for some t. Furthermore, since It (X) is a prime ideal,

It (X) = I(Rad S) ‚ Rad I(S) ‚ Rad I(Rad S) = It (X),

proving I(Rad S) = Rad I(S) and the third claim.

It only remains to show the implication “ =’” in the ¬rst statement for the prop-

erty “primary”. We shall however prove this implication completely without using that

the ideals It (X) are prime ideals, obtaining a new proof of the latter fact for ¬elds of

characteristic 0.

148 11. Representation Theory

(11.17) Lemma. The associated prime ideals of a G-stable ideal are themselves

G-stable.

Before proving (11.17) we conclude the proof of (11.16). Suppose that S is primary.

It follows from the lemma and the arguments above that the associated prime ideals

of I(S) are among the ideals It (X) (regardless whether these ideals are known to be

prime). In order to obtain a contradiction we assume that I(S) is not primary. Then

Rad(I(S)) = It (X), say, and I(S) has another associated prime ideal Iu (X). The latter

annihilates the ideal J = I(S) : Iu (X) modulo I(S), J ‚ It (X) and J containing I(S)

strictly. Since Iu (X) and I(S) are G-stable, J is G-stable, too, and thus contains an

irreducible G-submodule M„ , „ ∈ S. So Λ(u) Λ„ ∈ I(S), (u)„ ∈ S, (u) ∈ Rad S, „ ∈ S,

/ / /

contradicting the hypothesis that S be primary. Let now S be even prime. Then I(S) =

It (X) for some t, and It (X) is at least primary by what has just been shown. If it is not

prime, then Rad It (X) = Iu (X) for some u < t, implying (u)k ∈ S for some k; however

(u) ∈ S. ”

/

Proof of (11.17): Let P1 , . . . , Pq be the associated prime ideals. The action of

an element g permutes the set {P1 , . . . , Pq }. Let

Ai = { g ∈ G : g(P1 ) = Pi }.

Then Ai © Aj = … for i = j, and A1 ∪ · · · ∪ Aq = G. As an a¬ne algebraic set G is

connected, and the Ai are Zariski-closed subsets of G. Now A1 = …, and so g(P1 ) = P1

for all g ∈ G. (Cf. [HR], Lemma 10.3 for an explicit statement of the arguments just

used.) ”

Let I = Mσ be a G-stable ideal. Let T be the set of minimal elements of S with

σ∈S

respect to ‚. Then by (11.14) one has

I= Iσ ,

σ∈T

and this description is obviously irredundant. Furthermore T must be ¬nite. The (radical

and, therefore,) prime ideals among the G-stable ideals have been determined already,

and it remains to characterize the primary ones.

(11.18) Proposition. Let K be a ¬eld of characteristic 0, and I ‚ K[X] a G-

stable ideal written irredundantly as I = σ∈T Iσ as above. Let t be the shortest length

of a row of any of the diagrams σ ∈ T .

(a) Then I is primary if and only if some σ ∈ T is rectangular of width t, i.e. σ =

(t, . . . , t). In this case Rad I = It (X).

(b) Iσ is primary if and only if σ is rectangular.

Proof: In view of (11.16) one has to show that the given condition is equivalent

to S = { σ : Λσ ∈ I } being a primary D-ideal. Suppose ¬rst that S is primary, and let

σ ∈ T , σ = (s1 , . . . , sp ), sp = t. Then (s1 , . . . , sp’1 ) ∈ S, but (s1 , . . . , sp’1 )(t) ∈ S,

/

and so (t) ∈ S for some k. Now there is a „ ∈ T such that (t)k ⊃ „ , and „ must be

k

rectangular of width t. Conversely assume that there is a σ ∈ T , σ rectangular of width

t. Then Rad S = { „ : „ ⊃ (t) }. If „1 „2 ∈ S, then at least one of them has a row of length

≥ t, and so is in Rad S. (b) is a special case of (a). ”

149

E. U -Invariants and Algebras Generated by Minors

An immediate consequence:

(11.19) Corollary. Let σ be a diagram and t1 , . . . , tk , t1 > · · · > tk , the numbers

which occur as row lengths of σ. Let σi be the largest rectangular diagram of width ti

contained in σ. Then Iσ = Iσ1 © · · · © Iσk is an irredundant primary decomposition, and

It1 (X), . . . , Itk (X) are the associated prime ideals of Iσ .

The preceding discussion may suggest that the G-invariant ideals are very manage-

able objects. To some extent, however, this impression is deceptive: we do not have a

description in terms of generators, say. Vice versa, it can be quite di¬cult to describe the

D-ideal corresponding to a very “concrete” ideal, for example a power of I t (X), without

having a primary decomposition. In principle this can be done by purely representation-

theoretic methods, cf. the following remarks.

(11.20) Remarks. (a) The ¬rst problem remaining open in the consideration above

is how to determine the D-ideal S such that Iσ I„ = Mρ . It has been solved in [Wh],

ρ∈S

at the expense of more representation theory.

(b) Another problem is the primary decomposition of an arbitrary G-invariant ideal.

There is at least an algorithm by which it can be computed, cf. [DEP.1], p. 153.

(c) The combination of the methods mentioned in (a) and (b) should lead to a

primary decomposition of the powers of the ideals It (X), including the determination of

the symbolic powers (in characteristic 0). In [DEP.1] one ¬nds a “mixed” approach: the

symbolic powers are determined as in Section 10, whereas the proof of (11.2) is based

on representation theory. The reader consulting [DEP.1] should note that Lemma 6.2 of

[DEP.1] is covered by the case u = u of (10.10).

(d) A remarkable fact: I(σ) is the integral closure of Iσ , cf. [DEP.1], Section 8. This

is the key to the computation of the integral closure of an arbitrary G-stable ideal also

given in [DEP.1]. ”

E. U -Invariants and Algebras Generated by Minors

In Subsection 7.D we have outlined that certain properties of a K-algebra R are

inherited by the ring RG of invariants, the linear algebraic group G acting rationally on

R as a group of K-algebra automorphisms. Here we want to study a situation in which

the direction of inheritance can be reversed. Let B ‚ G a Borel subgroup and U the

radical of the maximal torus in B (cf. [Hm] and [Kr] for the notions of the theory of

algebraic groups). Then R shares important properties with the ring R U of U -invariants:

(11.21) Theorem. Let K be an algebraically closed ¬eld of characteristic 0, G a

(linearly) reductive linear algebraic group acting rationally on a ¬nitely generated K-

algebra R as a group of K-algebra automorphisms, and U as above.

(a) Then RU is a ¬nitely generated K-algebra.

(b) R is normal if and only if RU is normal.

(c) R has rational singularities if and only if R U has rational singularities.

Cf. [Hd] and [Gh] for (a), [LV] for (b), and [Bn] for (c); (a) and (b) are also proved

in [Kr].

In our case G = GL(m, K) — GL(n, K) acts on K[X], and a suitable subgroup U is

given by U’ (m, K) — U+ (n, K), the Borel subgroup being the direct product of the lower

150 11. Representation Theory

triangular matrices in GL(m, K) and the upper triangular matrices in GL(n, K), and the

maximal torus being formed by the direct product of the subgroup of diagonal matrices.

The results of Subsection C below make the computation of K[X]U a very easy problem.

There is of course nothing to learn about K[X] from (11.21), but we can simultaneously

study the induced action of G on Rr+1 (X), and even further that on the subalgebras of

Rr+1 (X) generated by the minors of ¬xed size t (of the matrix of residue classes).

(11.22) Proposition. Let K be a ¬eld of characteristic 0. Then the ring of U -

invariants of Rr+1 (X), 0 ¤ r ¤ min(m, n), is generated by the “initial” minors δk =

[1, . . . , k|1, . . . , k], 1 ¤ k ¤ r, and therefore a polynomial ring in r indeterminates.

Proof: Let Dr denote the set of diagrams with at most r columns. Since, as a

G-module,

Rr+1 (X) ∼ Mσ ,

=

σ∈Dr

the ¬rst statement is equivalent to: The subspace V of U -invariant elements in Mσ is

one-dimensional and generated by Λσ = (Kσ |Kσ ). The U+ (n, K)-variant of (11.7) implies

that V ‚ Lσ , and its U’ (m, K)-variant forces V ‚ σ L, hence V ‚ Lσ © σ L = KΛσ . On

the other hand Λσ is a U -invariant. The second statement is obvious: Every monomial

in the “initial” minors is standard. ”

In conjunction with (11.21) the preceding proposition yields a representation-theo-

retic proof of the normality of the rings Rr+1 (X), and a new proof for the rationality of

their singularities (cf. also (7.11)) including the Cohen-Macaulay property. (Normality

and the Cohen-Macaulay property descend if one restricts the ¬eld of coe¬cients.)

Let S ‚ Rr+1 (X) be the K-subalgebra generated by the t-minors, t ¬xed, 1 ¤ t ¤ r.

If t = 1, then S = Rr+1 (X), and if t = r, then S is a subalgebra of maximal minors,

cf. Subsection 9.A. These rings can be considered well-understood over every ring of

coe¬cients, as well as the case m = u = r, t = m ’ 1. Under the latter hypothesis S

is a polynomial ring over K, cf. (10.17), where it has also been pointed out that there

seems to be no characteristic-free approach to the rings S in general. Using the theory of

U -variants we can prove that all these rings S behave well in characteristic 0. However,

we have to draw heavily upon the theory of rings generated by monomials, as developped

in [Ho.2], and the results of [Ke.5] and [Bt].

Let Y1 , . . . , Yu generate the free commutative semigroup N in u variables, the com-

position written multiplicatively. We consider the elements of N as monomials in the

variables Y1 , . . . , Yu . Let M be a subsemigroup.

(a) M is called normal ([Ho.2]) if it is ¬nitely generated and if the equation πν k = µk for

elements π, ν, µ ∈ M implies that π = ρk for some ρ ∈ M . It is called a full subsemigroup

([Ho.2]), if πν = µ for ν, µ ∈ M , π ∈ N implies that π ∈ M . If M is a ¬nitely generated

radical subsemigroup (π k ∈ M =’ π ∈ M ) then M is certainly normal.

(b) A normal subsemigroup M of monomials can be embedded into a (possibly di¬erent)

free commutative semigroup N generated by variables Z1 , . . . , Zv such that it is a full

subsemigroup of N . Cf. [Ho.2], Proposition 1.

(c) Let B be an arbitrary commutative ring, and M a full semigroup of N . Then the

B-submodule generated by all the monomials π ∈ N \ M is obviously a B[M ]-submodule

of B[Y1 , . . . , Yu ], and one has a Reynolds operator B[Y1 , . . . , Yu ] ’’ B[M ].

151

E. U -Invariants and Algebras Generated by Minors

(d) Let K be a ¬eld, and M a normal semigroup of monomials. Then K[M ] is normal

([Ho.2], Proposition 1) and a Cohen-Macaulay ring. This is the main result of [Ho.2],

but follows (now) directly from (b), (c), and [Ke.5], Theorem 0.2: Because of (b) and (c)

K[M ] is a ¬nitely generated pure subalgebra of K[Z1 , . . . , Zv ].

(e) If K is a ¬eld of characteristic 0, then the last-mentioned fact implies that K[M ] has

rational singularities by the main result of [Bt].

(11.23) Theorem. Let K be a ¬eld of characteristic 0, X an m — n matrix of

indeterminates over K, R = Rr+1 (X), 0 ¤ r ¤ min(m, n), and t an integer, 1 ¤ t ¤ r.

Furthermore let S be the subalgebra of R generated by the t-minors of the matrix of residue

classes. Then S is a normal Cohen-Macaulay domain. It has rational singularities if K

is algebraically closed.

Proof: In view of the preceding discussion it is enough to show that the ring of

U -invariants of S is of the form K[M ] for a normal semigroup M of monomials. Let A be

the ring of U -invariants of R. It is a polynomial ring in the “initial” minors δ1 , . . . , δr ∈ R

of X by (11.22).

Let J = It (X)/Ir+1 (X). Then, with Rj denoting the j-th homogeneous component

of R, one has

(Rjt © J j )

S=

j

Since Rjt and J j have a basis consisting of the standard monomials they contain,

the same is true for S, and consequently for A © S, the ring of U -invariants of S. So A © S

is of the form K[M ], M being a subsemigroup of monomials in δ1 , . . . , δr . (11.21),(a)

implies that M is ¬nitely generated. (This can also be proved directly.) Now

k k

δ1 1 , . . . , δr r ∈ S

if and only if

r

iki ≡ 0 mod t

(1)

i=1

and, with the notations introduced below (10.2) and above (10.9),

r r

1

ki γj (δi ) ≥ (

(2) iki )e(j, t), j = 1, . . . , r,

t i=1

i=1

because of (10.4) and (10.13). The monomials satisfying (1) certainly form a full subsemi-

group, and those satisfying (2) a radical subsemigroup. Being ¬nitely generated and the

intersection of a full subsemigroup and a radical subsemigroup, M is clearly normal. ”

We don™t see how to avoid the detour via the U -invariants in the proof of (11.23).

None of the extensions S ’ Rr+1 (X), S © A ’ A has a Reynolds operator in general;

they are not even pure extensions: there are ideals I in S such that IRr+1 (X) © S =

I. As an example take m, n ≥ 3, t = 2. Then [1|1]2 ∈ S, [1 2 3|1 2 3]2 ∈ S and

/

2 2

[1|1] [1 2 3|1 2 3] ∈ S. In particular the application of Hochster™s results on normal

semigroups seems to be essential.

152 11. Representation Theory

F. Comments and References

We cannot comment adequately on the representation-theoretic context of Theorem

(11.11) here, instead we refer the reader to [ABW.2] (where the decomposition of K[X]

is derived in a di¬erent way), the introduction of [DEP.1], and [Gn]. Apart from some

details of the proofs, our treatment follows [DEP.1] closely. We have added (11.13) which

is only implicit in [DEP.1]. One of the main applications of (11.11) in [DEP.1] is the pri-

mary decomposition of products of determinantal ideals for which representation theory

is dispensable however, as seen in Section 10. We have already pointed to Whitehead™s

solution ([Wh]) of a problem left open in [DEP.1], cf. (11.20),(a).

References to the literature on U -invariants have been given in Subsection E. The

inclusion of the method of U -invariants has been suggested by Kraft™s book ([Kr]). The-

orem (11.23) seems to be new.

12. Principal Radical Systems

All the results in the Sections 4 “ 11 depend on standard monomial theory, and

therefore have a combinatorial ¬‚avour. The ¬rst (published) proof of the perfection of

determinantal ideals, given by Hochster and Eagon in [HE.2], avoids the use of standard

monomials. It is “pure” commutative algebra, and may to some extent be rated simpler

than the ASL approach. It has been employed in the investigation of other classes of

ideals, too, and is of principal importance. Therefore we develop it in detail, although

we cannot derive essential new results about determinantal ideals.

The proof of perfection uses the same inductive reasoning as the proof in Section 5. It

is based on two auxiliary arguments: (i) A certain element x is not a zero-divisor modulo

an ideal I; (ii) an ideal I is represented as I = I1 © I2 (with additional information on

I1 + I2 ). Whereas the validity of these auxiliary arguments is quite obvious in the ASL

approach (the hard part being the veri¬cation of the ASL axioms), their demonstration

is the central problem now. It is only natural to consider (i) and (ii) as problems on

(primary or) prime decomposition. As pointed out in Section 7, generic points are readily

constructed. So the crucial problem is to show that the ideals under consideration are

radical ideals, and this is done by means of an inductive scheme called a principal radical

system.

A. A Propedeutic Example. Principal Radical Systems

In order to seperate the pattern of the proof from its combinatorial details we discuss

an example ¬rst, the ideal I2 (X), X an m — n matrix of indeterminates. The main goal

is to prove its perfection (from which further properties can be derived by localization

arguments, cf. (2.10) “ (2.12)):

(1) The ideal J1 = I2 (X) is perfect of grade (m ’ 1)(n ’ 1).

It follows from (3.2) and (3.3) that it is enough to consider noetherian domains B

as rings of coe¬cients. Let A = B[X]. Auxiliary ideals are

J2 = I2 (X) + AX11 ,

m

J3 = I2 (X) + AXi1 ,

i=1

n

J4 = I2 (X) + AX1j .

j=1