In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$.

We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a product of linear factors involving generalised exponents. Our refinement gives a uniform proof and generalisation of a recent theorem of the second author.

This paper concerns free function theory. Free maps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of $g$ noncommuting variables is a function on $g$-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series.

In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invariant-theoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involution-free counterparts.

Harm Derksen made a conjecture concerning degree bounds for the syzygies of rings of polynomial invariants in the non-modular case [Degree bounds for syzygies of invariants, Adv. Math. 185 (2004), 207–214]. We provide counterexamples to this conjecture, but also prove a slightly weakened version. We also prove some general results that give degree bounds on the homology of complexes and of $\text{Tor}\,$ groups.

It is known that there exists a canonical system for every finite real reflection group. In a previous paper, the first and the third authors obtained an explicit formula for a canonical system. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

Abstract Let G be a linear algebraic group over an algebraically closed field 𝕜 acting rationally on a G-module V with its null-cone. Let δ(G, V) and σ(G, V) denote the minimal number d such that for every and , respectively, there exists a homogeneous invariant f of positive degree at most d such that f(v) ≠ 0. Then δ(G) and σ(G) denote the supremum of these numbers taken over all G-modules V. For positive characteristics, we show that δ(G) = ∞ for any subgroup G of GL2(𝕜) that contains an infinite unipotent group, and σ(G) is finite if and only if G is finite. In characteristic zero, δ(G) = 1 for any group G, and we show that if σ(G) is finite, then G0 is unipotent. Our results also lead to a more elementary proof that βsep(G) is finite if and only if G is finite.

We study bounds on nilpotence in H*(BG), the mod p cohomology of the classifying space of a compact Lie group G. Part of this is a report of our previous work on this problem, updated to reflect the consequences of Peter Symonds's recent verification of Dave Benson's Regularity Conjecture. New results are given for finite p-groups, leading to good bounds on nilpotence in H*(BP) determined by the subgroup structure of the p-group P.

For a split semisimple Chevalley group scheme G with Lie algebra over an arbitrary base scheme S, we consider the quotient of by the adjoint action of G. We study in detail the structure of over S. Given a maximal torus T with Lie algebra and associated Weyl group W, we show that the Chevalley morphism π : /W → /G is an isomorphism except for the group Sp2n over a base with 2-torsion. In this case this morphism is only dominant and we compute it explicitly. We compute the adjoint quotient in some other classical cases, yielding examples where the formation of the quotient → /G commutes, or does not commute, with base change on S.

We formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial function algebras on matrices are replaced by free algebras, while the universal cosovereign Hopf algebras play the role of the general linear group.

AMS 2000 Mathematics subject classification: Primary 16W30

For most of the finite subgroups of $\text{SL(3,}\,\text{C)}$ we give explicit formulae for the Molien series of the coinvariant algebras, generalizing McKay's formulae $\text{ }\!\![\!\!\text{ McKay99 }\!\!]\!\!\text{ }$ for subgroups of $\text{SU(2)}$. We also study the $G $-orbit Hilbert scheme $\text{Hil}{{\text{b}}^{G}}({{\mathbf{C}}^{3}})$ for any finite subgroup $G $ of $\text{SO(3)}$, which is known to be a minimal (crepant) resolution of the orbit space ${{\mathbf{C}}^{3}}/G$ . In this case the fiber over the origin of the Hilbert-Chow morphism from $\text{Hil}{{\text{b}}^{G}}({{\mathbf{C}}^{3}})$ to ${{\mathbf{C}}^{3}}/G$ consists of finitely many smooth rational curves, whose planar dual graph is identified with a certain subgraph of the representation graph of $G $. This is an $\text{SO(3)}$ version of the McKay correspondence in the $\text{SU(2)}$ case.

Let π : X [xlarr ] Y be a good quotient of a smooth variety X by a reductive algebraic group G and 1[les ]k≤ dim (Y) an integer. We prove that if, locally, any invariant horizontal differential k-form on X (resp. any regular differential k-form on Y) is a Kähler differential form on Y then codim(Ysing)>k+1. We also prove that the dualizing sheaf on Y is the sheaf of invariant horizontal dim(Y)-forms.

Let $X=\operatorname{Spec} B$ be an affine varietyover a field of arbitrary characteristic, and supposethat there exists an action of a unipotent group(possibly neither smooth nor connected). Thefundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action.Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of aGalois extension, $k\hookrightarrow B=K$, which iscyclic of degree $p=\text{ch} (k)$ (that is, such thattheunipotent group is $G={\Bbb Z}/p {\Bbb Z}$),an element of minimal degree becomes anArtin--Schreier radical, and the method for computing invariants gives, in particular, theexpression for any element of $K$ in terms of theseradicals, with an explicit formula. This replacesthe well-known formula of Lagrange (which is validonly when the degree of the extension and thecharacteristic are relatively prime) in the case of an extension of degree $p=\text{ch}(k)$. (2)In this paper we give an effectiveconstruction of a stable open subset where there is a quotient. In this sense we obtain an algebraic localcriterion for the existence of a quotient in aneighbourhood. It is proved (provided the variety isnormal) that, in the following cases, such an openset is the greatest one that admits a quotient:\begin{enumerate}\item[(a)] when the action is such that the orbits havedimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any actionof the additive group $G_a$;\item[(b)] in characteristic 0, when the action is proper(obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification:primary 14L30; secondary 14D25, 14D20.

We construct a correspondence between the set of partitions of a finite set M and the set of pairs of walks to the same vertex on a graph giving the Bratteli diagram of the partition algebra on M. This is the precise analogue of the correspondence between the set of permutations of a finite set and the set of pairs of Young tableaux of the same shape, called the Robinson–Schensted correspondence.

We consider the rings of invariants RG, where R is the symmetric algebra of a tensor product between two vector spaces over the field Fp andG = Un × Um. A polynomial algebra is constructed and these invariants provide Chern classes for the modular cohomology of Un+m.

Let V be a vector space of dimension n over a field k of characteristic p. Let G ⊆ Gl(V) be a finite group with p-Sylow subgroup P. G and P act on the symmetric algebra R of V. Denote the respective rings of invariants by RG and Rp. We show that if Rp is Cohen-Macaulay (CM) so also is RG, generalizing a result of M. Hochster and J. A. Eagon. If P is normal in G and G is generated by P and pseudo-reflections, we show that if RG is CM so also is Rp. However, in general, RG may even be polynomial with Rp not CM. Finally, we give a procedure for determining a set of generators for RG given a set of generators for Rp.