We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group-based models such as the Jukes-Cantor and Kimura models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices. The main novelty is that our results yield generators of the full ideal rather than an ideal which only defines the model set-theoretically. Set-up and theoremsIn phylogenetics, tree models have been introduced to describe the evolution of a number of species from a distant common ancestor. Given suitably aligned strings of nucleotides of n species alive today, one assumes that the individual positions in J. Draisma has been supported by DIAMANT, an NWO mathematics cluster and J. Kuttler by an NSERC Discovery Grant.

Let G be a semi-simple algebraic group over C, B a Borel subgroup of G and T a maximal torus in B. A beautiful unpublished result of Dale Peterson says that if G is simply laced, then every rationally smooth point of a Schubert variety X in G/B is nonsingular in X. The purpose of this paper is to generalize this result to arbitrary T -stable subvarieties of G/B, the only restriction being that G contains no G 2 factors. A key idea in Peterson's proof is to deform the tangent space Ty(X) to X at a nonsingular T -fixed point y along the orbit of y under a root subgroup in B, which is open in a T -invariant curve C (we say a T -curve) in X. In more generality, if a T -variety X is nonsingular along the open T -orbit in a T -curve C and x ∈ C T , we may consider the limit τ C (X, x) of the tangent spaces Tz(X) as z approaches x along C. We call τ C (X, x) the Peterson translate of X at x along C. Peterson showed that a Schubert variety X in G/B, where G is semi-simple, is nonsingular at x ∈ X T as long as all τ C (X, x) coincide for all such good T -curves..Our first result generalizes this theorem to any irreducible T -variety X, provided the fixed point x is attractive under much weaker hypotheses. We then prove that if X is a T -variety in G/B where G contains no G 2 factors, then every Peterson translate τ C (X, x) is contained in the linear span Θx(X) of the reduced tangent cone to X at x. (This fails when G = G 2 .) Combining these two results leads to our characterization of the smooth T -fixed points of such X. In particular, we show that if G is simply laced, then X is nonsingular at a T -fixed point x if and only if it is rationally smooth at x and x lies on at least two good T -curves Peterson's ADE result is an immediate consequence of this. In addition, we obtain a uniform description of the nonsingular T -fixed points of a Schubert variety in G/B modulo the G 2 restriction. In particular, a Schubert variety X in such a G/B is nonsingular if and only if all the reduced tangent cones of X are linear.Finally, we also obtain versions of the above results for all algebraic homogeneous spaces G/P modulo the G 2 restrictions.

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this statement for tensors of arbitrary dimension, where matrices correspond to two-dimensional tensors. More specifically, we prove that for each k there exists an upper bound d = d(k) such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d, regardless of the dimension of the tensor and regardless of its sizes in each dimension. Our proof involves passing to an infinite-dimensional limit of tensor powers of a vector space, whose elements we dub infinite-dimensional tensors, and exploiting the symmetries of this limit in crucial way.Comment: Corrected the formulation of Corollary 1.4---thanks to Sonja Petrovic

We show that a linear subspace of a reductive Lie algebra g that consists of nilpotent elements has dimension at most 1 2 (dim g−rk g), and that any nilpotent subspace attaining this upper bound is equal to the nilradical of a Borel subalgebra of g. This generalizes a classical theorem of Gerstenhaber, which states this fact for the algebra of (n×n)-matrices. Results and methodA classical theorem of Gerstenhaber [Ger58] states that any vector space consisting of nilpotent (n × n)-matrices has dimension at most n 2 , and that any such space attaining this maximal possible dimension is conjugate to the space of upper triangular matrices. We will prove the following generalization of this result. The difficult part of the theorem is the statement that the subspaces consisting of ad-nilpotent elements of maximal dimension d := 1 2 (dim g − rk g) are nilpotent subalgebras and are therefore all conjugate. The maximality assumption is essential here: there exist subspaces consisting of ad-nilpotent elements that are not contained in nilpotent subalgebras (see Examples 1-A.2 below). The classification of these 'nilpotent subspaces' is widely open, even in the case of (n × n)-matrices. For some partial results see [Ger59a,Ger59b,Ger62,Fas97,CRT98, MOR91].Gerstenhaber proves his theorem under the mild assumption that the ground field K has at least n elements. We will prove a version of the Main Theorem which holds in arbitrary characteristic (see Theorem 1), but still assumes that K is algebraically closed. To formulate this more technical result, let G be a connected reductive algebraic group over K and denote its Lie algebra by g. Fix a Borel subgroup B of G and a maximal torus

Let G be a reductive algebraic group over an algebraically closed field K of characteristic zero. Let π : g r → X = g r / /G be the categorical quotient where g is the adjoint representation of G and r is a suitably large integer (in general r ≥ 5, but for many cases r ≥ 3 or even r ≥ 2 suffices). We show that every automorphism ϕ of X lifts to a map Φ : g r → g r commuting with π. As an application we consider the action of ϕ on the Luna stratification of X.

Let G → GL(V ) will be a finite-dimensional linear representation of a reductive linear algebraic group G on a finite-dimensional vector space V , defined over an algebraically closed field of characteristic zero. The categorical quotient V // G carries a natural stratification, due to D. Luna. This paper addresses the following questions:(i) Is the Luna stratification of X intrinsic? That is, does every automorphism of V // G map each stratum to another stratum?(ii) Are the individual Luna strata in X intrinsic? That is, does every automorphism of V // G maps each stratum to itself?In general, the Luna stratification is not intrinsic. Nevertheless, we give positive answers to questions (i) and (ii) for large classes of interesting representations.

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Let G be a semi-simple algebraic group over C, B a Borel subgroup of G, T a maximal torus in B and P a parabolic in G containing B. In a previous work [7], the authors classified the singular T -fixed points of an irreducible T -stable subvariety X of the generalized flag variety G/P . It turns out that under the restriction that G doesn't contain any G2-factors, the key geometric invariant determining the singular T -fixed points of X is the linear span Θx(X) of the reduced tangent cone to X at a T -fixed point x. The goal of this paper is to describe this invariant at the maximal singular T -fixed points when X a Schubert variety in G/B and G doesn't contain any G2-factors. We first decribe Θx(X) solely in terms of Peterson translates, which were the main tool in [7]. Then, taking a further look at the Peterson translates (with the G2-restriction),we are able to describe Θx(X) in terms of its isotropy submodule and the Bruhat graph of X at x. This refinement gives a purely root theoretic description, which should be useful for computations. Finally, still with the G2-restriction, these considerations lead us to a non-recursive algorithm for X's singular locus solely involving only the root system of (G, T ) and the Bruhat graph of X. IntroductionLet G be a semi-simple algebraic group over an arbitrary algebraically closed field k, and suppose T ⊂ B ⊂ P are respectively a maximal torus, a Borel subgroup and an arbitrary standard parabolic in G. Each G/P , including G/B, is a projective G-variety with only finitely many B-orbits. Every B-orbit contains a unique T -fixed point x ∈ (G/P ) T , and these cells define an affine paving of G/P . If x ∈ (G/P ) T , then the closure of the B-orbit Bx is called the Schubert variety in G/P associated to x. This Schubert variety will be denoted throughout by X(x). We will use the well known fact that the T -fixed points in G/B are in one to one correspondence with the elements of the Weyl group W = N G (T )/T , so we don't distinguish between elements of W and fixed points in G/B.Schubert varieties are in general singular, and it's an old problem, inspired by a classical paper [8] of Chevalley, to describe their singular loci (or, equivalently, their smooth points). A related problem, with interesting consequences in representation theory, is to determine The first author was partially supported by the Natural Sciences and Engineering Research Council of Canada The second author was partially supported by the SNF (Schweizerischer Nationalfonds) 2 the locus of rationally smooth points of a Schubert variety (cf.[9]). In fact, if G is defined over C and simply laced (i.e. every simple factor is of type A, D or E), then all rationally smooth points of any Schubert variety in G/P are in fact smooth (see [7]). In this paper, we consider the singular locus of a Schubert variety in an arbitrary G/P , where G does not contain any G 2 -factors. Our results are an outgrowth of [7], where we used Peterson translates (defined below) to characterize the T -fixed points in the sing...