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138 11. Representation Theory

is a tableau of shape (5, 4, 2, 1) on {1, . . . , 7}. A bitableau (ОЈ|T) of shape Пѓ is a pair of
tableaux ОЈ, T of shape Пѓ. In the following it is always understood that ОЈ 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(ОЈ) of a tableau ОЈ is the function that counts the number of occurences
of a number in a tableau:

c(ОЈ)(u) = |{ (i, j) : ОЈ(i, j) = u }|.

The content c(ОЈ|T) of a bitableau is the pair (c(ОЈ)|c(T)).
To each bitableau we associate an element of K[X] in the following manner:

(ОЈ|T) = Оґ1 . . . Оґp

where p is the number of rows of ОЈ (or T) and Оґi the determinant of the matrix

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

Up to sign, (ОЈ|T) (as an element of K[X]) is a monomial which has the same shape as
(ОЈ|T).
A tableau is called standard if its rows are increasing (i.e. ОЈ(i, j) < ОЈ(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 (ОЈ|T) (satisfying the restrictions due to the size
of X) has a representation

ai (ОЈi |Ti ), ai в€€ K, ai = 0, (ОЈi |Ti ) standard.
(ОЈ|T) =
i

Furthermore c(ОЈ|T) = c(ОЈi |Ti ) for all i.

The last statement can be derived from the straightening procedure: an application
of a PlВЁcker 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(ОЈ))(u) = (c(ОЈi ))(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 reп¬Ѓne (11.3) by extending the partial order в‰¤ from diagrams to tableaux,
cf. [DEP.1].)
The tableaux ОЈ of shape Пѓ on {1, . . . , m} are partially ordered in a natural way by
component-wise comparison. The smallest tableau KПѓ with respect to this partial order
139
B. Bitableaux and the Straightening Law Revisited

is called the initial tableau of shape Пѓ, while the maximal one KПѓ is called п¬Ѓnal :

... ... m
mв€’4 mв€’3 mв€’2 mв€’1
1 2 3 4 5

... ... m
mв€’3 mв€’2 mв€’1
1 2 3

... ... m
KПѓ = KПѓ = mв€’3 mв€’2 mв€’1
1 2

... ... m
mв€’1
1

. ...
.
.

A bitableau is left (right) initial if it is of the form (KПѓ |T) ((ОЈ|KПѓ )). The left and right
п¬Ѓnal bitableaux are deп¬Ѓned correspondingly. We put

О›Пѓ = (KПѓ |KПѓ ) О›Пѓ = (KПѓ |KПѓ )
and

where KПѓ is of course a п¬Ѓnal tableau on {1, . . . , n} if it appears on the right side of a
bitableau. A last piece of notation:
(Пѓ)
I>
is the K-subspace generated by all (standard) monomials of shape > Пѓ (and thus an
ideal).
(11.4) Lemma. (a) A left initial bitableau (KПѓ |T) has a standard representation

(KПѓ |T) = ai (KПѓ |Ti ).
i

Analogous statements hold for вЂњrightвЂќ in place of вЂњleftвЂќ and п¬Ѓnal bitableaux.
(b) With the notations of (a) one has for every tableau ОЈ of shape Пѓ:

(Пѓ)
(ОЈ|T) в‰Ў ai (ОЈ|Ti ) mod I> .
i

j bj (ОЈj |KПѓ )
(c) If furthermore (ОЈ|KПѓ ) = is the standard representation of (ОЈ|KПѓ ), then

(Пѓ)
(ОЈ|T) в‰Ў ai bj (ОЈj |Ti ) mod I> .
i,j

Proof: (a) KПѓ is the only standard tableau which has content c(KПѓ ). 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 ОЈ and T are increasing, and we can switch to the language of monomials.
Let
(KПѓ |T) = Оґ1 . . . Оґp , Оґi в€€ в€†(X), |Оґi | в‰Ґ |Оґi+1 |.
140 11. Representation Theory

In the proof of (4.1) every monomial has been assigned a вЂњweightвЂќ. This is increased by an
application of the straightening law: if one replaces a product Оґi Оґj 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

Вµ = Оґ1 Оґ2 = [a1 , . . . , au |b1 , . . . , bu ][c1 , . . . , cv |d1 , . . . , dv ], u в‰Ґ v.

The corresponding left initial monomial is

Оµ1 Оµ2 = [1, . . . , u|b1 , . . . , bu ][1, . . . , v|d1 , . . . , dv ].

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

Оµ1 Оµ2 = [b1 , . . . , bu , n + 1, . . . , n + m в€’ u][d1 , . . . , dv , n + 1, . . . , n + m в€’ v].

Assume that bi в‰¤ di for i = 1, . . . , k, but bk+1 > dk+1 . In order to straighten the
product Оµ1 Оµ2 one applies a PlВЁcker relation from (4.4) as in (4.5). The вЂњsameвЂќ PlВЁcker
u u
relation is applied to Оґ 1 Оґ 2 , and the crucial point is to show that any formal term on
the right hand side of the relation for Оµ1 Оµ2 which drops out for this choice of minors,
also drops out for Оґ 1 Оґ 2 or gives a term of shape > |Оґ1 Оґ2 | back in K[X]. Observe in the
following that the indices n + 1, . . . , n + m в€’ v of the second factor are not involved in the
exchange of indices within the PlВЁcker relation. If a formal term vanishes for Оµ1 Оµ2 because
u
of a coincidence among the indices b1 , . . . , bu , d1 , . . . , dv , it also vanishes for Оґ 1 Оґ 2 . If it is
zero because of a coincidence among the indices n + 1, . . . , n + m в€’ u, n + 1, . . . , n + m в€’ v
then one of the indices n + 1, . . . , n + m в€’ u must have travelled from the вЂњleftвЂќ factor to
the вЂњrightвЂќ factor. Of course this term may drop out for Оґ 1 Оґ 2 , too; if not, it forces the
вЂњrightвЂќ 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 в€€ K such that

Оµ1 Оµ2 = fi Оѕi1 Оѕi2 ,
i
(|Оґ Оґ2 |)
mod I> 1
Оґ1 Оґ2 в‰Ў fi vi1 vi2 ,
i

Оѕij has the same row part as Оµj , vij has the same row part as Оґj , and the column parts of
Оѕij and vij coincide, j = 1, 2; furthermore the column parts of Оѕi1 and Оѕi2 are comparable
in the п¬Ѓrst k + 1 positions. Therefore induction on k п¬Ѓnishes the case p = 2.
Now we deal with the general case for p. Suppose that the column parts of Оґk and
Оґk+1 are incomparable. Let (KПѓ |T) = Оµ1 . . . Оµp . Then the (column parts of) Оµk and Оµk+1
are incomparable, too, and we substitute the standard representation of Оµk Оµk+1 into the
product Оµ1 . . . Оµp :
Оµ1 . . . Оµp = fi Оµ1 . . . Оµkв€’1 Оѕi1 Оѕi2 Оµk+2 . . . Оµp .
i
141
C. The Decomposition of K[X] into Irreducible G-Submodules

From the case p = 2:
(|Оґ Оґk+1 |)
mod Оґ1 . . . Оґkв€’1 Оґk+2 . . . Оґp I> k
Оґ1 . . . Оґp в‰Ў fi Оґ1 . . . Оґkв€’1 vi1 vi2 Оґk+2 . . . Оґp
i

where the row and the column parts of Оµk , Оµk+1 , Оґk , Оґk+1 , Оѕij , vij are related as above.
Since
(|Оґ Оґ |) (Пѓ)
Оґ1 . . . Оґkв€’1 Оґk+2 . . . Оґp I> k k+1 вЉ‚ I> ,
the result follows by induction on the вЂњweightвЂќ as indicated already.
Part (c) results immediately from (b) and its вЂњrightвЂќ analogue. вЂ”

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

Let now LПѓ be the K-subspace of K[X] generated by all the right initial bitableaux
of shape Пѓ, and Пѓ L the corresponding object for вЂњleftвЂќ. LПѓ is certainly a GL(m, K)-
submodule of K[X], and Пѓ L is a GL(n, K)-submodule. Letting G act by

(g, h)(x1 вЉ— x2 ) = g(x1 ) вЉ— h(x2 ), x1 в€€ LПѓ , x2 в€€ Пѓ L, g в€€ GL(m, K), h в€€ GL(n, K),

one makes LПѓ вЉ— Пѓ L a G-module.
(11.5) Theorem. (a) Пѓ L has a basis given by the standard bitableaux (KПѓ |T). A
corresponding statement holds for LПѓ .
(Пѓ)
(b) There is a G-isomorphism П• : LПѓ вЉ— Пѓ L в€’в†’ I(Пѓ) /I> such that
(Пѓ)
П•((ОЈ|KПѓ ) вЉ— (KПѓ |T)) = (ОЈ|T) + I>

for all tableaux ОЈ, T of shape Пѓ.
Proof: Part (a) is proved by part (a) of Lemma (11.4). Restricting the formula in
(b) to the standard tableaux one therefore deп¬Ѓnes an isomorphism of K-vector spaces,
whereupon the formula is valid for all tableaux because of part (c) of (11.4). Evidently
(Пѓ)
П• is compatible with the actions of G on LПѓ вЉ— Пѓ L and I(Пѓ) /I> resp. вЂ”
Next we analyze the structure of Пѓ L and LПѓ . On the grounds of symmetry it is
enough to consider Пѓ L. For reasons which will become apparent in Subsection E below,
it is useful to investigate the action of the subgroup Uв€’ (n, K) which consists of the lower
triangular matrices with the entry 1 on all diagonal positions. The subgroup U+ (n, K)
is deп¬Ѓned analogously.
Again the crucial argument is given as a lemma.
(11.6) Lemma. Let K be an inп¬Ѓnite п¬Ѓeld. Then every nontrivial Uв€’ (n, K)-sub-
module of K[X] contains a nonzero element of the K-subspace generated by all (standard)
right п¬Ѓnal bitableaux.
Proof: For the proof of the linear independence of the standard monomials in
G(X) we have studied the eп¬Ђect of the elementary transformation О±,
О± О± О±
Xst в€’в†’ Xst Xsi0 в€’в†’ Xsi0 + W Xsj0 , W в€’в†’ W,
for t = i0 ,

on a linear combination aВµ Вµ of standard monomials, cf. 4.C; W is a new indeter-
Вµв€€S
142 11. Representation Theory

minate, and (i0 , j0 ) is the lexicographically smallest special pair occuring in the factors
of the monomials Вµ в€€ S. Let now Оґ = [a1 , . . . , at |b1 , . . . , bt ] в€€ в€†(X). Then we say (in
this proof) that (i, j), i < j, is column-special for Оґ if i в€€ {b1 , . . . , bt }, j в€€ {b1 , . . . , bt }.
/
Consider an element Вµв€€S aВµ Вµ of K[X] given in its standard representation. If j > n
for every column-special pair appearing in the factors of the monomials Вµ в€€ S, then
each Вµ в€€ S corresponds to a right п¬Ѓnal bitableau. Otherwise we take (i0 , j0 ) to be the
lexicographically smallest of all the column-special pairs (i, j) with j в‰¤ n (as far as they
occur вЂњin SвЂќ). As in Section 4 we deп¬Ѓne

Оґ if (i0 , j0 ) is not column-special for Оґ,
О¦(Оґ) =
Оґ with i0 replaced by j0 in the column part (and ordered again) otherwise,
(Вµ = Оґ1 . . . Оґu , Оґi в€€ в€†(X)),
О¦(Вµ) = О¦(Оґ1 ) . . . О¦(Оґu )
v(Вµ) = |{k : (i0 , j0 ) is column-special for Оґk }|.

We leave it to the reader to check that the analogue of Lemma (4.8) holds:
(a) Let Оі, Оґ в€€ в€†(X) be factors of Вµ в€€ S. If Оі в‰¤ Оґ, then О¦(Оі) в‰¤ О¦(Оґ).
(b) For Вµ в€€ S the monomial О¦(Вµ) is again standard.
(c) Let Вµ, ОЅ в€€ S such that v(Вµ) = v(ОЅ). If Вµ = ОЅ, then О¦(Вµ) = О¦(ОЅ).
Put v0 = max{ v(Вµ) : Вµ в€€ S }. Then for the elementary transformation О± above one has
as in Section 4
v0 в€’1
v0
W i yi
aВµ Вµ) = В±W yi в€€ K[X],
О±( aВµ О¦(Вµ) + ,
i=0
Вµв€€S Вµв€€S0

where S0 = { Вµ в€€ S : v(Вµ) = v0 } is nonempty and the standard monomials О¦(Вµ), Вµ в€€ S0 ,
are pairwise distinct. Furthermore the lexicographically smallest among all the column-
special pairs (i, j), j в‰¤ n, occuring вЂњin S0 вЂќ is greater than (i0 , j0 ), provided there is left
such a pair.
Now we replace the indeterminate W by an element w в€€ K, obtaining an element
О±w в€€ Uв€’ (n, K), and
v0 в€’1
v0
w i yi
aВµ Вµ) = В±w yi в€€ K[X].
О±w ( aВµ О¦(Вµ) + ,
i=0
Вµв€€S Вµв€€S0

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