Classical Invariant Theory Research Articles

Catalytic Oxidation-like Nuclear Nano-fusion; Fractal Involving of Room Temperature Magnetically Induced μ-Catalyzed Fusion

The nuclear fusion reaction can be catalyzed in a suitable fusion fuel by muons (heavy electrons). “For the fractal relations, ranging from DNA knots to solar neutrino flux signals”, ever derived of scale-invariant properties distinguished between classical invariant theory & quantum invariant theory subfactors. Accompanying isomorphic & Connes FusionTensor Product retrieved to μ-catalyzed fusion where surroundings of room temperature fusion driven by the balance in mtDNA fusion & fission. On behalf of the nanometer dimension of the radius of heavy electrons & wavelength of UV-light, it assumed that muons can be produced by oxidation-like decay when UV-light impinging water.

Non-Lorentzian IIB supergravity from a polynomial realization of SL(2, ℝ)

We derive the action and symmetries of the bosonic sector of non-Lorentzian IIB supergravity by taking the non-relativistic string limit. We find that the bosonic field content is extended by a Lagrange multiplier that implements a restriction on the Ramond-Ramond fluxes. We show that the SL(2, ℝ) transformation rules of non-Lorentzian IIB supergravity form a novel, nonlinear polynomial realization. Using classical invariant theory of polynomial equations and binary forms, we will develop a general formalism describing the polynomial realization of SL(2, ℝ) and apply it to the special case of non-Lorentzian IIB supergravity. Using the same formalism, we classify all the relevant SL(2, ℝ) invariants. Invoking other bosonic symmetries, such as the local boost and dilatation symmetry, we show how the bosonic part of the non-Lorentzian IIB supergravity action is formed uniquely from these SL(2, ℝ) invariants. This work also points towards the concept of a non-Lorentzian bootstrap, where bosonic symmetries in non-Lorentzian supergravity are used to bootstrap the bosonic dynamics in Lorentzian supergravity, without considering the fermions.

Graphical methods and rings of invariants on the symmetric algebra

Abstract Let G be a complex classical group, and let V be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of G-invariant polynomial functions on the space $\mathcal P^m(V)$ of degree-m homogeneous polynomial functions on V. In this paper, we replace $\mathcal P^m(V)$ with the full polynomial algebra $\mathcal P(V)$ . As a result, the invariant ring is no longer finitely generated. Hence, instead of seeking generators, we aim to write down linear bases for bigraded components. Indeed, when G is of sufficiently high rank, we realize these bases as sets of graphs with prescribed number of vertices and edges. When the rank of G is small, there arise complicated linear dependencies among the graphs, but we remedy this setback via representation theory: in particular, we determine the dimension of an arbitrary component in terms of branching multiplicities from the general linear group to the symmetric group. We thereby obtain an expression for the bigraded Hilbert series of the ring of invariants on $\mathcal P(V)$ . We conclude with examples using our graphical notation, several of which recover classical results.

Rational local unitary invariants of symmetrically mixed states of two qubits

We compute the field of rational local unitary invariants for locally maximally mixed states and symmetrically mixed states of two qubits. In both cases, we prove that the field of rational invariants is purely transcendental. We also construct explicit geometric quotients and prove that they are always rational. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.

On the GL(n)-module structure of Lie nilpotent associative relatively free algebras

Let K〈X〉 denote the free associative algebra generated by a set X={x1,…,xn} over a field K of characteristic 0. Let Ip, for p≥2, denote the two-sided ideal in K〈X〉 generated by all commutators of the form [u1,…,up], where u1,…,up∈K〈X〉. We discuss the GL(n,K)-module structure of the quotient K〈X〉/Ip+1 for all p≥1 under the standard diagonal action. We give a bound on the values of partitions λ such that the irreducible GL(n,K)-module Vλ appears in the decomposition of K〈X〉/Ip+1 as a GL(n,K)-module. As an application, we take K=C and we consider the algebra of invariants (C〈X〉/Ip+1)G for G=SL(n,C), O(n,C), SO(n,C), or Sp(2s,C) (for n=2s). By a theorem of Domokos and Drensky, (C〈X〉/Ip+1)G is finitely generated. We give an upper bound on the degree of generators of (C〈X〉/Ip+1)G in a minimal generating set. In a similar way, we consider also the algebra of invariants (C〈X〉/Ip+1)G, where G=UT(n,C), and give an upper bound on the degree of generators in a minimal generating set. These results provide useful information about the invariants in C〈X〉G from the point of view of Classical Invariant Theory. In particular, for all G as above we give a criterion when a G-invariant of C〈X〉 belongs to Ip.

3D geometric moment invariants from the point of view of the classical invariant theory

The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory.Using the remarkable fact that the complex groups $SO(3,\mathbb{C})$ and $SL(2,\mathbb{C})$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory.
 We give a precise statement of the 3D geometric invariant moments computation, intro\-ducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2,\mathbb{C})$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3,\mathbb{C})$ to equivalent action of the complex Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in thefields of image analysis and pattern recognition.

Uniqueness of Local, Analytic Solutions to Singular ODEs

We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem to an equilibrium problem of a regular ODE. Then, we apply classical invariant manifold theory. We demonstrate that the class of ODEs under consideration captures models which describe the shape of axially symmetric surfaces which are closed on one side. Our main result guarantees smoothness at the tip of the surface.

Racah Algebras, the Centralizer $$Z_n({{{\mathfrak {s}}}{{\mathfrak {l}}}}_2)$$ and Its Hilbert–Poincaré Series

The higher rank Racah algebra R(n) introduced in De Bie et al. (J Phys A Math Theor 51(2):025203, 2017. arXiv:1610.02638 ) is recalled. A quotient of this algebra by central elements, which we call the special Racah algebra sR(n), is then introduced. Using the results from classical invariant theory, this sR(n) algebra is shown to be isomorphic to the centralizer $$Z_{n}({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ of the diagonal embedding of $$U({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ in $$U({{\mathfrak {s}}}{{\mathfrak {l}}}_2)^{\otimes n}$$ . This leads to a first and novel presentation of the centralizer $$Z_{n}({{\mathfrak {s}}}{{\mathfrak {l}}}_2)$$ in terms of generators and defining relations. An explicit formula of its Hilbert–Poincaré series is also obtained and studied. The extension of the results to the study of the special Askey–Wilson algebra and its higher rank generalizations is discussed.

Symmetric polynomials in free associative algebras

By a result of Margarete Wolf in 1936, we know that the algebra $K\langle X_d\rangle^{Sym(d)}$ of symmetric polynomials in noncommuting variables is not finitely generated. In 1984, Koryukin proved that if we equip the homogeneous component of degree $n$ with the additional action of $Sym(n)$ by permuting the positions of the variables, then the algebra of invariants $K\langle X_d\rangle^G$ of every reductive group $G$ is finitely generated. First, we make a short comparison between classical invariant theory of finite groups and its noncommutative counterpart. Then, we expose briefly the results of Wolf. Finally, we present the main result of our paper, which is, over a field of characteristic 0 or of characteristic $p>d$, the algebra $K\langle X_d\rangle^{Sym(d)}$ with the action of Koryukin is generated by the elementary symmetric polynomials.

Self-orthogonal codes over a non-unital ring and combinatorial matrices

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.

Symmetry of Dressed Photon

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters n≥r≥s on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with n=4,r=2,s=1.

The degree of Stiefel manifolds

We compute the degree of Stiefel manifolds, that is, the variety of orthonormal frames in a finite dimensional vector space. Our approach employs techniques from classical algebraic geometry, algebraic combinatorics, and classical invariant theory.

Cocharacters for the weak polynomial identities of the Lie algebra of 3 × 3 skew-symmetric matrices

Let so3(K) be the Lie algebra of 3×3 skew-symmetric matrices over a field K of characteristic 0. The ideal I(M3(K),so3(K)) of the weak polynomial identities of the pair (M3(K),so3(K)) consists of the elements f(x1,…,xn) of the free associative algebra K〈X〉 with the property that f(a1,…,an)=0 in the algebra M3(K) of all 3×3 matrices for all a1,…,an∈so3(K). The generators of I(M3(K),so3(K)) were found by Razmyslov in the 1980s. In this paper the cocharacter sequence of I(M3(K),so3(K)) is computed. In other words, the GLp(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 SO3(K)-equivariant polynomial maps from the space of p-tuples of 3×3 skew-symmetric matrices into M3(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 SO3(K)-invariants of triples of 3×3 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 NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS

AbstractBy work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp (S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and the classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

Complete set of translation invariant measurements with Lipschitz bounds

In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in C n are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep learning. Despite an impressive body of literature on the subject, most representations available are either not stable due to the presence of high frequencies or non discriminative. In the present paper, we construct low dimensional representations of signals in C n that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete Z m -invariant representations for any m ∈ N . Our construction yields a stable, discriminative transform with semi-explicit Lipschitz bounds on the dimension; this is particularly relevant for applications. Using some tools from Algebraic Geometry, we define a high dimensional homogeneous function that is injective. We then exploit the projective character of this embedding and see that the target space can be reduced significantly by using a generic linear transformation. Finally, we introduce the notion of non-parallel map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.

2D Geometric Moment Invariants from the Point of View of the Classical Invariant Theory

The aim of this paper is to clear up the question of the connection between the geometric moment invariants and the invariant theory, considering a problem of describing the 2D geometric moment invariants as a problem of the classical invariant theory. We give a precise statement of the problem of computation of the 2D geometric invariant moments, introducing the notions of the algebras of simultaneous 2D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $$\hbox {SO}(2)$$ -invariants of several binary forms. Also, to simplify the calculating of the invariants, we proceed from an action of Lie group $$\hbox {SO}(2)$$ to an action of its Lie algebra $${{\mathfrak {so}}}_2$$ . Though the 2D geometric moments are not as effective as the orthogonal ones are, the author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.

The monic rank

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone $X$. We show that the monic rank is finite and greater than or equal to the usual $X$-rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to B. Shapiro which states that a binary form of degree $d\cdot e$ is the sum of $d$ $d$-th powers of forms of degree $e$. Furthermore, in the case where $X$ is the cone of highest weight vectors in an irreducible representation---this includes the well-known cases of tensor rank and symmetric rank---we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.

Archimedean non-vanishing, cohomological test vectors, and standard L-functions of $${\mathrm {GL}}_{2n}$$: real case

The standard L-functions of $${\mathrm {GL}}_{2n}$$ expressed in terms of the Friedberg-Jacquet global zeta integrals have better structure for arithmetic applications, due to the relation of the linear periods with the modular symbols. The most technical obstacles towards such arithmetic applications are (1) non-vanishing of modular symbols at infinity and (2) the existence or construction of uniform cohomological test vectors. Problem (1) is also called the non-vanishing hypothesis at infinity, which was proved by Sun [Duke Math J 168(1):85–126, (2019), Theorem 5.1], by establishing the existence of certain cohomological test vectors. In this paper, we explicitly construct an archimedean local integral that produces a new type of a twisted linear functional $$\Lambda _{s,\chi }$$ , which, when evaluated with our explicitly constructed cohomological vector, is equal to the local twisted standard L-function $$L(s,\pi \otimes \chi )$$ for all complex values s. With the relations between linear models and Shalika models, we establish (1) with an explicitly constructed cohomological vector using classical invariant theory, and hence proves the non-vanishing results of Sun [24, Theorem 5.1] via a completely different method.

An algebraic independence result related to a conjecture of Dixmier on binary form invariants

In order to better understand the structure of classical rings of invariants for binary forms, Dixmier proposed, as a conjectural homogeneous system of parameters, an explicit collection of invariants previously studied by Hilbert. We generalize Dixmier's collection and show that a particular subfamily is algebraically independent. Our proof relies on showing certain alternating sums of products of binomial coefficients are nonzero. Along the way we provide a very elementary proof \`a la Racah, namely, only using the Chu-Vandermonde Theorem, for Dixon's Summation Theorem. We also provide explicit computations of invariants, for the binary octavic, which can serve as ideal introductory examples to Gordan's 1868 method in classical invariant theory.

Quantum Howe duality and invariant polynomials

We construct two examples of q-deformed classical Howe dual pairs (sl(2,C),so(3,C)) and (sl(2,C),sl(n,C)). Moreover, we obtain a noncommutative version of the first fundamental theorem of classical invariant theory. Our approach to these dualities differs from [9] and [11]. Furthermore, we solve the tensor product decomposition problem for Verma modules over Uq(sl(2,C)) provided q is not a root of unity.