G-invariant Polynomials Research Articles

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.

Flat connections and cohomology invariants

The main goal of this article is to construct some geometric invariants for the topology of the set F of flat connections on a principal G-bundle P→M. Although the characteristic classes of principal bundles are trivial when F≠∅, their classical Chern–Weil construction can still be exploited to define a homomorphism from the set of homology classes of maps S→F to the cohomology group H2r−k(M,R), where S is null-cobordant (k−1)-manifold, once a G-invariant polynomial p of degree r on Lie(G) is fixed. For S=Sk−1, this gives a homomorphism πk−1(F)→H2r−k(M,R). The map is shown to be globally gauge invariant and furthermore it descends to the moduli space of flat connections F/ Gau P, modulo cohomology with integer coefficients. The construction is also adapted to complex manifolds. In this case, one works with the set F0,2 of connections with vanishing (0, 2)-part of the curvature, and the Dolbeault cohomology. Some examples and applications are presented.

Separation Theorems for Group Invariant Polynomials

We study the existence of separation theorems by polynomials that are invariant under a group action. We show that if G is a finite subgroup of \(\textit{GL}(n,{\mathbb {C}})\), K is a set in \({\mathbb {C}}^{n}\) that is invariant under the action of G and z is a point in \({\mathbb {C}}^{n}\setminus K\) that can be separated from K by a polynomial Q, then z can be separated from K by a G-invariant polynomial P. Furthermore, if Q is homogeneous then P can be chosen to be homogeneous. As a particular case, if K is a symmetric polynomially convex compact set in \({\mathbb {C}}^{n}\) and \(z\notin K\) then there exists a symmetric polynomial that separates z and K.

Constructing the SU(2) × U(1) Orbit Space for Qutrit Mixed States

The orbit space of the group G := SU(2) × U(1) ⊂ U(3) acting adjointly on the state space of a three-level quantum system is discussed. The semialgebraic structure of is determined within the Procesi–Schwarz method. Using an integrity basis for the ring of G-invariant polynomials , the set of constraints on the Casimir invariants of the group U(3) coming from the positivity requirement Grad(z) ≥ 0 for the Procesi–Schwarz gradient matrix is analyzed in detail.

Note on Cubature Formulae and Designs Obtained from Group Orbits

Abstract In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

Flat bases of invariant polynomials and P̂-matrices of E7 and E8

Let G be a compact group of linear transformations of a Euclidean space V. The G-invariant C∞ functions can be expressed as C∞ functions of a finite basic set of G-invariant homogeneous polynomials, sometimes called an integrity basis. The mathematical description of the orbit space V/G depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semidefiniteness conditions of the P̂-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of G-invariant homogeneous polynomials forming an integrity basis is not unique, so it is not unique the mathematical description of the orbit space too. If G is an irreducible finite reflection group, Saito et al. [Commun. Algebra 8, 373 (1980)] characterized some special basic sets of G-invariant homogeneous polynomials that they called flat. They also found explicitly the flat basic sets of invariant homogeneous polynomials of all the irreducible finite reflection groups except of the two largest groups E7 and E8. In this paper the flat basic sets of invariant homogeneous polynomials of E7 and E8 and the corresponding P̂-matrices are determined explicitly. Using the results here reported one is able to determine easily the P̂-matrices corresponding to any other integrity basis of E7 or E8. From the P̂-matrices one may then write down the equations and inequalities defining the orbit spaces of E7 and E8 relatively to a flat basis or to any other integrity basis. The results here obtained may be employed concretely to study analytically the symmetry breaking in all theories where the symmetry group is one of the finite reflection groups E7 and E8 or one of the Lie groups E7 and E8 in their adjoint representations.

Invariant functions in Denjoy–Carleman classes

Let V be a real finite dimensional representation of a compact Lie group G. It is well known that the algebra \({\mathbb{R}[V]^G}\) of G-invariant polynomials on V is finitely generated, say by σ 1, . . . , σ p . Schwarz (Topology 14:63–68, 1975) proved that each G-invariant C ∞-function f on V has the form f = F(σ 1, . . . , σ p ) for a C ∞-function F on \({\mathbb{R}^p}\). We investigate this representation within the framework of Denjoy–Carleman classes. One can in general not expect that f and F lie in the same Denjoy–Carleman class C M (with M = (M k )). For finite groups G and (more generally) for polar representations V, we show that for each G-invariant f of class C M there is an F of class C N such that f = F(σ 1, . . . , σ p ), if N is strongly regular and satisfies$$\sup_{k \in \mathbb{N}_{ > 0}}\left(\frac{M_{km}}{N_k}\right)^{\frac{1}{k}} < \infty,$$where m is an (explicitly known) integer depending only on the representation. In particular, each G-invariant (1 + δ)-Gevrey function f (with δ > 0) has the form f = F(σ 1, . . . , σ p ) for a (1 + δm)-Gevrey function F. Applications to equivariant functions and basic differential forms are given.

Chern–Simons forms on associated bundles, and boundary terms

Let E be a principle bundle over a compact manifold M with compact structural group G. For any G-invariant polynomial P, the transgressive forms $$TP(\omega)$$ defined by Chern and Simons in (Ann. Math. 99:48–69, 1974) are shown to extend to forms $$\Phi P(\omega)$$ on associated bundles B with fiber a quotient F = G/H of the group. These forms satisfy a heterotic formula $$d\Phi P(\omega)=P(\Omega)-P(\Psi),$$ relating the characteristic form $$P(\Omega)$$ to a fiber-curvature characteristic form. For certain natural bundles B, $$P(\Psi)=0$$ , giving a true transgressive form on the associated bundle, which leads to the standard obstruction properties of characteristic classes as well as natural expressions for boundary terms. These forms also yield new secondary characteristic classes giving refined information about the associated bundles B.

About the cover: Invariants

It carries a natural action of G = SL2(C) via linear substitutions of the variables. In 1868 P. Gordan proved that the algebra C[V ] of G-invariant polynomials on V has a finite basis, that the module of algebraic relations between the generators (called syzygies) is also finitely generated, and provided explicit upper bounds for the degrees of the generators and relations. Previously, Cayley (1821-1895) had solved the problem for forms of degree 3 and 4 in two variables.

On the Dimension of Coinvariants of Permutation Representations

Let \(f (Z) \in {\Bbb F}[Z]\) be a univariate, separable polynomial of degree n with roots x1,…,xn in some algebraic closure \(\bar{\Bbb F}\) of the ground field \({\Bbb F}\). It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on \({\Bbb F}^n\) by permutation of the coordinates and each \(P \in {\Bbb F}[X_1, \ldots,X_n]^G\) defines a relation P − P(x1,…,xn) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants \({\Bbb F}[X_1,\ldots,X_n]_G\) has dimension ‖G‖ over \({\Bbb F}\). To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σn acting via the tautological permutation representation.

On polarizations in invariant theory

Given a reductive algebraic group G and a finite dimensional algebraic G-module V, we study how close is the algebra of G-invariant polynomials on V ⊕ n to the subalgebra generated by polarizations of G-invariant polynomials on V. We address this problem in a more general setting of G-actions on arbitrary affine varieties.

Constructive Axiomatic Approach to the Determination of the Orbit Spaces of Coregular Compact Linear Groups

We review the proposal of a constructive axiomatic approach to the determination of the orbit spaces of all the real compact linear groups, obtained through the computation of a metric matrix \(\widehat{P}(p)\) , which is defined only in terms of the scalar products between the gradients ∂p1(x),. . .,∂pq(x) of the elements of a minimal integrity basis (MIB) for the ring ℝ[ℝn]G of G-invariant polynomials. The domain of semi-positivity of \(\widehat{P}(p)\) is known to realize the orbit space ℝn/G of G as a semi-algebraic variety in the space ℝq spanned by the variables p1,. . .,pq.

Symmetric Surfaces with Many Singularities#

Let G ⊂ SO(4) denote a finite subgroup containing the Heisenberg group. In this paper we classify all such groups, we find the dimension of the spaces of G-invariant polynomials and we give equations for the generators whenever the space has dimension two. Then we complete the study of the corresponding G-invariant pencils of surfaces in ℙ3 which we started in Sarti [Sarti, A. (2000). Pencils of symmetric surfaces in ℙ3(C). J. Algebra 246:429–452]. It turns out that we have five more pencils, two of them containing surfaces with nodes.

Lie-Poincaré transformations and a reduction criterion in Landau theory

In the Landau theory of phase transitions one considers an effective potential Φ whose symmetry group G and degree d depend on the system under consideration; generally speaking, Φ is the most general G-invariant polynomial of degree d. When such a Φ turns out to be too complicate for a direct analysis, it is essential to be able to drop unessential terms, i.e., to apply a simplifying criterion. Criteria based on singularity theory exist and have a rigorous foundation, but are often very difficult to apply in practice. Here we consider a simplifying criterion (as stated by Gufan) and rigorously justify it on the basis of classical Lie-Poincaré theory as far as one deals with fixed values of the control parameter(s) in the Landau potential; when one considers a range of values, in particular near a phase transition, the criterion has to be accordingly partially modified, as we discuss. We consider some specific cases of group G as examples, and study in detail the application to the Sergienko–Gufan–Urazhdin model for highly piezoelectric perovskites.

Finite-dimensional representations of invariant differential operators, II

In Part I of this paper [G.W. Schwarz, Finite-dimensional representations of invariant differential operators, J. Algebra 258 (2002) 160–204] we considered the representation theory of the algebra B:=D(g)G, where G=SL3(C) and D(g)G denotes the algebra of G-invariant polynomial differential operators on the Lie algebra g of G. We also considered the representation theory of the subalgebra A of B, where A is generated by the invariant functions O(g)G⊂B and the invariant constant coefficient differential operators S(g)G⊂B. Among other things, we found that the finite-dimensional representations of A and B are completely reducible, and we could reduce the study of the finite-dimensional irreducible representations of B to those of A. Irreducible finite-dimensional representations of A are quotients of “Verma modules.” We found sufficient conditions for the irreducible quotients of Verma modules to be finite-dimensional, and we conjectured that these sufficient conditions are also necessary. In this paper we establish the conjecture, giving a complete classification of the finite-dimensional representations of A and B.

Milnor fiber complexes for the exceptional Shephard groups

The symmetry group of a regular real polytope is a finite Coxeter group. The intersection of the unit sphere with the reflecting hyperplanes of the corresponding Coxeter arrangement induces a simplicial triangulation of the sphere, called the Coxeter complex. A Shephard group G is the symmetry group of a regular complex polytope. Orlik has shown that there can be associated to G a real simplicial complex which possesses many properties analogous to the Coxeter complex. Let f 1 be the G-invariant polynomial of minimal positive degree, and let F= f 1 −1(1) be its Milnor fiber. Orlik showed that there is a complex Γ which is an equivariant strong deformation retract of F, is G-stable, is stratified by the associated reflection arrangement, and satisfies specific cell-counting formulas. His proof is existential; it does not give an explicit method of constructing the complex, though Orlik and Solomon did explicitly construct complexes for one infinite family of Shephard groups. Here we describe the construction of complexes for the remaining 15 “exceptional” Shephard groups.

A note on the complete integrability of the Hitchin system

We consider the Hitchin map \( {\cal H}\!: T^{*}{\cal M}_{G}\longrightarrow {\cal K},\) defined on the cotangent bundle of the moduli space of stable principal G-bundles over a compact Riemann surface, G being any reductive complex Lie group. In [5] Hitchin showed that the \(m=\dim{\cal M}_{G}\) components of \(\cal H\) are analytic functions that commute with respect to the natural Poisson structure on \(T^{*}{\cal M}_{G}\). In order to do this, Hitchin used a description of \(T^{*}{\cal M}_{G}\) as a Marsden-Weinstein quotient \(\mu ^{-1}(0)/{\cal G}, \mu \) being the moment map for the action of the Gauge group \(\cal G\) on the infinite dimensional space of stable holomorphic structures on a fixed principal G-bundle. In this note we obtain the same result as an application of a few elementary properties satisfied by the homogeneous Ad G-invariant polynomials on the Lie algebra of G.

On symmetric semialgebraic sets and orbit spaces

For a symmetric (= invariant under the action of a compact Lie group G) semialgebraic basic set C, described by s polynomial inequalities, we show, that C can also be written by s + 1 G-invariant polynomials. We also describe orbit spaces for the action o

Computing Bases for Rings of Permutation-invariant Polynomials

Let R be a commutative ring with 1, let R [ X 1,…, X n ] be the polynomial ring in X 1,…, X n over R and let G be an arbitrary group of permutations of { X 1,…, X n }. The paper presents an algorithm for computing a small finite basis B of the R-algebra of G-invariant polynomials and a polynomial representation of an arbitrary G-invariant polynomial in R [ X 1,…, X n ] as a polynomial in the polynomials of the finite basis B. The algorithm works independently of the ground ring R, and the basis B contains only polynomials of total degree ≤ max{ n, n( n - 1)/2}, independent of the size of the permutation group G.

G-invariant polynomial extensions of Lie algebras in quantum many-body physics

A new class of Lie-algebraic structures gd is revealed in some multi-particle processes of quantum physics having internal symmetry groups Ginv. They are extensions of some Lie algebras h (via coset construction) by G-invariant h-tensors v which are polynomials in boson operators. Applications of these algebras are briefly discussed for solving spectral and evolution tasks with gd as their dynamic symmetry algebras.