is a tableau of shape (5, 4, 2, 1) on {1, . . . , 7}. A bitableau (Σ|T) of shape σ is a pair of

tableaux Σ, T of shape σ. In the following it is always understood that Σ is a tableau

on {1, . . . , m}, while T has values in {1, . . . , n}. It is clear what is meant by a row of a

tableau.

The content c(Σ) of a tableau Σ is the function that counts the number of occurences

of a number in a tableau:

c(Σ)(u) = |{ (i, j) : Σ(i, j) = u }|.

The content c(Σ|T) of a bitableau is the pair (c(Σ)|c(T)).

To each bitableau we associate an element of K[X] in the following manner:

(Σ|T) = δ1 . . . δp

where p is the number of rows of Σ (or T) and δi the determinant of the matrix

[Σ(i, j)|T(i, k)] : j = 1, . . . , sp , k = 1, . . . , sp .

Up to sign, (Σ|T) (as an element of K[X]) is a monomial which has the same shape as

(Σ|T).

A tableau is called standard if its rows are increasing (i.e. Σ(i, j) < Σ(i, k), T(i, j) <

T(i, k) for j < k) and its columns are non-decreasing. A standard bitableau is a pair of

standard tableaux. Obviously the standard bitableaux are in bijective correspondence

with the standard monomials. Therefore we can reformulate part of the ASL axioms for

K[X] in the language of tableaux. On this occasion we note an additional property of

the standard representation:

(11.3) Theorem. Each bitableau (Σ|T) (satisfying the restrictions due to the size

of X) has a representation

ai (Σi |Ti ), ai ∈ K, ai = 0, (Σi |Ti ) standard.

(Σ|T) =

i

Furthermore c(Σ|T) = c(Σi |Ti ) for all i.

The last statement can be derived from the straightening procedure: an application

of a Pl¨cker relation only renders an exchange of indices, none gets lost and none is

u

created (neither in G(X) nor in K[X]). A direct proof: In order to test the equality

(c(Σ))(u) = (c(Σi ))(u) one multiplies the row u of X by a new indeterminate W and

exploits the linear independence of the standard bitableaux (monomials) over K[W ] (One

can further re¬ne (11.3) by extending the partial order ¤ from diagrams to tableaux,

cf. [DEP.1].)

The tableaux Σ of shape σ on {1, . . . , m} are partially ordered in a natural way by

component-wise comparison. The smallest tableau Kσ with respect to this partial order

139

B. Bitableaux and the Straightening Law Revisited

is called the initial tableau of shape σ, while the maximal one Kσ is called ¬nal :

... ... m

m’4 m’3 m’2 m’1

1 2 3 4 5

... ... m

m’3 m’2 m’1

1 2 3

... ... m

Kσ = Kσ = m’3 m’2 m’1

1 2

... ... m

m’1

1

. ...

.

.

A bitableau is left (right) initial if it is of the form (Kσ |T) ((Σ|Kσ )). The left and right

¬nal bitableaux are de¬ned correspondingly. We put

Λσ = (Kσ |Kσ ) Λσ = (Kσ |Kσ )

and

where Kσ is of course a ¬nal tableau on {1, . . . , n} if it appears on the right side of a

bitableau. A last piece of notation:

(σ)

I>

is the K-subspace generated by all (standard) monomials of shape > σ (and thus an

ideal).

(11.4) Lemma. (a) A left initial bitableau (Kσ |T) has a standard representation

(Kσ |T) = ai (Kσ |Ti ).

i

Analogous statements hold for “right” in place of “left” and ¬nal bitableaux.

(b) With the notations of (a) one has for every tableau Σ of shape σ:

(σ)

(Σ|T) ≡ ai (Σ|Ti ) mod I> .

i

j bj (Σj |Kσ )

(c) If furthermore (Σ|Kσ ) = is the standard representation of (Σ|Kσ ), then

(σ)

(Σ|T) ≡ ai bj (Σj |Ti ) mod I> .

i,j

Proof: (a) Kσ is the only standard tableau which has content c(Kσ ). So part (a)

is a trivial consequence of (11.3).

(b) holds trivially if T is a standard tableau. If not, we may certainly assume that

the rows of Σ and T are increasing, and we can switch to the language of monomials.

Let

(Kσ |T) = δ1 . . . δp , δi ∈ ∆(X), |δi | ≥ |δi+1 |.

140 11. Representation Theory

In the proof of (4.1) every monomial has been assigned a “weight”. This is increased by an

application of the straightening law: if one replaces a product δi δj of incomparable minors

by its standard representation and expands the resulting expression, the monomials on

the right hand side have a higher weight. This leads to a proof by induction as we shall

see below.

First we deal with the crucial case p = 2. Let

µ = δ1 δ2 = [a1 , . . . , au |b1 , . . . , bu ][c1 , . . . , cv |d1 , . . . , dv ], u ≥ v.

The corresponding left initial monomial is

µ1 µ2 = [1, . . . , u|b1 , . . . , bu ][1, . . . , v|d1 , . . . , dv ].

As in Section 4 we relate K[X] to G(X). Then

µ1 µ2 = [b1 , . . . , bu , n + 1, . . . , n + m ’ u][d1 , . . . , dv , n + 1, . . . , n + m ’ v].

Assume that bi ¤ di for i = 1, . . . , k, but bk+1 > dk+1 . In order to straighten the

product µ1 µ2 one applies a Pl¨cker relation from (4.4) as in (4.5). The “same” Pl¨cker

u u

relation is applied to δ 1 δ 2 , and the crucial point is to show that any formal term on

the right hand side of the relation for µ1 µ2 which drops out for this choice of minors,

also drops out for δ 1 δ 2 or gives a term of shape > |δ1 δ2 | back in K[X]. Observe in the

following that the indices n + 1, . . . , n + m ’ v of the second factor are not involved in the

exchange of indices within the Pl¨cker relation. If a formal term vanishes for µ1 µ2 because

u

of a coincidence among the indices b1 , . . . , bu , d1 , . . . , dv , it also vanishes for δ 1 δ 2 . If it is

zero because of a coincidence among the indices n + 1, . . . , n + m ’ u, n + 1, . . . , n + m ’ v

then one of the indices n + 1, . . . , n + m ’ u must have travelled from the “left” factor to

the “right” factor. Of course this term may drop out for δ 1 δ 2 , too; if not, it forces the

“right” factor to be a minor of smaller size in K[X], as desired.

The preceding arguments have proved the following assertion: There are elements

fi ∈ K such that

µ1 µ2 = fi ξi1 ξi2 ,

i

(|δ δ2 |)

mod I> 1

δ1 δ2 ≡ fi vi1 vi2 ,

i

ξij has the same row part as µj , vij has the same row part as δj , and the column parts of

ξij and vij coincide, j = 1, 2; furthermore the column parts of ξi1 and ξi2 are comparable

in the ¬rst k + 1 positions. Therefore induction on k ¬nishes the case p = 2.

Now we deal with the general case for p. Suppose that the column parts of δk and

δk+1 are incomparable. Let (Kσ |T) = µ1 . . . µp . Then the (column parts of) µk and µk+1

are incomparable, too, and we substitute the standard representation of µk µk+1 into the

product µ1 . . . µp :

µ1 . . . µp = fi µ1 . . . µk’1 ξi1 ξi2 µk+2 . . . µp .

i

141

C. The Decomposition of K[X] into Irreducible G-Submodules

From the case p = 2:

(|δ δk+1 |)

mod δ1 . . . δk’1 δk+2 . . . δp I> k

δ1 . . . δp ≡ fi δ1 . . . δk’1 vi1 vi2 δk+2 . . . δp

i

where the row and the column parts of µk , µk+1 , δk , δk+1 , ξij , vij are related as above.

Since

(|δ δ |) (σ)

δ1 . . . δk’1 δk+2 . . . δp I> k k+1 ‚ I> ,

the result follows by induction on the “weight” as indicated already.

Part (c) results immediately from (b) and its “right” analogue. ”

C. The Decomposition of K[X] into Irreducible G-Submodules

Let now Lσ be the K-subspace of K[X] generated by all the right initial bitableaux

of shape σ, and σ L the corresponding object for “left”. Lσ is certainly a GL(m, K)-

submodule of K[X], and σ L is a GL(n, K)-submodule. Letting G act by

(g, h)(x1 — x2 ) = g(x1 ) — h(x2 ), x1 ∈ Lσ , x2 ∈ σ L, g ∈ GL(m, K), h ∈ GL(n, K),

one makes Lσ — σ L a G-module.

(11.5) Theorem. (a) σ L has a basis given by the standard bitableaux (Kσ |T). A

corresponding statement holds for Lσ .

(σ)

(b) There is a G-isomorphism • : Lσ — σ L ’’ I(σ) /I> such that

(σ)

•((Σ|Kσ ) — (Kσ |T)) = (Σ|T) + I>

for all tableaux Σ, T of shape σ.

Proof: Part (a) is proved by part (a) of Lemma (11.4). Restricting the formula in

(b) to the standard tableaux one therefore de¬nes an isomorphism of K-vector spaces,

whereupon the formula is valid for all tableaux because of part (c) of (11.4). Evidently

(σ)

• is compatible with the actions of G on Lσ — σ L and I(σ) /I> resp. ”

Next we analyze the structure of σ L and Lσ . On the grounds of symmetry it is

enough to consider σ L. For reasons which will become apparent in Subsection E below,

it is useful to investigate the action of the subgroup U’ (n, K) which consists of the lower

triangular matrices with the entry 1 on all diagonal positions. The subgroup U+ (n, K)

is de¬ned analogously.

Again the crucial argument is given as a lemma.

(11.6) Lemma. Let K be an in¬nite ¬eld. Then every nontrivial U’ (n, K)-sub-

module of K[X] contains a nonzero element of the K-subspace generated by all (standard)

right ¬nal bitableaux.

Proof: For the proof of the linear independence of the standard monomials in

G(X) we have studied the e¬ect of the elementary transformation ±,

± ± ±

Xst ’’ Xst Xsi0 ’’ Xsi0 + W Xsj0 , W ’’ W,

for t = i0 ,

on a linear combination aµ µ of standard monomials, cf. 4.C; W is a new indeter-

µ∈S

142 11. Representation Theory

minate, and (i0 , j0 ) is the lexicographically smallest special pair occuring in the factors

of the monomials µ ∈ S. Let now δ = [a1 , . . . , at |b1 , . . . , bt ] ∈ ∆(X). Then we say (in

this proof) that (i, j), i < j, is column-special for δ if i ∈ {b1 , . . . , bt }, j ∈ {b1 , . . . , bt }.

/

Consider an element µ∈S aµ µ of K[X] given in its standard representation. If j > n

for every column-special pair appearing in the factors of the monomials µ ∈ S, then

each µ ∈ S corresponds to a right ¬nal bitableau. Otherwise we take (i0 , j0 ) to be the

lexicographically smallest of all the column-special pairs (i, j) with j ¤ n (as far as they

occur “in S”). As in Section 4 we de¬ne

δ if (i0 , j0 ) is not column-special for δ,

¦(δ) =

δ with i0 replaced by j0 in the column part (and ordered again) otherwise,

(µ = δ1 . . . δu , δi ∈ ∆(X)),

¦(µ) = ¦(δ1 ) . . . ¦(δu )

v(µ) = |{k : (i0 , j0 ) is column-special for δk }|.

We leave it to the reader to check that the analogue of Lemma (4.8) holds:

(a) Let γ, δ ∈ ∆(X) be factors of µ ∈ S. If γ ¤ δ, then ¦(γ) ¤ ¦(δ).

(b) For µ ∈ S the monomial ¦(µ) is again standard.

(c) Let µ, ν ∈ S such that v(µ) = v(ν). If µ = ν, then ¦(µ) = ¦(ν).

Put v0 = max{ v(µ) : µ ∈ S }. Then for the elementary transformation ± above one has

as in Section 4

v0 ’1

v0

W i yi

aµ µ) = ±W yi ∈ K[X],

±( aµ ¦(µ) + ,

i=0

µ∈S µ∈S0

where S0 = { µ ∈ S : v(µ) = v0 } is nonempty and the standard monomials ¦(µ), µ ∈ S0 ,

are pairwise distinct. Furthermore the lexicographically smallest among all the column-

special pairs (i, j), j ¤ n, occuring “in S0 ” is greater than (i0 , j0 ), provided there is left

such a pair.

Now we replace the indeterminate W by an element w ∈ K, obtaining an element

±w ∈ U’ (n, K), and

v0 ’1

v0

w i yi

aµ µ) = ±w yi ∈ K[X].

±w ( aµ ¦(µ) + ,

i=0

µ∈S µ∈S0