u

[1 2][3 4] ’ [1 3][2 4] + [1 4][2 3] = 0

42 4. Algebras with Straightening Law on Posets of Minors

(corresponding to k = 1, a1 = 1, l = 3, (c1 , c2 , c3 ) = (2, 3, 4)). It is, solved for [1 4][2 3],

the single straightening relation for this case. The Pl¨cker relation

u

[1 4 6][2 3 5] + [1 2 4][3 5 6] ’ [1 3 4][2 5 6] + [1 2 6][3 4 5]

’ [1 3 6][2 4 5] ’ [1 2 3][4 5 6] = 0

corresponds, after reordering the columns, to k = 1, a1 = 1, l = 3, b3 = 5, (c1 , . . . , c4 ) =

(4, 6, 2, 3). It is not a straightening relation, the ¬rst product is the worst “twisted” one,

however: for it incomparability results from the second position already, whereas, for the

fourth and ¬fth term, the ¬rst two positions are comparable. They are straightened by

the Pl¨cker relations

u

[1 2 6][3 4 5] ’ [1 2 3][4 5 6] + [1 2 4][3 5 6] ’ [1 2 5][3 4 6] = 0,

[1 3 6][2 4 5] + [1 2 3][4 5 6] + [1 3 4][2 5 6] ’ [1 3 5][2 4 6] = 0.

After substitution we ¬nally obtain

[1 4 6][2 3 5] = ’[1 2 3][4 5 6] ’ [1 2 5][3 4 6] + [1 3 5][2 4 6].

This stepwise straightening where at each step the number of comparable positions is

shifted up by one, works in general:

(4.5) Lemma. Let [a1 , . . . , am ], [b1 , . . . , bm ] ∈ “(X), ai ¤ bi for i = 1, . . . , k,

ak+1 > bk+1 (k may be 0). We put

l = k + 2, s = m + 1, (c1 , . . . , cs ) = (ak+1 , . . . , am , b1 , . . . , bk+1 ).

Then, in the Pl¨cker relation corresponding to these data, all the terms

u

[d1 , . . . , dm ][e1 , . . . , em ] = 0 and di¬erent from [a1 , . . . , am ][b1 , . . . , bm ]

have the following properties (after arranging the indices in ascending order):

(i) [d1 , . . . , dm ] ¤ [a1 , . . . , am ] (ii) d1 ¤ e1 , . . . , dk+1 ¤ ek+1 .

and

Proof: Since b1 < · · · < bk+1 < ak+1 < · · · < am , [d1 , . . . , dm ] arises from

[a1 , . . . , am ] by a replacement of some of the ai by smaller indices. This implies (i) and

di ¤ ei for i = 1, . . . , k. Furthermore dk+1 ∈ {a1 , . . . , ak , b1 , . . . , bk+1 }, so dk+1 ¤ bk+1 ,

and ek+1 ∈ {ak+1 , . . . , am , bk+1 , . . . , bm }, so bk+1 ¤ ek+1 . ”

After these preparations we return to the proof of Theorem (4.3). It follows im-

mediately from (4.5) by induction on k that every product ±β of minors ±, β ∈ “(X)

can be expressed by a linear combination of standard monomials δµ, δ, µ ∈ “(X) such

that δ ¤ ±, δ ¤ µ. In order to show that this representation satis¬es condition (H2 ), we

assume that the standard monomials are linearly independent. When a product ±β of

incomparable minors is given, we ¬rst straighten it in the order ±β obtaining a repre-

sentation in which δ ¤ ± for all standard monomials occuring. Then we straighten it in

the order β± obtaining a representation in which δ ¤ β always. By linear independence

both representations coincide, and (H2 ) follows.

43

C. The Linear Independence of the Standard Monomials in G(X)

The reader may wonder whether one needs linear independence of standard monomi-

als in proving (H2 ). The following example indicates the main di¬culty in deriving (H2 )

directly from Lemma (4.5): Applying (4.5) once in order to “straighten” the product

[1 5 6][2 3 4] (with (c1 , . . . , c4 ) = (5, 6, 2, 3)) one gets an intermediate result containing

the product [1 3 5][2 4 6], a standard monomial violating the condition in (H2 )!

When one reverses the partial order on “(X), the set of standard monomials remains

unchanged. Reversing the partial order corresponds to reversing the sequence of columns

of X which may be viewed as an automorphism of B[X] and G(X). This automorphism

maps the elements of “(X) to the minors of the new matrix (up to sign). Therefore

G(X) is an ASL with respect to the reverse order on “(X), too, and the straightening

relations must satisfy (H2 ) and the dual condition simultaneously:

ai ∈ Z, ±, β, γi , δi ∈ “(X), γi ¤ δi , γi ¤ ±, β, δi ≥ ±, β.

±β = a i γ i δi ,

We call an ASL on Π symmetric if it is an ASL with respect to the reverse order on Π,

too. Thus we may state:

(4.6) Corollary. G(X) is a symmetric ASL on “(X).

G(X) was de¬ned as a subalgebra of B[X]. As a consequence of Theorem (4.3) and

Proposition (4.2) one gets a representation:

(4.7) Corollary. Let B be a commutative ring, X an m — n-matrix of indetermi-

nates over B, m ¤ n, and “(X) the set of m-minors of X. Then G(X), the B-subalgebra

of B[X] generated by “(X), is the residue class ring of B[Tγ : γ ∈ “(X)] modulo the

ideal generated by the elements corresponding to the Pl¨cker relations with s = m + 1

u

and a1 ¤ · · · ¤ am , b1 ¤ · · · ¤ bm .

In fact, by (4.2), G(X) is de¬ned by the straightening relations, and these were

obtained by iterated applications of the Pl¨cker relations mentioned. It follows from

u

(4.7) that the Pl¨cker relations generate the de¬ning ideal of the Grassmann variety

u

n

Gm (K ), K an algebraically closed ¬eld, and that G(X) is isomorphic to its homogeneous

coordinate ring. A particular consequence of (4.7) (actually an abstract consequence of

(4.2) and (4.3)): G(X) arises from the corresponding object over Z by extension of

coe¬cients. This will be needed soon.

C. The Linear Independence of the Standard Monomials in G(X)

It remains to prove the linear independence of the standard monomials in G(X).

For simplicity we write

i ∈ [a1 , . . . , am ] ⇐’ i = aj for some j.

We say that (i, j), i < j, is a special pair for [a1 , . . . , am ] if i ∈ [a1 , . . . , am ], j ∈ /

[a1 , . . . , am ], and that (i, j) is extraspecial for [a1 , . . . , am ] if (i, j) is the lexicographically

smallest special pair for [a1 , . . . , am ]. Let a ¬nite subset S = … of the set of standard

monomials be given, (i0 , j0 ) being the lexicographically smallest pair which is extraspe-

cial for some factor of some µ ∈ S. We prove the linear independence of the standard

monomials by descending induction on (i0 , j0 ).

44 4. Algebras with Straightening Law on Posets of Minors

The greatest possible extraspecial pair is (n ’ m + 1, n + 1). If (i0 , j0 ) = (n ’ m + 1,

n + 1), then S consists only of powers of [n ’ m + 1, . . . , n] which certainly are linearly

independent. Suppose that (i0 , j0 ) is smaller than (n ’ m + 1, n + 1). Then i0 ¤ n ’ m,

j0 ¤ n.

For δ ∈ “(X), µ ∈ S let

δ if (i0 , j0 ) is not special for δ,

¦(δ) =

δ with i0 replaced by j0 (and ordered again) otherwise,

¦(µ) = ¦(δ1 ) . . . ¦(δu ) (µ = δ1 . . . δu , δi ∈ “(X)),

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

Note: If δ ∈ “(X) is a factor of µ ∈ S and (i0 , j0 ) is special for δ, then (i0 , j0 ) is

extraspecial. The purpose of ¦ is to push up (i0 , j0 ): For such a minor δ, the extraspecial

pair of ¦(δ) is greater than (i0 , j0 ). In the following lemma the elements of S should be

considered formal monomials.

(4.8) Lemma. (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).

Proof: (a) If (i0 , j0 ) is not special for γ, then ¦(γ) = γ ¤ δ ¤ ¦(δ). Let (i0 , j0 ) be

special for γ. Then, by choice of (i0 , j0 ),

γ = [i0 , i0 + 1, . . . , j0 ’ 1, gk , . . . , gm ], gk > j 0 ,

¦(γ) = [i0 + 1, . . . , j0 , gk , . . . , gm ].

Since δ ≥ γ, δ starts with an element ≥ i0 . If it starts with i0 , then

δ = [i0 , . . . , j0 ’ 1, dk , . . . , dm ], dk ≥ g k > j 0 ,

and ¦(δ) = [i0 + 1, . . . , j0 , dk , . . . , dm ] ≥ ¦(γ); otherwise ¦(δ) = δ ≥ ¦(γ), since δ starts

with an element ≥ i0 +1, its elements increase by at least one and from position k upward

nothing has changed.

(b) follows directly from (a). For (c) one may assume that µ = γ1 . . . γt , ν = δ1 . . . δt ,

γ1 ¤ · · · ¤ γt , δ1 ¤ · · · ¤ δt . If ¦(µ) = ¦(ν), then, by virtue of (a), ¦(γi ) = ¦(δi ) for

i = 1, . . . , t.

Suppose ¬rst that (i0 , j0 ) is special for γi if and only if it is special for δi , i = 1, . . . , t.

Then ¦(γi ) = ¦(δi ) if γi = δi , so ¦(µ) = ¦(ν). Otherwise there are r, s such that (i0 , j0 )

is special for γr and δs , and not special for γs and δr . One may assume s < r, hence

γs < γr . Then

γr = [i0 , . . . , j0 ’ 1, gk , . . . , gm ],

δs = [i0 , . . . , j0 ’ 1, dk , . . . , dm ].

If ¦(γs ) = ¦(δs ) then γs = ¦(γs ) = ¦(δs ) = [i0 +1, . . . , j0 , . . . ], contradicting γs < γr . ”

Suppose that µ∈S aµ µ = 0. We extend the ring B[X] by adjoining a new indeter-

minate W and consider an automorphism ± of B[X][W ]:

±|B = id, ±(W ) = W, ±(Xst ) = Xst if t = i0 , ±(Xui0 ) = Xsi0 + W Xsj0 .

45

D. B[X] as an ASL

On the matrix X this automorphism acts as an elementary transformation adding the

W -fold of column j0 to column i0 . For a minor δ ∈ “(X) one has

δ if (i0 , j0 ) is not special for δ,

±(δ) =

δ ± W ¦(δ) if (i0 , j0 ) is special for δ,

and for a monomial µ ∈ S

±(µ) = ±W v(µ) ¦(µ) + terms of lower degree in W.

Let v0 = max{v(µ) : µ ∈ S} and S0 = {µ ∈ S : v(µ) = v0 }. Then v0 ≥ 1, S0 = …, and

aµ ±(µ) = ±W v0 aµ ¦(µ) + yv0 ’1 W v0 ’1 + · · · + y0 , yi ∈ G(X).

0=

µ∈S µ∈S0

Therefore µ∈S0 aµ ¦(µ) = 0. As observed above, the lexicographically smallest special

pair for the monomials ¦(µ) is greater than (i0 , j0 ). By virtue of Lemma (4.8) the

monomials ¦(µ), µ ∈ S0 , are pairwise distinct standard monomials. The inductive

hypothesis on (i0 , j0 ) now implies

aµ = 0 for µ ∈ S0 ,

and by induction on v0 (or |S|) we conclude that the standard monomials µ ∈ S are

linearly independent. The proof of Theorem (4.3) is complete. ”

We want to illustrate the last part of the proof by means of an example. Let m = 2

and suppose that

a1 [1 2][3 4] + a2 [1 2][1 4] + a3 [1 3]2 + a4 [1 3][1 4] = 0.

Then (i0 , j0 ) = (1, 2), v0 = 2, S0 = {[1 3]2 , [1 3][1 4]}, ±([1 3]) = [1 3] + W [2 3],

±([1 4]) = [1 4] + W [2 4]. The highest degree in W is 2, and

a3 [2 3]2 + a4 [2 3][2 4] = 0,

the smallest special pair now being (2, 3).

D. B[X] as an ASL

Now the polynomial ring B[X] itself will be considered. We build the matrix X

from X and m new columns attached to the right side of X:

«

··· ···

X11 X1n X1n+1 X1n+m

¬. ·

. . .

X = . . . . .

. . . .

··· ···

Xm1 Xmn Xmn+1 Xmn+m

46 4. Algebras with Straightening Law on Posets of Minors

Then we map B[X] onto B[X] by sending every element of X to the corresponding

element of the following matrix:

«

X11 · · · X1n 0 ··· ··· 0 1

.

¬ ·

. .

. .. ..

¬ 0·

.

¬. . ·.

. . . . .

¬. .·

. . .. .. ..

¬. . . .·

¬ .·

. .

.. .. .

0 .

··· ··· ···

Xm1 Xmn 1 0 0

Let • : G(X) ’ B[X] be the induced homomorphism, δ = [b1 , . . . , bm ] ∈ “(X), “(X)

denoting the set of m-minors of X, of course. Then, for δ = [n + 1, . . . , n + m],

•(δ) = ±[a1 , . . . , at |b1 , . . . , bt ]

(—)

where t = max{i : bi ¤ n} and a1 , . . . , at have been chosen such that

{a1 , . . . , at , n + m + 1 ’ bm , . . . , n + m + 1 ’ bt+1 } = {1, . . . , m}.

For combinatorial purpose we write •(δ) = [a1 , . . . , at |b1 , . . . , bt ] whenever one of the

equations (—) is satis¬ed. The minor [n + 1, . . . , n + m] is mapped to

µ = (’1)m(m’1)/2 ,

and • maps “(X) \ {[n + 1, . . . , n + m]} bijectively onto

∆(X),

the set of all minors of X. In particular the B-algebra homomorphism • is surjective,

and, as we shall see in Lemma (4.10), [n + 1, . . . , n + m] ’ µ generates Ker •. This fact

almost immediately implies that B[X] is an ASL on ∆(X), ∆(X) inheriting its order

from “(X). The map • is chosen such that the inherited order is just the “natural” order

on ∆(X): Let

[a1 , . . . , au |b1 , . . . , bu ] ¤ [c1 , . . . , cv |d1 , . . . , dv ]

⇐’ u ≥ v, a 1 ¤ c 1 , . . . , av ¤ c v , b1 ¤ d 1 , . . . , bv ¤ d v .

(4.9) Lemma. Let γ, δ ∈ “(X) \ {[n + 1, . . . , n + m]}. Then (disregarding signs)

γ ¤ δ if and only if •(γ) ¤ •(δ).