The Cohen–Macaulay property of separating invariants of finite groups

In the case of finite groups, a separating algebra is a subalgebra of the ring of invariants which separates the orbits. Although separating algebras are often better behaved than the ring of invariants, we show that many of the criteria which imply the ring of invariants is non-Cohen–Macaulay actually imply that no graded separating algebra is Cohen–Macaulay. For example, we show that, over a field of positive characteristic p, given sufficiently many copies of a faithful modular representation, no graded separating algebra is Cohen–Macaulay. Furthermore, we show that, for a p-group, the existence of a Cohen–Macaulay graded separating algebra implies the group is generated by bireections. Additionally, we give an example which shows that Cohen–Macaulay separating algebras can occur when the ring of invariants is not Cohen–Macaulay.     

Rings of invariants for modular representations of elementary abelian <Emphasis Type="Italic">p</Emphasis>-groups

We initiate a study of the rings of invariants of modular representations of elementary abelian p-groups. With a few notable exceptions, the modular representation theory of an elementary abelian p-group is wild. However, for a given dimension, it is possible to parameterise the representations. We describe parameterisations for modular representations of dimension two and of dimension three. We compute the ring of invariants for all two-dimensional representations; these rings are generated by two algebraically independent elements. We compute the ring of invariants of the symmetric square of a two-dimensional representation; these rings are hypersurfaces. We compute the ring of invariants for all three-dimensional representations of rank at most three; these rings are complete intersections with embedding dimension at most five. We conjecture that the ring of invariants for any three-dimensional representation of an elementary abelian p-group is a complete intersection.     

Actions of Lie Superalgebras on Reduced Rings

In this paper, we look at the question of whether the subring of invariants is always nontrivial when a finite dimensional Hopf algebra acts on a reduced ring. Affirmative answers where given by Kharchenko for group algebras and by Beidar and Grzeszczuk for finite dimensional restricted Lie algebras. Our main result is Theorem 13 If R is a graded-reduced ring of characteristic p > 2 acted on by a finitely generated restricted K-Lie superalgebra L, then . We can then use Theorem 13 to prove Corollary 15 Let R be a reduced algebra over a field K of characteristic p > 2 acted on by a finite dimensional restricted K-Lie superalgebra L and let H = u(L)#G, where G is the group of order 2 with the natural action on L. If R ^H satisfies a polynomial identity of degree d, then R satisfies a polynomial identity of degree dN, where N is the dimension of H. Presented by Donald S. Passman.     

Vector invariants for the two-dimensional modular representation of a cyclic group of prime order

In this paper, we study the vector invariants of the 2-dimensional indecomposable representation V₂ of the cyclic group, Cp, of order p over a field F of characteristic p, FCp[mV₂]. This ring of invariants was first studied by David Richman (1990) [20] who showed that the ring required a generator of degree m(p−1), thus demonstrating that the result of Noether in characteristic 0 (that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group) does not extend to the modular case. He also conjectured that a certain set of invariants was a generating set with a proof in the case p=2. This conjecture was proved by Campbell and Hughes (1997) in [3]. Later, Shank and Wehlau (2002) in [24] determined which elements in Richman's generating set were redundant thereby producing a minimal generating set.We give a new proof of the result of Campbell and Hughes, Shank and Wehlau giving a minimal algebra generating set for the ring of invariants FCp[mV₂]. In fact, our proof does much more. We show that our minimal generating set is also a SAGBI basis for FCp[mV₂]. Further, our results provide a procedure for finding an explicit decomposition of F[mV₂] into a direct sum of indecomposable Cp-modules. Finally, noting that our representation of Cp on V₂ is as the p-Sylow subgroup of SL₂(Fp), we describe a generating set for the ring of invariants F[mV₂]^SL₂(Fp) and show that (p+m−2)(p−1) is an upper bound for the Noether number, for m>2.     

On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are, respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dimH. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.     

The invariants of modular indecomposable representations of {{\mathbb Z}_{p^2}}

We consider the invariant ring for an indecomposable representation of a cyclic group of order p ² over a field of characteristic p. We describe a set of -algebra generators of this ring of invariants, and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants. This bound, as a polynomial in p, is of degree two.     

When is a ring of torus invariants a polynomial ring?

Let ρ:T→GL(V) be a finite dimensional rational representation of a torus over an algebraically closed fieldk. We give necessary and sufficient conditions on the arrangement of the weights ofV within the character lattice ofT for the ring of invariants,k[V] ^T , to have a homogeneous system of parameters consisting of monomials (Theorem 4.1). Using this we give two simple constructive criteria each of which gives necessary and sufficient conditions fork[V] ^T to be a polynomial ring (Theorem 5.8 and Theorem 5.10). Research supported in part by NSERC Grant OGP 137522     

On nonpositively curved Euclidean submanifolds: splitting results

In this article, we prove that a n–dimensional, non–positively curved Euclidean submanifold with codimension p and with minimal index of relative nullity is (in an open dense subset) locally the product of p hypersurfaces. Received: October 21, 1997     

On a system of basis invariants of the groupE 7

We give a new proof that the Pogorelov ring of the group E₇ coincides with the ring of invariants of this group. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 135–139     

Degree bounds for type-A weight rings and Gelfand–Tsetlin semigroups

A weight ring in type A is the coordinate ring of the GIT quotient of the variety of flags in ℂ ⁿ modulo a twisted action of the maximal torus in SL(n,ℂ). We show that any weight ring in type A is generated by elements of degree strictly less than the Krull dimension, which is at worst O(n ²). On the other hand, we show that the associated semigroup of Gelfand–Tsetlin patterns can have an essential generator of degree exponential in n.     

Quantum hyperbolic invariants for diffeomorphisms of small surfaces

An earlier article [Bonahon, F., Liu, X. B.: Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol., 11, 889-937 (2007)] introduced new invariants for pseudo-Anosov diffeomorphisms of surface, based on the representation theory of the quantum Teichmu¨ller space. We explicitly compute these quantum hyperbolic invariants in the case of the 1-puncture torus and the 4-puncture sphere.     

The Noether numbers for cyclic groups of prime order

The Noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants. We compute the Noether number for an arbitrary representation of a cyclic group of prime order, and as a consequence prove the “2p−3 conjecture.”     

On the Differential Geometric Characterization of The Lee Models

This paper pertains to the J-Hermitian geometry of model domains introduced by Lee (Mich. Math. J. 54(1), 179–206, 2006; J. Reine Angew. Math. 623, 123–160, 2008). We first construct a Hermitian invariant metric on the Lee model and show that the invariant metric actually coincides with the Kobayashi-Royden metric, thus demonstrating an uncommon phenomenon that the Kobayashi-Royden metric is J-Hermitian in this case. Then we follow Cartan’s differential-form approach and find differential-geometric invariants, including torsion invariants, of the Lee model equipped with this J-Hermitian Kobayashi-Royden metric, and present a theorem that characterizes the Lee model by those invariants, up to J-holomorphic isometric equivalence. We also present an all dimensional analysis of the asymptotic behavior of the Kobayashi metric near the strongly pseudoconvex boundary points of domains in almost complex manifolds.     

Invariants of PSL2(F11)Acting on C5

after 92:consult AMS membership list In this paper we give enerators and relations for the ring of invariants of an ducible dimensional complex representation of the finite simple group PSL ₂(F₁₁). We also discuss the geometry of the invariants.

“Es ist nicht meine Absicht, alle ungeandert bleibenden ganzen Funktionen der y zu mitzuteilen; dies wurde jedenfalls eine weitldufige and vielleicht eine schwierige Aufgabe sein.”

Felix Klein ([K],§4, p.146 of Ges. Math. Abh., Bd. 3)     

Matroids and quotients of spheres

For any linear quotient of a sphere, where is an elementary abelian p–group, there is a corresponding representable matroid which only depends on the isometry class of X. When p is 2 or 3 this correspondence induces a bijection between isometry classes of linear quotients of spheres by elementary abelian p–groups, and matroids representable over Not only do the matroids give a great deal of information about the geometry and topology of the quotient spaces, but the topology of the quotient spaces point to new insights into some familiar matroid invariants. These include a generalization of the Crapo–Rota critical problem inequality and an unexpected relationship between and whether or not the matroid is affine. Received: 7 February 2001; in final form: 30 October 2001/ Published online: 29 April 2002     

Nonlinear subelliptic equations

A finitely generated quadratic module or preordering in the real polynomial ring is called stable, if it admits a certain degree bound on the sums of squares in the representation of polynomials. Stability, first defined explicitly in Powers and Scheiderer (Adv Geom 1, 71–88, 2001), is a very useful property. It often implies that the quadratic module is closed; furthermore, it helps settling the Moment Problem, solves the Membership Problem for quadratic modules and allows applications of methods from optimization to represent nonnegative polynomials. We provide sufficient conditions for finitely generated quadratic modules in real polynomial rings of several variables to be stable. These conditions can be checked easily. For a certain class of semi-algebraic sets, we obtain that the nonexistence of bounded polynomials implies stability of every corresponding quadratic module. As stability often implies the non-solvability of the Moment Problem, this complements the result from Schmüdgen (J Reine Angew Math 558, 225–234, 2003), which uses bounded polynomials to check the solvability of the Moment Problem by dimensional induction. We also use stability to generalize a result on the Invariant Moment Problem from Cimpric et al. (Trans Am Math Soc, to appear).     

An <Emphasis Type="Italic">R</Emphasis> = <Emphasis Type="Italic">T</Emphasis> theorem for imaginary quadratic fields

We prove the modularity of certain residually reducible p-adic Galois representations of an imaginary quadratic field assuming the uniqueness of the residual representation. We obtain an R = T theorem using a new commutative algebra criterion that might be of independent interest. To apply the criterion, one needs to show that the quotient of the universal deformation ring R by its ideal of reducibility is cyclic Artinian of order no greater than the order of the congruence module T/J, where J is an Eisenstein ideal in the local Hecke algebra T. The inequality is proven by applying the Main conjecture of Iwasawa Theory for Hecke characters and using a result of Berger [Compos Math 145(3):603–632, 2009]. This strengthens our previous result [Berger and Klosin, J Inst Math Jussieu 8(4):669–692, 2009] to include the cases of an arbitrary p-adic valuation of the L-value, in particular, cases when R is not a discrete valuation ring. As a consequence we show that the Eisenstein ideal is principal and that T is a complete intersection.     

Galois ring extensions and localized modular rings of invariants of p-groups

We apply recent results on Galois-ring extensions and trace surjective algebras to analyze dehomogenized modular invariant rings of finite p-groups, as well as related localizations. We describe criteria for the dehomogenized invariant ring to be polynomial or at least regular and we show that for regular affine algebras with possibly non-linear action by a p-group, the singular locus of the invariant ring is contained in the variety of the transfer ideal. If V is the regular module of an arbitrary finite p-group, or V is any faithful representation of a cyclic p-group, we show that there is a suitable invariant linear form, inverting which renders the ring of invariants into a “localized polynomial ring” with dehomogenization being a polynomial ring. This is in surprising contrast to the fact that for a faithful representation of a cyclic group of order larger than p, the ring of invariants itself cannot be a polynomial ring by a result of Serre. Our results here generalize observations made by Richman [R] and by Campbell and Chuai [CCH].     

Pseudoconvex regions of finite D’Angelo type in four dimensional almost complex manifolds

Let D be a J-pseudoconvex region in a smooth almost complex manifold (M, J) of real dimension four. We construct a local peak J-plurisubharmonic function at every point p ∈ bD of finite D’Angelo type. As applications we give local estimates of the Kobayashi pseudometric, implying the local Kobayashi hyperbolicity of D at p. In case the point p is of D’Angelo type less than or equal to four, or the approach is nontangential, we provide sharp estimates of the Kobayashi pseudometric.     

Density of Lipschitz mappings in the class of Sobolev mappings between metric spaces

We prove that Lipschitz mappings are dense in the Newtonian–Sobolev classes N ^1,p (X, Y) of mappings from spaces X supporting p-Poincaré inequalities into a finite Lipschitz polyhedron Y if and only if Y is [p]-connected, π ₁(Y) = π ₂(Y) = · · · = π _[p](Y) = 0, where [p] is the largest integer less than or equal to p. This work was supported by the NSF grant DMS-0500966.