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This is the first in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field K , we construct a standard monomial basis for the arc space of the determinantal variety over K. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group. [ABSTRACT FROM AUTHOR]

We develop an invariant theory of quasi-split ı quantum groups U n ı of type AIII on a tensor space associated to ı Howe dualities. The first and second fundamental theorems for U n ı -invariants are derived. [ABSTRACT FROM AUTHOR]

We described a minimal separating set for the algebra of O 2 + (F q) -invariant polynomial functions of m -tuples of two-dimensional vectors over a finite field F q. [ABSTRACT FROM AUTHOR]

Consider a Noetherian domain R and a finite group G ⊆ G l n (R). We prove that if the ring of invariants R [ x 1 , ... , x n ] G is a Cohen-Macaulay ring, then it is generated as an R -algebra by elements of degree at most max ⁡ (| G | , n (| G | − 1)). As an intermediate result we also show that if R is a Noetherian local ring with infinite residue field then such a ring of invariants of a finite group G over R contains a homogeneous system of parameters consisting of elements of degree at most | G |. [ABSTRACT FROM AUTHOR]

We consider a finite permutation group acting naturally on a vector space V over a field . A well-known theorem of Göbel asserts that the corresponding ring of invariants [ V ] G is generated by the invariants of degree at most dim V 2 . In this paper, we show that if the characteristic of is zero, then the top degree of vector coinvariants [ V m ] G is also bounded above by dim V 2 , which implies the degree bound dim V 2 + 1 for the ring of vector invariants [ V m ] G . So, Göbel's bound almost holds for vector invariants in characteristic zero as well. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) $ \mathcal {V} $ V , and let $ {\Bbbk [\mathcal {V}]^{G}} $ k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set $ S \subseteq {\Bbbk [\mathcal {V}]^{G}} $ S ⊆ k [ V ] G is a set of polynomials with the property that for all $ v,w \in \mathcal {V} $ v , w ∈ V , if there exists $ f \in {\Bbbk [\mathcal {V}]^{G}} $ f ∈ k [ V ] G separating $v$ and $w$, then there exists $ f \in S $ f ∈ S separating $v$ and $w$. In this article, we consider the action of $ G = \operatorname {GL}_2(\mathbb {C}) $ G = GL 2 ⁡ (C) on the $ \mathbb {C} $ C -vector space $ {\mathcal {M}}_2^n $ M 2 n of n-tuples of $ 2 \times 2 $ 2 × 2 matrices by simultaneous conjugation. Minimal generating sets $ S_n $ S n of $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G are well known and $ |S_n| = \frac 16(n^3+11n) $ | S n | = 1 6 (n 3 + 11 n). In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for $ 2 \times 2 $ 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all $ n \geq ~1 $ n ≥ 1 , $ S_n $ S n is a minimal separating set by inclusion, i.e. that no proper subset of $ S_n $ S n is a separating set. This does not necessarily mean that $ S_n $ S n has minimum cardinality among all separating sets for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G . Our main result shows that any separating set for $ \mathbb {C}[{\mathcal {M}}_2^n]^G $ C [ M 2 n ] G has cardinality $ \geq ~5n-5 $ ≥ 5 n − 5. In particular, there is no separating set of size $ \dim (\mathbb {C}[{\mathcal {M}}_2^n]^G) = 4n-3 $ dim ⁡ (C [ M 2 n ] G) = 4 n − 3 for $ n \geq ~3 $ n ≥ 3. Further, $ S_3 $ S 3 has indeed minimum cardinality as a separating set, but for $ n \geq ~4 $ n ≥ 4 there may exist a smaller separating set than $ S_n $ S n . We show that a smaller separating set does in fact exist for all $ n \geq ~5 $ n ≥ 5. We also prove similar results for the left–right action of $ \operatorname {SL}_2(\mathbb {C}) \times \operatorname {SL}_2(\mathbb {C}) $ SL 2 ⁡ (C) × SL 2 ⁡ (C) on $ {\mathcal {M}}_2^n $ M 2 n . [ABSTRACT FROM AUTHOR]

A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras. [ABSTRACT FROM AUTHOR]

Let p be an odd prime and let Mn be the extra-special p-group of order p2n+1(n ⩾ 1) and exponent p². We completely compute the mod p Margolis homology of the image ImRes(A,Mn) for every maximal elementary abelian p-subgroup A of Mn. [ABSTRACT FROM AUTHOR]

We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the A n , B n , and D n case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2 d stabilizes with n > 2 d. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating v and w , then there exists f ∈ S separating v and w. In this article we consider the action of G = SL 2 (C) × SL 2 (C) on the C -vector space M 2 , 2 n of n -tuples of 2 × 2 matrices by multiplication on the left and the right. Minimal generating sets S n of C [ M 2 , 2 n ] G are known, and | S n | = 1 24 (n 4 − 6 n 3 + 23 n 2 + 6 n). In recent work, Domokos [8] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 , 2 n ] G. Our main result shows that any separating set for C [ M 2 , 2 n ] G has cardinality ≥ 5 n − 9. In particular, there is no separating set of size dim ⁡ (C [ M 2 n ] G) = 4 n − 6 for n ≥ 4. Further, S 4 has indeed minimum cardinality as a separating set, but for n ≥ 5 there may exist a smaller separating set than S n. We also consider the action of G = SL l (C) on M l , n by left multiplication. In that case the algebra of invariants has a minimum generating set of size ( n l ) (the l × l minors of a generic matrix) and dimension l n − l 2 + 1. We show that a separating set for C [ M l , n ] G must have size at least (2 l − 2) n − 2 (l 2 − l). In particular, C [ M l , n ] G does not contain a separating set of size dim ⁡ (C [ M l , n ] G) for l ≥ 3 and n ≥ l + 2. We include an interpretation of our results in terms of representations of quivers, and make a conjecture generalising the Skowronski-Weyman theorem. [ABSTRACT FROM AUTHOR]

Let K 〈 X 〉 denote the free associative algebra generated by a set X = { x 1 , ... , x n } over a field K of characteristic 0. Let I p , for p ≥ 2 , denote the two-sided ideal in K 〈 X 〉 generated by all commutators of the form [ u 1 , ... , u p ] , where u 1 , ... , u p ∈ K 〈 X 〉. We discuss the GL (n , K) -module structure of the quotient K 〈 X 〉 / I p + 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 V λ appears in the decomposition of K 〈 X 〉 / I p + 1 as a GL (n , K) -module. As an application, we take K = C and we consider the algebra of invariants (C 〈 X 〉 / I p + 1) G for G = SL (n , C) , O (n , C) , SO (n , C) , or Sp (2 s , C) (for n = 2 s). By a theorem of Domokos and Drensky, (C 〈 X 〉 / I p + 1) G is finitely generated. We give an upper bound on the degree of generators of (C 〈 X 〉 / I p + 1) G in a minimal generating set. In a similar way, we consider also the algebra of invariants (C 〈 X 〉 / I p + 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 I p. [ABSTRACT FROM AUTHOR]

By using the Lewis–Riesenfeld invariants theory, we investigate the quantum dynamics of a two-dimensional (2D) time-dependent coupled oscillator. We introduce a unitary transformation and show the conditions under which the invariant operator is uncoupled to describe two simple harmonic oscillators with time-independent frequencies and unit masses. The decouplement allows us to easily obtain the corresponding eigenstates. The generalization to three-dimensional (3D) time-dependent coupled oscillator is briefly discussed where a diagonalized invariant, which is exactly the sum of three simple harmonic oscillators, is obtained under specific conditions on the parameters. [ABSTRACT FROM AUTHOR]

We provide a table of multiplication of the Poisson algebra on the minimal set of generators of the invariants of pairs of matrices of degree three. [ABSTRACT FROM AUTHOR]

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action and the size of a curve's symmetry group. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to different types of noise. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero. The K-vector space Vn of all binary forms f (x , y) = ∑ i = 0 n a i x i y n − i of degree n with coefficients in K carries a natural action of the group GL 2 (K) via substitutions of variables x and y. The invariant polynomials p (a 0 , ... , a n) under the action of certain subgroups of GL 2 (K) were studied intensively in the 19th century. If U is the subgroup of the upper triangular unipotent matrices, with the aid of heavy computer calculations, we know the algebra of U-invariant polynomials in K [ a 0 , ... , a n ] only for n ≤ 10. It appears to be a hopeless task to get much further along these lines. However, it is known that after inverting the U-invariant a0, the algebra has a very special form, namely it is generated by algebraically independent polynomials p 1 = a 0 , p 2 , ... , p n together with a 0 − 1 . In this note, we give an explicit set of such polynomials p i = p i (a 0 , ... , a i) which are of degrees 2 and 3. We also extend these results to U-invariants of several binary forms. [ABSTRACT FROM AUTHOR]

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating $v$ and $w$, then there exists f ∈ S separating $v$ and $w$. In this article, we consider the action of G = GL 2 ⁡ ( C ) on the C -vector space M 2 n of n-tuples of 2 × 2 matrices by simultaneous conjugation. Minimal generating sets S n of C [ M 2 n ] G are well known and | S n | = 1 6 ( n 3 + 11 n ) . In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 n ] G . Our main result shows that any separating set for C [ M 2 n ] G has cardinality ≥ 5 n − 5 . In particular, there is no separating set of size dim ⁡ ( C [ M 2 n ] G ) = 4 n − 3 for n ≥ 3 . Further, S 3 has indeed minimum cardinality as a separating set, but for n ≥ 4 there may exist a smaller separating set than S n . We show that a smaller separating set does in fact exist for all n ≥ 5 . We also prove similar results for the left–right action of SL 2 ⁡ ( C ) × SL 2 ⁡ ( C ) on M 2 n . [ABSTRACT FROM AUTHOR]

The article is devoted to studying the Singer transfer. The image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ for the infinite real projective space is proved to be a module over the image of the transfer Tr * for the sphere. Further, the algebraic Kahn–Priddy homomorphism is shown to be an epimorphism from Im ⁡ Tr * ℝ ⁢ ℙ ∞ onto Im ⁡ Tr * in positive stems. The indecomposable elements h ^ i for i ≥ 1 and c ^ i , d ^ i , e ^ i , f ^ i , p ^ i for i ≥ 0 are in the image of the Singer transfer Tr * ℝ ⁢ ℙ ∞ , whereas the ones g ^ i for i ≥ 1 and D ^ 3 ⁢ (i) , p ^ i ′ for i ≥ 0 are not in its image. This transfer is shown to be not monomorphic in every positive homological degree. The transfer behavior is also investigated near "critical elements". The squaring operation on the domain of Tr * ℝ ⁢ ℙ ∞ is proved to be eventually isomorphic. This phenomenon leads to the so-called "stability" of the Singer transfer for the infinite real projective space under the iterated squaring operation. [ABSTRACT FROM AUTHOR]

The defining relation on the minimal set of generators of the invariants of pairs of matrices of degree three has been established by Nakamoto. Furthermore, it has been simplified with another minimal set of generators by Aslaksen, Drensky and Sadikova using methods coming from representation theory. This work is devoted to obtain this defining relation in a completely different and novel way. We use the Poisson algebra structure on the ring of invariants of pairs of matrices to do so. [ABSTRACT FROM AUTHOR]

A classical result of Klein states that, given a finite primitive group G ⊆ S L 2 (C) , there exists a hypergeometric equation such that every second order LODE whose differential Galois group is isomorphic to G is projectively equivalent to the pullback by a rational map of this hypergeometric equation. In this paper, we generalize this result. We show that, given a finite primitive group G ⊆ S L n (C) , there exist a positive integer d = d (G) and a standard equation such that every LODE whose differential Galois group is isomorphic to G is gauge equivalent, over a field extension F of degree d , to an equation projectively equivalent to the pullback by a map in F of this standard equation. For n = 3 , these standard equations can be chosen to be hypergeometric. [ABSTRACT FROM AUTHOR]

Let GL(n) = GL(n, C) denote the complex general linear group and let G GL(n) be one of the classical complex subgroups O(n), SO(n), and Sp(2k) (in the case n = 2k). We take a finite dimensional polynomial GL(n) - module W and consider the symmetric algebra S(W). Extending previous results for G = SL(n), we develop a method for determining the Hilbert series H(S(W)G, t) of the algebra of invariants S(W)G. Our method is based on simple algebraic computations and can be easily realized using popular software packages. Then we give many explicit examples for computing H(S(W)G, t). As an application, we consider the question of regularity of the algebra S(W)O(n). For n = 2 and n = 3 we give a complete list of modules W, so that if S(W)O(n) is regular then W is in this list. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants (S2V)G and (2V)G, where V = Cn denotes the standard GL(n) -module. [ABSTRACT FROM AUTHOR]

Much of the captivating numerology surrounding finite reflection groups stems from Solomon's celebrated 1963 theorem describing invariant differential forms. Invariant differential derivations also exhibit fascinating numerology over the complex numbers linked to rational Catalan combinatorics. We explore the analogous theory over arbitrary fields, in particular, when the characteristic of the underlying field divides the order of the acting reflection group and the conclusion of Solomon's Theorem may fail. Using results of Broer and Chuai, we give a Saito criterion (Jacobian criterion) for finding a basis of differential derivations invariant under a finite group that distinguishes certain cases over fields of characteristic 2. We show that the reflecting hyperplanes lie in a single orbit and demonstrate a duality of exponents and coexponents when the transvection root spaces of a reflection group are maximal. A set of basic derivations are used to construct a basis of invariant differential derivations with a twisted wedging in this case. We obtain explicit bases for the special linear groups SL (n , q) and general linear groups GL (n , q) , and all groups in between. [ABSTRACT FROM AUTHOR]

We use the k[V]-module generator of the dual module of the polynomial ring k[V] over its subring of invariants of a finite group to construct modular invariants and show that it behaves better than the transfer homomorphism. [ABSTRACT FROM AUTHOR]

Let M = G / H be an (n + 1) -dimensional homogeneous manifold and J k (n , M) = : J k be the manifold of k -jets of hypersurfaces of M. The Lie group G acts naturally on each J k . A G -invariant partial differential equation of order k for hypersurfaces of M (i.e., with n independent variables and 1 dependent one) is defined as a G -invariant hypersurface ℰ ⊂ J k . We describe a general method for constructing such invariant partial differential equations for k ≥ 2. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup H (k − 1) of the (k − 1) -prolonged action of G. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space n + 1 and in the conformal space n + 1 . Our method works under some mild assumptions on the action of G , namely: A1) the group G must have an open orbit in J k − 1 , and A2) the stabilizer H (k − 1) ⊂ G of the fiber J k → J k − 1 must factorize via the group of translations of the fiber itself. [ABSTRACT FROM AUTHOR]

The possibility of interstellar scientific missions, even if only to explore the local Interstellar Medium (ISM), has received increased consideration in recent years. Due to the immense distances involved, an interstellar spacecraft must be capable of autonomous navigation. Many autonomous navigation strategies exist, but nearly all of these depend in some way on star observations and an ability to match these observations to a catalog of known stars. This is generally accomplished by computing and matching a star pattern descriptor that is invariant to changes in the spacecraft attitude. Unlike missions within the Solar System, an interstellar spacecraft will require a catalog that treats stars as 3D points rather than directions. This is problematic, since it has long been known that view invariants do not exist for sets of 3D points—making the classical method (or any modifications thereof) completely ineffectual for star identification in interstellar space. Moreover, other effects for a spacecraft traveling at relativistic speeds – such as stellar aberration and the relativistic Doppler effect – can negatively affect star identification, navigation observability, and sensor design. These challenges of star identification during interstellar flight are discussed and some possible solutions are suggested. Numerical examples are provided for a variety of motivating mission concepts. • Interstellar missions are expected to rely on star observations as part of their navigation system. • Interstellar missions require 3D star catalogs, which represents a departure from conventional star catalogs. • If stars in the galaxy must be modeled as 3D points, there are no pose invariant descriptors that can be used to index star patterns. • We may obtain star pattern invariants by constraining the allowable motion of the interstellar spacecraft. [ABSTRACT FROM AUTHOR]

An approach to exact quantum solutions of the time-dependent two energy level Jaynes-Cummings model with an imaginary photon process is represented in this work. The Lewis-Riesenfeld invariant treatment and the unitary transformation method are used for this purpose. The original Schrödinger equation is reduced to an equivalent solvable one through unitary transformations by using suitable unitary operators. The reduced equation corresponds to a simpler Hamiltonian which is written as a linear combination of the generators of the reduced-dimensional SU(2) algebra. A Hermitian invariant operator is constructed based on the same algebraic formulation and its instantaneous eigenfunctions are obtained. By utilizing such eigenfunctions, the complete quantum wave functions of the system are evaluated. Such wave functions are necessary when we analyze the quantum characteristics of the system. [ABSTRACT FROM AUTHOR]

In this paper, we introduce some new polynomials associated to linear codes over F q. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over F q , and give the MacWilliams type identity for it. We also give the notion of generalized q -colored t -designs. As an application of the generalized q -colored t -designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over F q. Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type III and Type IV codes. [ABSTRACT FROM AUTHOR]

An analogue of Brylinski's knot beta function is defined for a compactly supported (Schwartz) distribution T on d -dimensional Euclidean space. This is a holomorphic function on a right half-plane. If T is a (uniform) double-layer on a compact smooth hypersurface, then the beta function has an analytic continuation to the complex plane as a meromorphic function, and the residues are integrals of invariants of the second fundamental form. The first few residues are computed when d = 2 and d = 3. [ABSTRACT FROM AUTHOR]

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation. [ABSTRACT FROM AUTHOR]

The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the SO (3) and O (3) symmetries is presented. This paper is a follow-up to our previous work that dealt with a set of two dimensional vectors under the action of the SO (2) and O (2) groups (Dhont and Zhilinskií in J Phys A Math Theor 46:455202, 2013). The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non-free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for O (3) are deduced from the SO (3) ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time. [ABSTRACT FROM AUTHOR]

Let θ : G ↪ GL ⁢ (n , F) be a representation of a finite group 𝐺 over the field 𝔽 and F[V] ⊗F⁢ [V]GF[V] the associated equivariant coinvariant algebra. The purpose of this manuscript is to determine the associated prime ideals of F[V] ⊗F⁢ [V]GF[V] for representations 𝜃 which are defined over a finite field. We show that the only possible embedded prime ideal is the maximal ideal completing the delineation of the associated prime ideals of an equivariant coinvariant algebra for which descriptions of the minimal primes are already in the literature. In the first part of this manuscript we develop some tools particular to the case where 𝔽 is a Galois field using the Steenrod algebra P* of a Galois field 𝔽 culminating in a version of W. Krull's Going Down Theorem for the inclusion F⁢ [V] ↪ F⁢ [V] ⊗F[V]GF[V] of either of the tensor factors and we then apply this result to determine the height and the coheight of all the P*-invariant prime ideals in F[V] ⊗F⁢ [V]GF[V]. Since it has long been known that the associated prime ideals in F[V] ⊗F⁢ [V]GF[V] must be P* -invariant, our main result is an easy consequence. As indicated above, our main result is that the associated prime ideals of F[V] ⊗F⁢ [V]GF[V] for 𝔽 a Galois field are either minimal or the maximal ideal, meaning the ideal consisting of all forms of strictly positive degree. [ABSTRACT FROM AUTHOR]

Quantum dynamical properties of a general time-dependent coupled oscillator are investigated based on the theory of two-dimensional (2D) dynamical invariants. The quantum dynamical invariant of the system satisfies the Liouville–von Neumann equation and it coincides with its classical counterpart. The mathematical formula of this invariant involves a cross term which couples the two oscillators mutually. However, we show that, by introducing two pairs of annihilation and creation operators, it is possible to uncouple the original invariant operator so that it becomes the one that describes two independent subsystems. The eigenvalue problem of this decoupled quantum invariant can be solved by using a unitary transformation approach. Through this procedure, we eventually obtain the eigenfunctions of the invariant operator and the wave functions of the system in the Fock state. The wave functions that we have developed are necessary in studying the basic quantum characteristics of the system. In order to show the validity of our theory, we apply our consequences to the derivation of the fluctuations of canonical variables and the uncertainty products for a particular 2D oscillatory system whose masses are exponentially increasing. [ABSTRACT FROM AUTHOR]

In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K U Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U = End (W ⊕ k) and Γ = GL (W) and on the case where U = End (V ⊗ W) and Γ = GL (V) × GL (W) , though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series, using Mathematica (see the appendix by Dejan Govc). [ABSTRACT FROM AUTHOR]

In this paper, we present a reciprocity on finite abelian groups involving zero-sum sequences. Let G and H be finite abelian groups with (| G|, | H|) = 1. For any positive integer m, let sansM (G,m) denote the set of all zero-sum sequences over G of length m. We have the reciprocity | sansM (G, | H|)| = | sansM (H, | G|)|. Moreover, we provide a combinatorial interpretation of the above reciprocity using ideas from rational Catalan combinatorics. We also present and explain some other symmetric relationships on finite abelian groups with methods from invariant theory. Among others, we partially answer a question proposed by Panyushev in a generalized version. [ABSTRACT FROM AUTHOR]

The ring of covariants of binary form is isomorphic to the ring of regular symmetric bracket polynomials in homogenized roots, which are obtained by symmetrization of regular bracket polynomials. In this paper, we give an algorithm for computing all cyclically standard bracket monomials of arbitrary degree. These monomials form a basis of the vector space of the regular bracket polynomials of a given degree. We also characterize the elemental bracket monomial of degree 2 in terms of graph theory. [ABSTRACT FROM AUTHOR]

This article studies separating invariants for the ring of multisymmetric polynomials in m sets of n variables over an arbitrary field K. We prove that in order to obtain separating sets it is enough to consider polynomials that depend only on ⌊n/2⌋ + 1 sets of these variables. This improves a general result by Domokos about separating invariants. In addition, for n ≤ 4 we explicitly give minimal separating sets (with respect to inclusion) for all m in case char(K) = 0 or char(K) > n. [ABSTRACT FROM AUTHOR]

We consider the algebra of invariants of d -tuples of n × n matrices under the action of the orthogonal group by simultaneous conjugation over an infinite field of characteristic p different from two. It is well known that this algebra is generated by the coefficients of the characteristic polynomial of all products of generic and transpose generic n × n matrices. We establish that in case 0 < p ≤ n the maximal degree of indecomposable invariants tends to infinity as d tends to infinity. In other words, there does not exist a constant C (n) such that it only depends on n and the considered algebra of invariants is generated by elements of degree less than C (n) for any d. This result is well-known in case of the action of the general linear group. On the other hand, for the rest of p the given phenomenon does not hold. We investigate the same problem for the cases of symmetric and skew-symmetric matrices. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of canonical coordinates for generic co-adjoint orbits on triangular groups. These orbits correspond to a set of full Plancherel measure on the associated dual groups. This generalizes a well-known coordinatization of co-adjoint orbits of a minimal (non-generic) type originally discovered by Flaschka. The latter had strong connections to the classical Toda lattice and its associated Poisson geometry. Our results develop connections with the Full Kostant–Toda lattice and its Poisson geometry. This leads to novel insights relating the details of Plancherel theorems for Borel Lie groups to the invariant theory for Borels and their subgroups. We also discuss some implications for the quantum integrability of the Full Kostant Toda lattice. • New links explored between dual groups of Lie groups and their Poisson geometry. • Dixmier–Pukanszky operator of Plancherel theory related to Toda invariant theory. • Canonical coordinates established for symplectic leaves of full Toda systems. [ABSTRACT FROM AUTHOR]

The algebra O (N n d) G L n of G L n -invariants of d -tuple of n × n nilpotent matrices with respect to the action by simultaneous conjugation is generated by the traces of products of nilpotent generic matrices in the case of an algebraically closed field of characteristic zero. We describe a minimal separating set for this algebra in case n = d = 3. [ABSTRACT FROM AUTHOR]

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form (G a − × G a −) ⋊ (G a (r) × (Z / p) s). In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-supergroup schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely (G a − × G a −) ⋊ G a (1) and (G a − × G a −) ⋊ Z / p. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Lewis, Reiner, and Stanton conjectured a Hilbert series for a space of invariants under an action of finite general linear groups using (q , t) -binomial coefficients. This work gives an analog in positive characteristic of theorems relating various Catalan numbers to the representation theory of rational Cherednik algebras. They consider a finite general linear group as a reflection group acting on the quotient of a polynomial ring by iterated powers of the irrelevant ideal under the Frobenius map. We prove a variant of their conjecture in the local case, when the group acting fixes a reflecting hyperplane. [ABSTRACT FROM AUTHOR]

Let k (x) = k (x 1 , ... , x n) be the rational function field, and k ⊂ L ⊂ k (x) an intermediate field. Hilbert's fourteenth problem asks whether the k -algebra L ∩ k x 1 , ... , x n is finitely generated. Various counterexamples to this problem were already given, but there are still many open cases. In this paper, we study the problem in terms of the field-theoretic properties of L. Our result implies that, for any S ⊂ k x 1 , ... , x n − 1 with k (S) ≠ k (x 1 , ... , x n − 1) and tr. deg k k (S) ≥ 2 , there exists σ ∈ Aut k k (x) such that σ (k (S ∪ { x n })) is a counterexample to Hilbert's fourteenth problem. This implies the existence of several interesting new counterexamples, such as one with k (x) : L = 2. [ABSTRACT FROM AUTHOR]

For the standard 2 7 -dimensional representation V of the exceptional group G of type E 6 we prove that (S L (V) , G) is a Donkin pair if and only if the characteristic of a ground field is greater than 1 3. We also develop an elementary approach to describe submodule structure of any exterior power of V. [ABSTRACT FROM AUTHOR]

In this paper we consider self-dual NRT codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman metric (NRT metric) and their shape enumerators as defined by Barg and Park. We use polynomial invariant theory to describe the shape enumerator of a binary self-dual NRT code, even self-dual NRT code, and weak doubly even self-dual NRT code in $ {M}_{ {n},2}(\mathbb {F}_{2})$. Motivated by these results, we describe the number of invariant polynomials that we must find to describe the shape enumerator of a self-dual NRT code in $ {M}_{ {n}, {s}}(\mathbb {F}_{2})$. We define the ordered flip of a matrix $ {A}\in {M}_{ {k},{ { ns}}}(\mathbb {F}_{ {q}})$ and present some constructions of self-dual NRT codes over $\mathbb {F}_{ {q}}$. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two. [ABSTRACT FROM AUTHOR]

The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of k [ x 1 , ... , x n ] under the natural action of a finite subgroup of S L n (k) is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras A. We develop the notion of a homological determinant of an automorphism of A, then use the homological determinant to study actions of finite groups G on A. We give a sufficient condition so that the invariant ring AG has finite injective dimension and satisfies the generalized Gorenstein condition. More precisely, let A be a noetherian N -graded generalized Gorenstein algebra with finite global dimension. Suppose all elements of G fix the idempotents of A and act with trivial homological determinant. Then the invariant ring AG is generalized Gorenstein. [ABSTRACT FROM AUTHOR]

Let V , W {V,W} be representations of a cyclic group G of prime order p over a field 𝕜 {\Bbbk} of characteristic p. The module of covariants 𝕜 ⁢ [ V , W ] G {\Bbbk[V,W]^{G}} is the set of G-equivariant polynomial maps V → W {V\rightarrow W} , and is a module over 𝕜 ⁢ [ V ] G {\Bbbk[V]^{G}}. We give a formula for the Noether bound β ⁢ (𝕜 ⁢ [ V , W ] G , 𝕜 ⁢ [ V ] G) {\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})} , i.e. the minimal degree d such that 𝕜 ⁢ [ V , W ] G {\Bbbk[V,W]^{G}} is generated over 𝕜 ⁢ [ V ] G {\Bbbk[V]^{G}} by elements of degree at most d. [ABSTRACT FROM AUTHOR]

In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over Q is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (Ivanyos et al. in Comput Complex 26(3):717–763, 2017) (whether or not randomization is allowed). The algorithm efficiently solves the "word problem" for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004), who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits 2004). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp–Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular, we explain how some of these sources played an important role in the development of Gurvits' algorithm and in our analysis of it here. [ABSTRACT FROM AUTHOR]

This note provides a set of separating invariants for the ring of vector invariants K [ V 2 ] S n of two copies of the natural Sn-representation V = K n over a field of characteristic 0. This set is much smaller than generating sets of K [ V 2 ] S n . For n ≤ 4 , we show that this set is minimal with respect to inclusion among all separating sets. [ABSTRACT FROM AUTHOR]

We extend the Dodonov–Malkin–Man'ko–Trifonov (DMMT) invariant method (Malkin et al. Phys. Rev. D 2, 1371 1, J. Math. Phys. 14, 576 2; Dodonov et al. Int. J. Theor. Phys. 14, 37 3; Dodonov and Man'ko Phys. Rev. A 20, 550 4) to time-dependent pseudo-fermionic systems by introducing ladder invariant operators (time-dependent integrals of motion) which play the role of time-dependent pseudo-fermionic operators and constructing time-dependent pseudo-fermionic coherent states (PFCS) for such systems. As illustrative example, we study in details the time-dependent parity-time-symmetric two-level system under synchronous combined modulations. We explicitly determine time-dependent pseudo-fermionic operators and construct time-dependent PFCS for this physical system. We show that our approach can be extended to time-dependent pseudo-bosonic systems. [ABSTRACT FROM AUTHOR]