In
mathematics Haboush's theorem, often still referred to as the Mumford conjecture, states that for any
semisimple algebraic group
In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
''G'' over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'', and for any linear representation ρ of ''G'' on a ''K''-
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'', given ''v'' ≠ 0 in ''V'' that is fixed by the action of ''G'', there is a
''G''-invariant polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
''F'' on ''V'', without constant term, such that
:''F''(''v'') ≠ 0.
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
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his 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 t ...
, 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 theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
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 of ''G'', there is an invariant homogeneous polynomial that does not vanish on it.
*For any finitely generated ''K'' algebra on which ''G'' act rationally, 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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
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 ''E''
λ+''n''ρ⊗''E''
''n''ρ, where ρ is the Weyl vector for a choice of simple roots of ''T'', ''n'' is a positive integer, and ''E''
μ is the space of sections of the
line bundle over ''G''/''B'' corresponding to a character μ of ''T'', where ''B'' is a
Borel subgroup
In 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 general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
containing ''T''.
*If ''n'' is sufficiently large then ''E''
''n''ρ 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 formula
In 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 . There is a closely related formula for the ch ...
, and for ''n'' large enough that the line bundle over ''G''/''B'' is
very ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of a ...
, ''E''
''n''ρ has the same dimension as in characteristic 0.
*If ''q''=''p''
''r'' for a positive integer ''r'', and ''n''=''q''−1, then ''E''
''n''ρ contains the
Steinberg representation of ''G''(F
''q'') of dimension ''q''
''N''. (Here F
''q'' ⊂ ''K'' is the finite field of order ''q''.) The Steinberg representation is an irreducible representation of ''G''(F
''q'') and therefore of ''G''(''K''), and for ''r'' large enough it has the same dimension as ''E''
''n''ρ, so there are infinitely many values of ''n'' such that ''E''
''n''ρ is irreducible.
*If ''E''
''n''ρ is irreducible it is isomorphic to its dual, so ''E''
''n''ρ⊗''E''
''n''ρ is isomorphic to End(''E''
''n''ρ). 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.
References
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*
*Mumford, D.; Fogarty, J.; Kirwan, F. ''Geometric invariant theory''. Third edition.
Ergebnisse der Mathematik und ihrer Grenzgebiete
''Ergebnisse der Mathematik und ihrer Grenzgebiete''/''A Series of Modern Surveys in Mathematics'' is a series of scholarly monographs published by Springer Science+Business Media. The title literally means "Results in mathematics and related area ...
(2) (Results in Mathematics and Related Areas (2)), 34. Springer-Verlag, Berlin, 1994. xiv+292 pp.
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*{{cite journal , last1 = Seshadri , first1 = C.S. , year = 1977 , title = Geometric reductivity over arbitrary base , journal =
Advances in Mathematics , volume = 26 , issue = 3, pages = 225–274 , doi=10.1016/0001-8708(77)90041-x, doi-access = free
Representation theory of algebraic groups
Invariant theory
Theorems in representation theory
Conjectures that have been proved