F(2, 2).

The usual inductive technique (cf. (10.1)) allows one to conclude that

[1, . . . , t ’ 1|1, . . . , t ’ 1][1, . . . , t ’ 2, t, t + 1, t + 2|1, . . . , t ’ 2, t, t + 1, t + 2] ∈ F(t, t)

/

if t ≥ 2, and X is at least a (t + 2) — (t + 2) matrix. Therefore the list of exceptional cases

in which (10.13) holds without an assumption on the characteristic of B, is complete as

given in (d) above.

In (10.17) we shall see that the preceding computations shed some light on the

structure of the subalgebra of Z[X] generated by the t-minors of X. ”

132 10. Primary Decomposition

s

(h) Under the hypotheses of (10.13) the ideal It (X) is generated as a B-module

by standard monomials. As the preceding example shows, this is not true in general

over Z, and not even over a ¬eld: [1 2 3|1 2 3][4|4] appears with coe¬cient 1 in the

standard representation of [2 3|1 2][1 4|3 4], but [1 2 3|1 2 3][4|4] ∈ I2 (X)2 if B is a ¬eld

/

of characteristic 2 and X (at least) a 4 — 4 matrix. ”

Some of the consequences of (10.13) hold without an assumption on the characteristic

of B.

(10.15) Proposition. Let B be a noetherian domain, X an m — n matrix of

indeterminates.

(a) Let t < min(m, n). Then the ideals Ij (X), 1 ¤ j ¤ t, are associated prime ideals of

It (X)s for s ≥ t.

(b) If B contains a ¬eld, then the associated prime ideals of It1 (X) . . . Its (X) are among

the ideals Ij (X), j = 1, . . . , max ti .

Proof: If B contains a ¬eld K, B[X]/It1 (X) . . . Its (X) is a ¬‚at B-algebra, since

B[X]/It1 (X) . . . Its (X) = (K[X]/It1 (X) . . . Its (X))—K B. The usual technique (involving

the ¬bers of B ’ B[X]/It1 (X) . . . Its (X)) reduces part (b) to the case in which B is a ¬eld

itself. One now observes that I1 (X) is a maximal ideal and applies the usual inductive

trick of inverting an element of X.

Part (a) is a statement about the localizations of B[X] with respect to the prime

ideals Ij (X). Inverting B \ {0} ¬rst we may assume that B is a ¬eld again and use part

(b). Since an element of X is not contained in any of the ideals Ij (X), 2 ¤ j ¤ t, it is

now enough to show that X11 , say, is a zero-divisor modulo It (X)s for s ≥ t. Let X be

the (t + 1) — (t + 1) submatrix corresponding to the ¬rst t + 1 rows and columns. One

has a natural inclusion B[X] ’ B[X] and a natural epimorphism B[X] ’ B[X] whose

composition is the identity on B[X]. As remarked above, the conclusion of (10.9) holds

for B[X] without an assumption on the characteristic of B. So X11 is a zero-divisor mod

s

It (X)s whence it is a zero-divisor modulo It (X) . ”

As a consequence of (10.15) one has grade(I1 (X), B[X]/It (X)s ) = 0 for 1 ¤ t <

min(m, n) and s ≥ t. By virtue of (9.23) this implies

grade I1 (X)GrIt (X) B[X] = 0.

One can say more:

(10.16) Proposition. Let B be a noetherian domain, X an m — n matrix of inde-

terminates, t an integer, 1 ¤ t ¤ min(m, n), I = It (X), and A = B[X].

(a) GrI A/I1 (X)GrI A is isomorphic to the B-subalgebra S generated by the t-minors of

X.

(b) If t < min(m, n), the ¬eld of fractions of B[X] is algebraic over the ¬eld of fractions

of S. Minors of size ≥ t are even integral over S.

(c) If t < min(m, n), I1 (X)GrI A is a minimal prime ideal.

Proof: Part (a) holds more generally for ideals J of B[X] which are generated by

homogeneous polynomials of constant degree. Then J k /I1 (X)J k is isomorphic to the

B-submodule of B[X] generated by the products of length k in the generators of J, and

133

C. Primary Decomposition of Products of Determinantal Ideals

these isomorphisms are compatible with the multiplications J k /I1 (X)J k —J p /I1 (X)J p ’

J k+p /I1 (X)J k+p .

It is enough to show (b) for (t + 1) — (t + 1) matrices X. The matrix Cof X of its

cofactors has entries in S and

(det X)t = det(Cof X) ∈ S,

proving the second statement and implying that the entries of X ’1 = (det X)’1 (Cof X)

are algebraic over the ¬eld of fractions of S. The extension ¬eld generated by them

contains the entries of X = (X ’1 )’1 , too, as one sees by taking cofactors again.

Part (c) concerns the localization of GrI A with respect to I1 (X)GrI A. Since fur-

thermore the inversion of B \ {0} commutes with the formation of the associated graded

ring, there is no harm in assuming that B is a ¬eld. Dimension can now be measured by

transcendence degree. Hence

dim GrI A = mn = dim GrI A/I1 (X)GrI A. ”

For 2 ¤ t < min(m, n) the ring GrIt (X) B[X] does not seem to have an attractive

2 t+1

structure. It is not even reduced: δ ∈ It (X)\It (X) for a (t+1)-minor δ, but δ t ∈ It (X)

(in arbitrary characteristic !), so (δ — )t = 0.

(10.17) Remark. We know the structure of S very precisely in the trivial case

t = 1, S = B[X], and the case t = min(m, n), S = G(X). Another case is completely

explained by (10.16),(b): m = n = t + 1. In this case the t-minors of X are algebraically

independent. As in the proof of (6.1) it is enough to consider a ¬eld of coe¬cients, for

which the algebraic independence follows from (10.16),(b). By representation-theoretic

methods we shall show in 11.E that S is a normal Cohen-Macaulay ring over a ¬eld of

characteristic zero. In fact, there seems to be no characteristic-free access to the rings S,

s

except in the special cases in which the primary decomposition of It (X) is independent

of characteristic. Let S0 be the Z-algebra of Z[X] generated by the t-minors of X.

The example in (10.14),(g) demonstrates that Z[X]/S0 is not Z-¬‚at in general. For

B = Z/2Z, X at least a 4 — 4 matrix, and t = 2 the natural epimorphism S0 /2S0 ’ S is

not injective, for its kernel contains the residue class of 2[1|1][2 3 4|2 3 4]. This element

is even nilpotent in S0 /2S0 ! Since

µ = [1|1][2 3 4|2 3 4] ’ [2|1][1 3 4|2 3 4] ∈ F(2, 2),

one has

2[1|1]2 [2 3 4|2 3 4]2 = 2([1|1][1 3 4|2 3 4])([2|1][2 3 4|2 3 4])

+ 2[1|1][2 3 4|2 3 4] µ ∈ S0

by (10.11), and therefore 4[1|1]2 [2 3 4|2 3 4]2 ∈ 2S0 .

134 10. Primary Decomposition

D. Comments and References

The symbolic powers of the ideals It (X) have ¬rst been computed by de Concini,

Eisenbud, and Procesi ([DEP.1], Section 7). We have reproduced their proof in Subsec-

tion A. In [DEP.2], Section 10 it has been indicated how to consider the symbolic graded

ring and the symbolic extended Rees algebra as an ASL.

The article [DEP.1] is the source for the primary decomposition of products of de-

terminantal ideals, too. Our proof of (10.9) seems to be new, however. Since it does

not depend on representation theory (di¬erent from the one in [DEP.1]), it allows us to

re¬ne the hypotheses on the characteristic of the ring of coe¬cients. We have followed

[DEP.1] essentially in the determination of the irredundant primary components.

Proposition (10.16) has been observed in [CN].

11. Representation Theory

Though some of the results of this section hold over quite general rings B of coef-

¬cients, we will assume throughout that B = K is a ¬eld which, in this introduction,

has characteristic 0. Let X be an m — n matrix of indeterminates, T1 ∈ GL(m, K),

T2 ∈ GL(n, K). Then the substitution

’1

X ’’ T1 XT2

induces a K-algebra automorphism of K[X], and K[X] becomes a G-module, G =

GL(m, K)—GL(n, K). The group G is linearly reductive, and K[X] has a decomposition

into irreducible G-submodules. This decomposition is our main objective. Furthermore

the G-stable ideals of K[X] will be determined in conjunction with the characterization

of the prime and primary ones among them. In the last subsection we will indicate that

important properties of the rings Rr+1 (X) and their subalgebras generated by minors of

a ¬xed size can be derived by the method of U -invariants, U being the unipotent radical

of the maximal torus in a Borel subgroup of G.

A. The Filtration of K[X] by the Intersections of Symbolic Powers

The determinantal ideals It (X), their products, and their symbolic powers are ob-

viously G-stable ideals. In this subsection we study a ¬ltration of K[X] by certain

intersections of the symbolic powers. This ¬ltration is an important tool in the investi-

gation of the G-structure of K[X]. In characteristic zero it coincides with the ¬ltration

by the products of the ideals It (X), cf. (11.2).

Whether a monomial µ = δ1 . . . δp , δi ∈ ∆(X), belongs to the symbolic power

(k)

It (X) only depends on the size of its factors δi : By virtue of (10.4)

p

(k)

µ ∈ It (X) ⇐’ γt (δi ) ≥ k

γt (µ) =

i=1

where

0 if δi is an s-minor, s < t,

γt (δi ) =

s’t+1 if δi is an s-minor, s ≥ t.

It will be very convenient to extend the notion of size from minors to monomials, for

which it is called shape. We arrange the factors δi such that their sizes form a non-

increasing sequence: δi is an si -minor and si ≥ sj if i ¤ j. The shape of µ is the

sequence

|µ| = (s1 , . . . , sp ).

136 11. Representation Theory

More pictorially, the shape of a monomial can be described by a (Young) diagram: The

diagram corresponding to a non-increasing (!) sequence σ = (s1 , . . . , sp ), simply denoted

by (s1 , . . . , sp ), is the subset

{ (i, j) ∈ N+— N+ : j ¤ si }

of N+— N+ . One can depict such a diagram as a sequence of rows of boxes. For example

(6, 4, 4, 1) is represented by:

If σ = (s1 , . . . , sp ) with sp = 0 we call s1 the number of the columns and p the number of

the rows of σ. It is tacitly understood that the diagrams considered in connection with

K[X] have at most min(m, n) columns.

Let σ = (s1 , . . . , sp ) and µ a monomial of shape σ. Without ambiguity we then

de¬ne γt (σ) by

γt (σ) = γt (µ).

Obviously γt (σ) is the number of “boxes” of σ in its t-th column or further right.

The ¬ltration we want to study is formed by the ideals

(γt (σ))

I(σ) = It (X) ,

t

σ running through the diagrams with at most min(m, n) columns. As noted above, for

a monomial µ one has

µ ∈ I(σ) ⇐’ γt (σ) ¤ γt (µ) for all t.

This motivates the introduction of a partial order on diagrams:

σ1 ¤ σ 2 ⇐’ γt (σ1 ) ¤ γt (σ2 ) for all t.

As subsets of N+— N+ , the diagrams are also partially ordered by the inclusion ‚.

It is clear that ¤ re¬nes ‚.

Using the new notations, we recapitulate the main properties of I(σ) :

(11.1) Proposition. (a) I(σ) is the K-subspace of K[X] generated by the monomi-

als µ of shape ≥ σ.

(b) I(σ) has a basis given by the standard monomials of shape ≥ σ.

(c) I(σ) is a G-submodule of K[X].

137

B. Bitableaux and the Straightening Law Revisited

For ¬elds of characteristic 0 the ideal I(σ) can also be described as a product of

determinantal ideals. Let Iσ be the ideal generated by all monomials of shape σ =

(s1 , . . . , sp ):

p

σ

I= Isi (X).

i=1

(11.2) Proposition. Let σ be a diagram.

(a) Iσ is the K-subspace generated by all monomials µ such that |µ| ⊃ σ. It is a G-

submodule of K[X].

(b) If char K = 0, then Iσ = I(σ) .

Proof: (a) is trivial and (b) follows at once from (10.9). ”

B. Bitableaux and the Straightening Law Revisited

Let ν be a monomial of shape σ. Since I(σ) has a basis of standard monomials, the

standard monomials µ appearing in the standard representation

aµ ∈ K, aµ = 0,

ν= aµ µ,

all have shape ≥ σ. This representation can be split into two parts:

ν= aµ µ + aµ µ.

|µ|=σ |µ|<σ

In order to analyze the G-structure of K[X], we need some information about the ¬rst

of these summands. In some sense it can be computed by a seperate consideration of the

“row part” and the “column part” of ν (cf. (11.4),(c)). For this purpose we need a more

¬‚exible notation.

Let σ be a diagram. A tableau Σ of shape σ on {1, . . . , m} is a map

Σ : σ ’’ {1, . . . , m}.

Pictorially, Σ “writes” a number between 1 and m into each of the “boxes” of σ. An

example:

3 2 6 7 7

1 4 5 6

2 3

1