Mumford conjecture
Encyclopedia
In mathematics
Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple
algebraic group
G over a field
K, and for any linear representation ρ of G on a K-vector space
V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial
F on V such that
The polynomial can be taken to be homogeneous
, in other words an element of a symmetric power of the dual of V, and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p.
When K has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of G implies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p was proved by W. J. , about a decade after the problem had been posed by David Mumford
, in the introduction to the first edition of his book Geometric Invariant Theory.
from characteristic 0, where they were already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra is also finitely generated.
Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1 on Y).
C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.
It follows from the work of , Haboush, and Popov that the following conditions are equivalent for an affine algebraic group G over a field K:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
Haboush's theorem, often still referred to as the Mumford conjecture, states that for any semisimple
Semisimple algebraic group
In mathematics, especially in the areas of abstract algebra and algebraic geometry studying linear algebraic groups, a semisimple algebraic group is a type of matrix group which behaves much like a semisimple Lie algebra or semisimple ring.- Definition :...
algebraic group
Algebraic group
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety...
G over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, and for any linear representation ρ of G on a K-vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
V, given v ≠ 0 in V that is fixed by the action of G, there is a G-invariant polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
F on V such that
- F(v) ≠ 0.
The polynomial can be taken to be homogeneous
Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose monomials with nonzero coefficients all have thesame total degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial...
, in other words an element of a symmetric power of the dual of V, and if the characteristic is p>0 the degree of the polynomial can be taken to be a power of p.
When K has characteristic 0 this was well known; in fact Weyl's theorem on the complete reducibility of the representations of G implies that F can even be taken to be linear. Mumford's conjecture about the extension to prime characteristic p was proved by W. J. , about a decade after the problem had been posed by David Mumford
David Mumford
David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
, in the introduction to the first edition of his book Geometric Invariant Theory.
Applications
Haboush's theorem can be used to generalize results of geometric invariant theoryGeometric invariant theory
In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
from characteristic 0, where they were already known, to characteristic p>0. In particular Nagata's earlier results together with Haboush's theorem show that if a reductive group (over an algebraically closed field) acts on a finitely generated algebra then the fixed subalgebra is also finitely generated.
Haboush's theorem implies that if G is a reductive algebraic group acting regularly on an affine algebraic variety, then disjoint closed invariant sets X and Y can be separated by an invariant function f (this means that f is 0 on X and 1 on Y).
C.S. Seshadri (1977) extended Haboush's theorem to reductive groups over schemes.
It follows from the work of , Haboush, and Popov that the following conditions are equivalent for an affine algebraic group G over a field K:
- G is reductive (its unipotent radical is trivial).
- For any non-zero invariant vector in a rational representation on G, there is an invariant homogeneous polynomial that does not vanish on it.
- For any finitely generated K algebra acted on rationally by G, the algebra of fixed elements is finitely generated.
Proof
The theorem is proved in several steps as follows:- We can assume that the group is defined over an algebraically closed field K of characteristic p>0.
- Finite groups are easy to deal with as one can just take a product over all elements, so one can reduce to the case of connected reductive groups (as the connected component has finite index). By taking a central extension which is harmless one can also assume the group G is simply connected.
- Let A(G) be the coordinate ring of G. This is a representation of G with G acting by left translations. Pick an element v′ of the dual of V that has value 1 on the invariant vector v. The map V to A(G) by sending w∈V to the element a∈A(G) with a(g) = v′(g(w)). This sends v to 1∈A(G), so we can assume that V⊂A(G) and v=1.
- The structure of the representation A(G) is given as follows. Pick a maximal torus T of G, and let it act on A(G) by right translations (so that it commutes with the action of G). Then A(G) splits as a sum over characters λ of T of the subrepresentations A(G)λ of elements transforming according to λ. So we can assume that V is contained in the T-invariant subspace A(G)λ of A(G).
- The representation A(G)λ is an increasing union of subrepresentations of the form Eλ+nρ⊗Enρ, where ρ is the Weyl vector for a choice of simple roots of T, n is a positive integer, and Eμ is the space of sections of the line bundleLine bundleIn mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
over G/B corresponding to a character μ of T, where B is a Borel subgroupBorel subgroupIn the theory of algebraic groups, a Borel subgroup of an algebraic group G is a maximal Zariski closed and connected solvable algebraic subgroup.For example, in the group GLn ,...
containing T. - If n is sufficiently large then Enρ has dimension (n+1)N where N is the number of positive roots. This is because in characteristic 0 the corresponding module has this dimension by the Weyl character formulaWeyl character formulaIn mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It was proved by ....
, and for n large enough that the line bundle over G/B is very ample, Enρ has the same dimension as in characteristic 0. - If q=pr for a positive integer r, and n=q−1, then Enρ contains the Steinberg representationSteinberg representationIn mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field...
of G(Fq) of dimension qN. (Here Fq ⊂ K is the finite field of order q.) The Steinberg representation is an irreducible representation of G(Fq) and therefore of G(K), and for r large enough it has the same dimension as Enρ, so there are infinitely many values of n such that Enρ is irreducible. - If Enρ is irreducible it is isomorphic to its dual, so Enρ⊗Enρ is isomorphic to End(Enρ). Therefore the T-invariant subspace A(G)λ of A(G) is an increasing union of subrepresentations of the form End(E) for representations E (of the form E(q−1)ρ)). However for representations of the form End(E) an invariant polynomial that separates 0 and 1 is given by the determinant. This completes the sketch of the proof of Haboush's theorem.