Invariant Theory Of Finite Groups Research Articles

Contemporary Galois theory and its classical problems

In this survey we outline the milestones of several classical problems in Galois theory: Noether’s problem, the inverse problem and the embedding problem. We demonstrate the connection between these problems in the bigger context of the invariant theory of finite groups. We also point out several new results of the author and his collaborators regarding Noether’s problem and the embedding problem.

Frobenius Manifolds on Orbits Spaces

The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group \({\mathcal {W}}\) acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.

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.

Depth and detection for Noetherian unstable algebras

For a connected Noetherian unstable algebra R R over the mod p p Steenrod algebra, we prove versions of theorems of Duflot and Carlson on the depth of R R , originally proved when R R is the mod p p cohomology ring of a finite group. This recovers the aforementioned results, and also proves versions of them when R R is the mod p p cohomology ring of a compact Lie group, a profinite group with Noetherian cohomology, a Kac–Moody group, a discrete group of finite virtual cohomological dimension, as well as for certain other discrete groups. More generally, our results apply to certain finitely generated unstable R R -modules. Moreover, we explain the results in the case of the p p -local compact groups of Broto, Levi, and Oliver, as well as in the modular invariant theory of finite groups.

Invariants of the action of a semisimple finite-dimensional Hopf algebra on special algebras

In this paper we extend classical results of the invariant theory of finite groups to the action of a finite-dimensional semisimple Hopf algebra H on a special algebra A, which is homomorphically mapped onto a commutative integral domain, and the kernel of this map contains no nonzero H-stable ideals. We prove that the algebra A is finitely generated as a module over a subalgebra of invariants, and the latter is finitely generated as a k-algebra. We give a counterexample to the finite generation of a non-semisimple Hopf algebra.

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups II

We continue our study of the homological properties of the purely inseparable extensions $ \mathrm{H} \hookrightarrow\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}}$ of integrally closed unstable Noetherian integral domains over the Steenrod algebra. It turns out that the projective dimension of $\mathrm {H}$ is a lower bound for the projective dimension of $\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}}$. Furthermore, $\operatorname{depth} (\mathrm{H}) \geq\operatorname{depth}(\!\!\sqrt[\mathcal{P}^*]{\mathrm{H}})$, where $\operatorname{depth}$ denotes the depth. Moreover, both algebras have the same global dimension. We apply these results to extension $\mathbb F[V_{\bullet}]^G \hookrightarrow\mathbb F[V]^G$ of rings of invariants.

Degree bounds—An invitation to postmodern invariant theory

This is an invitation to invariant theory of finite groups; a field where methods and results from a wide range of mathematics merge to form a new exciting blend. We use the particular problem of finding degree bounds to illustrate this.

Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions H ↪ H P ∗ \textrm {H}\hookrightarrow \sqrt [\mathscr {P}^*]{\textrm {H}} of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group G ≤ GL ( V ) G\le \textrm {GL}(V) and a vector space decomposition V = W 0 ⊕ W 1 ⊕ ⋯ ⊕ W e V=W_0\oplus W_1\oplus \dotsb \oplus W_e such that H ¯ = ( F [ W 0 ] ⊗ F [ W 1 ] p ⊗ ⋯ ⊗ F [ W e ] p e ) G \overline {\textrm {H}}=(\mathbb {F}[W_0] \otimes \mathbb {F}[W_1]^p\otimes \dotsb \otimes \mathbb {F}[W_e]^{p^e})^G and H P ∗ ¯ = F [ V ] G \overline {\sqrt [\mathscr {P}^*]{\textrm {H}}}=\mathbb {F}[V]^G , where ( − ) ¯ \overline {(-)} denotes the integral closure. Moreover, H \textrm {H} is Cohen-Macaulay if and only if H P ∗ \sqrt [\mathscr {P}^*]{\textrm {H}} is Cohen-Macaulay. Furthermore, H ¯ \overline {\textrm {H}} is polynomial if and only if H P ∗ \sqrt [\mathscr {P}^*]{\textrm {H}} is polynomial, and H P ∗ = F [ h 1 , … , h n ] \sqrt [\mathscr {P}^*]{\textrm {H}}=\mathbb {F}[h_1,\dotsc ,h_n] if and only if \[ H = F [ h 1 , … , h n 0 , h n 0 + 1 p , … , h n 1 p , h n 1 + 1 p 2 , … , h n e p e ] , \textrm {H}=\mathbb {F}[h_1,\dotsc ,h_{n_0},h_{n_0+1}^p,\dotsc ,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc ,h_{n_e}^{p^e}], \] where n e = n n_e=n and n i = dim F ⁡ ( W 0 ⊕ ⋯ ⊕ W i ) n_i=\dim _{\mathbb {F}}(W_0\oplus \dotsb \oplus W_i) .

Divisorial free modules of relative invariants on Krull domains

An effective construction of relative invariants plays an important role in the study of finite reflection groups (e.g., [J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Stud. Adv. Math., vol. 29, Cambridge Univ. Press, Cambridge, 1990]). Using a combinatorial method, R.P. Stanley (cf. [R.P. Stanley, Relative invariants of finite groups generated by pseudo-reflections, J. Algebra 49 (1977) 134–148; Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979) 475–511]) generalized such classical result to a criterion for a module Sym ( V ) χ of invariants of G relative to a character χ to be Sym ( V ) G -free of rank one in the case where G is any finite complex subgroup of GL ( V ) . His criterion is useful in invariant theory of finite groups and in combinatorics. In this paper we will study on relative invariants of a group G consisting of automorphisms of a Krull domain R from the view point of a generalized partial result on ramifications in number theory mentioned in [H. Nakajima, Reduced ramification indices of quotient morphisms under torus actions, J. Algebra 242 (2001) 536–549]. We will give a criterion for R χ to be a free R G -module of rank one for a 1-cocycle χ of G in the unit group U ( R ) , and consequently establish a criterion for module of relative invariants of a finite central extension of algebraic tori to be free, in terms of local characters, which is similar to one in [H. Nakajima, Relative invariants of finite groups, J. Algebra 79 (1982) 218–234; R.P. Stanley, Relative invariants of finite groups generated by pseudo-reflections, J. Algebra 49 (1977) 134–148].

The Direct Summand Property in Modular Invariant Theory

In nonmodular invariant theory of finite groups, the invariant ring is always a direct summand of the full polynomial ring. This is no longer generally true in modular invariant theory, but nevertheless interesting examples are known where this happens. We give useful characterisations of the direct summand property in terms of the image of a twisted transfer map. For example, for p-groups acting in characteristic p the direct summand property holds if and only if the image of the ordinary transfer is a principal ideal; and in that case the group is generated by transvections. We also extend some known results in the nonmodular case to where only the direct summand property is assumed, e.g., the invariant ring is always generated by its elements of degree at most the order of the group.

CONSTRUCTION OF A GLOBAL POTENTIAL ENERGY SURFACE FROM NOVEL AB INITIO MOLECULAR DYNAMICS FOR THE O(3P) + C3H3 REACTION

We present a global potential energy surface (PES) for the 2 A state of the O (3 P ) + C 3 H 3 radical reaction. The global PES is constructed mainly using direct ab initio molecular dynamics and further sampling is done using the Diffusion Monte Carlo method. The potential is fully invariant with respect to permutational symmetry of like atoms. Special techniques, based on invariant theory of finite groups, have been used to develop basis functions for fitting that display this symmetry. The resultant potential energy surface shows multiple reaction paths with six different product channels. The products of the reactions are CO + C 2 H 3 radicals H + C 3 H 2 O radicals (with two isomers, propynal and propa-1,2-dien-1-one) and OH + C 3 H 2 radicals (with three isomers, vinylidenecarbene, propargylene and cyclopropenylidene). Energies of the PES are in excellent agreement with ab initio energies for each stationary point, the reactants and the products. Most stationary points are fitted at the sub Kcal/mol level. The global potential surface represents all the stationary points and six different product channels correctly. Preliminary dynamics calculations show abstraction and insertion mechanisms for the OH + vinylidenecarbene channel and the H + propynal channel, respectively.

Invariants of finite Hopf algebras

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Topics considered include integrality over the invariant rings, properties of the canonical map between the prime spectra, orbital and stabilizer algebras, projectivity over the invariant rings, and descent of Cohen–Macaulayness.

Sharpening the generalized Noether bound in the invariant theory of finite groups

We consider linear representations of a finite group G on a finite dimensional vector space over a field F in which | G | is invertible. By a theorem due to E. Noether in char 0, and to Fleischmann and Fogarty in general, the ring of invariants is generated by homogeneous elements of degree at most | G |. Schmid, Domokos, and Hegedűs sharpened Noether's bound when G is not cyclic and char F =0. We prove that the sharpened bound holds over general fields: If G is not cyclic and | G | is invertible in F , then the ring of invariants is generated by elements of degree at most 3 4 ·|G| if | G | is even, and at most 5 8 ·|G| if | G | is odd.

Unstable Cohen–Macaulay Algebras

We characterize Cohen-Macaulay algebrasin the category Kfg of unstable Noetherian algebras over the Steenrod algebra via the depth of the P ∗ - invariant ideals . Thisallowsusto trans fer the Cohen-Macaulay property to suitable subalgebras. We apply this to rings of invariants of finite groups and to the P ∗ -inseparable closure. For over 50 years the Steenrod algebra P ∗ and algebras over it have proven to be decisive tools in algebraic topology. Applications occur in cohomology theory, invariant theory of finite groups, and most recently in algebraic geometry. This makes it desirable to have a completely algebraic (and not topological) approach to the subject. In this paper we follow the path begun in (12), (13), (14) and (16) of building a complete P ∗ -invariant commutative algebra. We find a P ∗ -invariant version of the classical characterization of Cohen-Macaulay algebras. This allows us to show that Cohen-Macaulayness is inherited by subalgebras provided that the P ∗ -invariant prime ideal spectra are in bijective correspondence and the extension is integral. This applies in particular to an unstable algebra and its P ∗ -inseparable closure. It also applies to rings of polynomial invariants of finite groups.

Noether's Bound in the Invariant Theory of Finite Groups and Vector Invariants of Iterated Wreath Products of Symmetric Groups

Noether's Bound in the Invariant Theory of Finite Groups and Vector Invariants of Iterated Wreath Products of Symmetric Groups

Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants

AbstractThis paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial p-group over a field of characteristic p, the ring of vector invariants ofmcopies of that representation is not Cohen-Macaulay for m ≥ 3. In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group G and a faithful representation of dimension n with n > 1, if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to n(|G| − 1). If the ring of invariants is a hypersurface, the upper bound can be improved to |G|.

Jacobi Polynomials, Type II Codes, and Designs

Jacobi polynomials were introduced by Ozeki in analogy with Jacobi forms of lattices. They are useful to compute coset weight enumerators, and weight enumerators of children. We determine them in most interesting cases in length at most 32, and in some cases in length 72. We use them to construct group divisible designs, packing designs, covering designs, and (t,r)-designs in the sense of Calderbank-Delsarte. A major tool is invariant theory of finite groups, in particular simultaneous invariants in the sense of Schur, polarization, and bivariate Molien series. A combinatorial interpretation of the Aronhold polarization operator is given. New rank parameters for spaces of coset weight distributions and Jacobi polynomials are introduced and studied here.

E. Noether's bound in the invariant theory of finite groups

E. Noether's bound in the invariant theory of finite groups

On the invariant theory of finite groups: Orbit polynomials and splitting principles

On the invariant theory of finite groups: Orbit polynomials and splitting principles

Counting $G$-extensions by discriminant

The problem of analyzing the number of number field extensions $L/K$ with bounded (relative) discriminant has been the subject of renewed interest in recent years, with significant advances made by Schmidt, Ellenberg-Venkatesh, Bhargava, Bhargava-Shankar-Wang, and others. In this paper, we use the geometry of numbers and invariant theory of finite groups, in a manner similar to Ellenberg and Venkatesh, to give an upper bound on the number of extensions $L/K$ with fixed degree, bounded relative discriminant, and specified Galois closure.