of the transfer of information between zero-sum theory and invariant theory is reversed. First it is shown how a presentation by generators and relations of the ring of invariants of an abelian group acting linearly on a finite dimensional vector space can be obtained from a presentation of the ring of invariants for the corresponding multiplicity free representation. This combined with a known degree bound for syzygies of rings of invariants, yields bounds on the presentation of a block monoid associated to a finite sequence of elements in an abelian group. The results have an equivalent formulation in terms of binomial ideals, but here the language of monoid congruences and the notion of catenary degree is used.
Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of $n$ by $n$ matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Furthermore, a characteristic free treatment is given to Kuzmin's lower bound for the nilpotency index.
It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the algebra of polynomial invariants of a group of linear transformations is rational, whenever this algebra is finitely generated. This basic principle is applied here to prove rationality of Hilbert series of algebras of invariants that are neither commutative nor finitely generated. Our main focus is on linear groups acting on certain factor algebras of the tensor algebra that arise naturally in the theory of polynomial identities.
It is shown that up to dimension four, the toric ideal of a quiver polytope is generated in degree two, with the only exception of the four-dimensional Birkhoff polytope. As a consequence, Bøgvad's conjecture holds for quiver polytopes of dimension at most four. In arbitrary dimension, the toric ideal of a compressed polytope is generated in degree two if the polytope has no neighbouring singular vertices. Furthermore, the toric ideal of a compressed polytope with at most one singular vertex has a quadratic Gröbner basis.
The best known method to give a lower bound for the Noether number of a given finite group is to use the fact that it is greater than or equal to the Noether number of any of the subgroups or factor groups. The results of the present paper show in particular that these inequalities are strict for proper subgroups or factor groups. This is established by studying the algebra of coinvariants of a representation induced from a representation of a subgroup.
A presentation by generators and relations of the $n$th symmetric power $B$ of a commutative algebra $A$ over a field of characteristic zero or greater than $n$ is given. This is applied to get information on a minimal homogeneous generating system of $B$ (in the graded case). The known result that in characteristic zero the algebra $B$ is isomorphic to the coordinate ring of the scheme of $n$-dimensional representations of $A$ is also recovered. The special case when $A$ is the two-variable polynomial algebra and $n=3$ is applied to find generators and relations of an algebra of invariants of the symmetric group of degree four that was studied in connection with the problem of classifying sets of four unit vectors in the Euclidean space.
This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better understanding of the multiplicative ideal theory of invariant rings, and connections between the Noether number and the Davenport constants of finite groups.
It is proved that the universal degree bound for separating polynomial invariants of a finite abelian group (in non-modular characteristic) is strictly smaller than the universal degree bound for generators of polynomial invariants, unless the goup is cyclic or is the direct product of $r$ even order cyclic groups where the number of two-element direct factors is not less than the integer part of the half of $r$. A characterization of separating sets of monomials is given in terms of zero-sum sequences over abelian groups.