Related papers: Vector invariants for two-dimensional orthogonal g…

Let $SL_{2}(F_{q})$ be the special linear group over a finite field $F_{q}$, $V$ be the 2-dimensional natural representation of $SL_{2}(F_{q})$ and $V^{\ast}$ be the dual representation. We denote by $F_{q}[V\oplus…

Let ${\mathbb{F}_{q}}$ be the finite field of order $q$. Let $G$ be one of the three groups ${\rm GL}(n, \mathbb{F}_q)$, ${\rm SL}(n, \mathbb{F}_q)$ or ${\rm U}(n, \mathbb{F}_q)$ and let $W$ be the standard $n$-dimensional representation of…

We described a minimal separating set for the algebra of $O(F_q)$-invariant polynomial functions of $m$-tuples of two-dimensional vectors over a finite field $F_q$.

Let $W$ be an $n$-dimensional vector space over a finite field $\mathbb{F}_q$ of any characteristic and $mW$ denote the direct sum of $m$ copies of $W$. Let $\mathbb{F}_q[mW]^{{\rm GL}(W)}$ and $\mathbb{F}_q(mW)^{{\rm GL}(W)}$ denote the…

Consider a finite dimensional vector space $V$ over a finite field $\mathbb{F}_q$. We give a minimal generating set for the ring of invariants $\mathbb{F}_q[V \oplus V^*]^{\text{GL}(V)}$, and show that this ring is a Gorenstein ring but is…

In this paper, we study the vector invariants, ${\bf{F}}[m V_2]^{C_p}$, of the 2-dimensional indecomposable representation $V_2$ of the cylic group, $C_p$, of order $p$ over a field ${\bf{F}}$ of characteristic $p$. This ring of invariants…

The first fundamental theorem of invariant theory for the orthosymplectic supergroup ${\rm OSp}(V)$ (where $V$ has superdimension $(m|2n)$) in the endomorphism algebra setting states that there is a surjective algebra homomorphism $F_r^r:…

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…

We first give a short intrinsic, diagrammatic proof of the First Fundamental Theorem of invariant theory (FFT) for the special orthogonal group $\text{SO}_m(\mathbb{C})$, given the FFT for $\text{O}_m(\mathbb{C})$. We then define, by means…

Let $p$ be an odd prime and $\mathbb{F}_p$ be the prime field of order $p$. Consider a $2$-dimensional orthogonal group $G$ over $\mathbb{F}_p$ acting on the standard representation $V$ and the dual space $V^*$. We compute the invariant…

We determine the minimal number of separating invariants for the invariant ring of a matrix group $G < \mathrm{GL}_n(\mathbb{F}_q)$ over the finite field $\mathbb{F}_q$. We show that this minimal number can be obtained with invariants of…

We consider a finite permutation group acting naturally on a vector space $V$ over a field $\Bbbk$. A well known theorem of G\"obel asserts that the corresponding ring of invariants $\Bbbk[V]^G$ is generated by invariants of degree at most…

We describe the ring of invariants for the finite orthogonal groups in odd dimension and even characteristic acting on the defining representation. We construct a minimal algebra generating set and describe the relations among the…

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…

We determine the rings of invariants in the symmetric algebra on the dual of a vector space V over the field of two elements, for the group G of orthogonal transformations preserving a non-singular quadratic form on V. The invariant ring is…

For $K$ an infinite field of characteristic other than two, consider the action of the special orthogonal group $\operatorname{SO}_t(K)$ on a polynomial ring via copies of the regular representation. When $K$ has characteristic zero,…

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…

We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation we compute the Noether number and give minimal generating sets for the Hilbert ideal and the field of…

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 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…