Related papers: Separating invariants for the basic G_a-actions

We prove that certain classical groups $G\subseteq {\rm GL}(d,\mathbb{R}^d)$ serve to characterize ordinary polynomials in $d$ real variables as elements of finite-dimensional subspaces of $C(\mathbb{R}^d)$ that are invariant by changes of…

Let $X$ be a smooth complex projective variety equipped with an action of a linear algebraic group $G$ over $\mathbb{C}$. Let $D$ be a reduced effective divisor on $X$ that is invariant under the $G$--action on $X$. Let $s_D$ be the…

A method based on the symbolic methods of the classical invariant theory is developed for a representation of elements of kernel of Weitzenb\"ok derivations.

Separable elements in Weyl groups are generalizations of the well-known class of separable permutations in symmetric groups. Gaetz and Gao showed that for any pair $(X,Y)$ of subsets of the symmetric group $\mathfrak{S}_n$, the…

We give explicit descriptions of rings of differential operators of toric face rings in characteristic $0$. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators…

We identify conditions giving large natural classes of partial differential operators for which it is possible to construct a complete set of Laplace invariants. In order to do that we investigate general properties of differential…

Let $G$ be a finite group and let $\pi(G)=\{p_1, p_2, \ldots, p_k\}$ be the set of prime divisors of $|G|$ for which $p_1<p_2<\cdots<p_k$. The Gruenberg-Kegel graph of $G$, denoted ${\rm GK}(G)$, is defined as follows: its vertex set is…

We describe the rings of invariants for the finite orthogonal groups of plus type in odd characteristic acting on the defining representations. We also describe the invariants of the corresponding Sylow subgroups in the defining…

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 $F_q$.

The fundamental representations of the special linear group ${\rm SL}_n$ over the complex numbers are the exterior powers of $\mathbb{C}^n$. We consider the invariant rings of sums of arbitrary many copies of these ${\rm SL}_n$-modules. The…

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras $sp(n,1)$. Our choice of these algebras is motivated by the fact that they belong to a…

In this paper we determine the number of the meaningful compositions of higher order of the differential operations and Gateaux directional derivative.

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

Using our recent results on eigenvalues of invariants associated to the Lie superalgebra gl(m|n), we use characteristic identities to derive explicit matrix element formulae for all gl(m|n) generators, particularly non-elementary…

Fractional supersymmetry denotes a generalisation of supersymmetry which may be constructed using a single real generalised Grassmann variable, $\theta = \bar{\theta}, \, \theta^n = 0$, for arbitrary integer $n = 2, 3, ...$. An explicit…

We propose a spectrum for a class of gauged non-compact G/Ad(H) WZNW models, including spectrally flowed images of highest, lowest, and mixed extremal weight modules. These are combined into blocks whose characters, due to the Lorentzian…

Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation of the hyperelliptic discriminant of X/S, and the valuation of the Mumford…

In this paper we introduce a notion of symplectic normal crossings divisor V and define the GW invariant of a symplectic manifold X relative such a divisor. Our definition includes normal crossings divisors from algebraic geometry. The…

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…

Much of the fascinating numerology surrounding finite reflection groups stems from Solomon's celebrated 1963 theorem describing invariant differential forms. Invariant differential derivations also exhibit interesting numerology over the…