µ∈S0 aµ ¦(µ) ∈ V , and now the hypothesis that K is in¬nite plays an essential role: If

v i

i=0 w xi = 0 for vectors x0 , . . . , xv in a K-vector space and all w ∈ K, then x0 = · · · =

xv = 0. Apply this to K[X]/V . ”

(11.7) Proposition. Let K be an in¬nite ¬eld.

(a) Then every nontrivial U’ (n, K)-submodule of σ L contains (Kσ |Kσ ).

(b) σ L is U’ (n, K)-indecomposable: it does not contain a nontrivial direct U’ (n, K)-

summand. All the more, it is GL(n, K)-indecomposable.

(c) If char K = 0, then σ L is GL(n, K)-irreducible: it does not contain a nontrivial proper

GL(n, K)-submodule.

Analogous statements hold for U+ (n, K), U’ (m, K), and U+ (m, K).

143

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

Proof: Part (a) follows directly from (11.4),(a) and the preceding lemma, (b) is

an immediate consequence of (a), and (c) results from (b) since GL(n, K) is linearly

reductive if char K = 0. ”

It is easy to see that (11.7),(c) is wrong in positive characteristic p. The case m = 2,

n = 1 provides a “universal” counterexample: Consider the subspace generated by the

p-th powers of the indeterminates; it is a GL(2, K)-subspace of Lσ , σ = (1, . . . , 1), because

of the mathematics-made-easy binomial formula for p-th powers in characteristic p.

(11.8) Corollary. Let K be an in¬nite ¬eld, and σ, „ diagrams. Then σ L and „ L

are isomorphic as GL(n, K)-modules if and only if σ = „ . An analogous statement holds

for Lσ , L„ , and GL(m, K).

Proof: One has to show that σ can be reconstructed from the GL(n, K)-action

on σ L. The K-module of U’ (n, K)-invariants of σ L is one-dimensional, generated by

(Kσ |Kσ ) because of (11.7),(a) and the de¬nition of σ L. Consider the subgroup D(n, K)

of diagonal matrices in GL(n, K), and let d1 , . . . , dn be the elements in the diagonal of

d ∈ D(n, K). Then

n

d’ei )(Kσ |Kσ )

d(Kσ |Kσ ) = (

(—) i

i=1

where ei is the multiplicity with which the column index i appears in Kσ . Conversely,

the exponents ei are uniquely determined by the equation (—) if d runs through D(n, K).

They in turn characterize σ. ”

(11.9) Remark. It is a fundamental theorem of representation theory that the

representations GL(m, K) ’’ GL(Lσ ), σ running through the diagrams with at most m

columns, are the only irreducible polynomial representations of GL(m, K) over a ¬eld of

characteristic zero. (A representation GL(m, K) ’’ GL(V ) is polynomial if it is given

by a polynomial map in the entries of the matrices in GL(m, K).) For the representation

theory of GL(m, K) the reader may consult [Gn]. ”

As a preliminary stage to the G-decomposition of K[X] we shall study its decom-

position over GL(m, K) and GL(n, K). Let H be a linearly reductive group and V an

H-module. Then V decomposes into the direct sum Vω , where Vω is the submodule

formed by the sum of all irreducible H-submodules of V which have a given isomorphism

type ω. Vω is called the isotypic component of type ω. (Of course Vω = 0 if none such

submodule occurs in V .) As a consequence every H-submodule U ‚ V decomposes into

Uω , Uω = U © V ω .

the direct sum

(11.10) Proposition. Let K be a ¬eld of characteristic 0, and Mσ denote a G-

(σ)

complement of I> in I(σ) .

(a) Then K[X] = Mσ , the sum being extended over the diagrams σ with at most

min(m, n) columns.

(b) Mσ is the isotypic GL(n, K)-component of K[X] of type Lσ , as well as the isotypic

GL(n, K)-component of K[X] of type σ L.

(σ)

(c) Therefore Mσ is the unique G-complement of I> in I(σ) .

144 11. Representation Theory

Proof: By (11.5) the GL(m, K)-module Mσ is a direct sum of GL(m, K)-modules

of type Lσ , and Lσ is irreducible, cf. (11.7),(c). Since Lσ and L„ are non-isomorphic

according to (11.8), one has

Mσ © M„ = 0,

„ =σ

as discussed above. The rest of (a) is a dimension argument: Let d be the degree of a

monomial of shape σ; since Lσ ‚ Mσ , the elements of Mσ have degree d, and Mσ has the

same dimension as the K-subspace Vσ generated by the standard monomials of shape σ,

(cf. (11.5) again). Furthermore the d-th homogeneous component of K[X] is the direct

sum of the subspaces Vσ , the sum being extended over all the diagrams σ with exactly d

boxes (and at most min(m, n) columns).

Parts (a) and (b) have been proved now, and the uniqueness of Mσ follows directly

from (b). ”

The main objective of this section is of course the following theorem which reveals

the G-structure of K[X] in characteristic 0.

(11.11) Theorem. Let K be a ¬eld of characteristic 0.

(a) Every nontrivial G-submodule V of Mσ contains Λσ = (Kσ |Kσ ) (and Λσ = (Kσ |Kσ )).

Therefore Mσ is irreducible as a G-module.

(b) The direct sum K[X] = Mσ is a decomposition into pairwise non-isomorphic

irreducible G-modules.

Proof: Only (a) needs a proof, since (b) follows directly from (a) and the preceding

L„ . By virtue of the U+ (n, K)-variant of (11.6) we have V ©

proposition. Let U =

L„ = 0. The only isotypic component L„ of U which can intersect Mσ nontrivially,

is Lσ . Thus Lσ ‚ V , and, a fortiori, Λσ ∈ V . (Here we use of course (11.7),(c) and

(11.8).) ”

(11.12) Remark. Let K be a ¬eld of characteristic 0. One should note that Mσ

is generated as a G-module by any left (or right) initial (or ¬nal) bitableau (Σ|T) of

shape σ since, as above, Lσ and σ L and their “¬nal” analogues are contained in Mσ . On

the other hand the G-module generated by an arbitrary bitableau of shape σ, even a

standard one, always contains Mσ , but may be bigger. It contains Mσ since it is part of

(σ)

I(σ) and not contained in I> . As an example, consider m = 2, n = 2, and let V be the

subspace of homogeneous polynomials of degree 2 in K[X]. Then

V = M(2) • M(1,1) ,

M(2) has the basis [1 2|1 2], and M(1,1) has the basis

i, j, k, r, s, t ∈ {1, 2},

[i|j][i|k], [r|t][s|t],

[1|1][2|2] + [2|1][1|2].

As a G-module V is generated by [1|1][2|2], for example. ”

In an application below it will be useful to have at least an upper approximation to

the G-module generated by a bitableau.

145

D. G-Invariant Ideals

(11.13) Proposition. Let (Σ|T) be a bitableau of shape σ, and S be the set of

diagrams „ (with at most min(m, n) columns) such that (i) „ ≥ σ, and (ii) there is a

standard bitableau (Σ |T ) of shape „ with the same content as (Σ|T). Then (the G-

submodule generated by) (Σ|T) is contained in M„ .

„ ∈S

Proof: Let S be the set of all diagrams „ such that „ ≥ σ and there is a standard

bitableau (Σ |T ) of shape „ with c(Σ ) = c(Σ). By symmetry it is enough to show that

(Σ|T) ∈ M„ .

„ ∈S

Descending inductively with respect to ¤, we may further suppose that all the bitableaux

(Σ |T ) of shape > σ and with c(Σ ) = c(Σ) are contained in

Nσ = M„ .

„ ∈S ,„ >σ

Let now V be the K-subspace generated by all bitableaux (Σ|T ), T a standard

tableau of shape σ. V is a GL(n, K)-submodule, as well as V + Nσ . Arguing inductively

via (11.4),(b) and (11.3) one has (Σ|T) ∈ V + Nσ . Because of (11.4),(b) again, the

assignment

(Kσ |Ξ) ’’ (Σ|Ξ), Ξ a standard tableau of shape σ,

de¬nes an isomorphism σ L ’’ (V + Nσ )/Nσ . Therefore

V + Nσ ∼ σ L • Nσ

=

as GL(n, K)-modules. Every GL(n, K)-submodule of K[X] of type σ L is contained in

Mσ , so (Σ|T) ∈ Mσ • Nσ as claimed. ”

At this point the reader should note that the attributes “initial” or “¬nal” could

always have been replaced by “nested”: A tableau Σ of shape σ = {s1 , . . . , sp } is nested

if

{ Σ(i, j) : 1 ¤ j ¤ si } ⊃ { Σ(k, j) : 1 ¤ j ¤ sk }

for all i, k = 1, . . . , p, i < k.

D. G-Invariant Ideals

In this section K has characteristic 0 throughout. In the decomposition K[X] =

Mσ the irreducible G-submodules are pairwise non-isomorphic. Therefore every G-

submodule of K[X] has the form

Mσ

σ∈S

for some subset S of the set of diagrams (with at most min(m, n) columns). As a

remarkable fact, the ideals among the G-submodules correspond to the ideals in the set

146 11. Representation Theory

of diagrams partially ordered by ⊃ : S is called a D-ideal if it satis¬es the following

condition:

σ ∈ S, „ ⊃ σ =’ „ ∈ S.

(11.14) Theorem. Let K be a ¬eld of characteristic 0. Then a G-submodule

Mσ is an ideal if and only if S is a D-ideal.

σ∈S

The theorem will follow at once from the description of the G-submodules Iσ corre-

sponding to the “principal” D-ideals given in (11.15): For a diagram σ we put

Iσ = M„ .

„ ⊃σ

Obviously

Iσ ‚ I(σ) = M„ .

„ ≥σ

The determinantal ideals It (X) are given as

It (X) = I(t) ,

where (t) is the diagram with a single row of t boxes. (One applies (11.14) or observes

that „ ⊃ (t) if and only if „ ≥ (t).)

(11.15) Proposition. Iσ is the ideal generated by Mσ . It is the smallest G-stable

ideal containing Λσ .

Proof: Since Λσ generates the G-module Mσ , it is enough to prove the ¬rst state-

ment. Let J = Mσ K[X]. Then J is a G-stable ideal. If we can show that Λ„ ∈ J for the

upper neighbours „ of σ with respect to ‚, then Iσ ‚ J follows by induction. An upper

neighbour of „ di¬ers from σ in exactly one row in which it has one more box (including

the case in which σ has no box in the pertaining row). Let σ = (s1 , . . . , sp ), allowing

sp = 0. Then „ = (t1 , . . . , tp ) with tk = sk + 1 for exactly one k, and ti = si otherwise.

We switch to monomials:

δi = [1, . . . , si |1, . . . , si ].

Λσ = δ 1 . . . δp ,

Then

Λ„ = ( δi )[1, . . . , sk + 1|1, . . . , sk + 1]

i=k

sk +1

±[j|sk + 1][1, . . . , j, . . . , sk + 1|1, . . . , sk ]

=( δi )

j=1

i=k

sk +1

±[j|sk + 1] (

= δi )[1, . . . , j, . . . , sk + 1|1, . . . , sk ] .

j=1 i=k

+ 1|1, . . . , sk ] is in Lσ ‚ Mσ .

The bitableau corresponding to ( i=k δi )[1, . . . , j, . . . , sk

147

D. G-Invariant Ideals

The converse inclusion J ‚ Iσ is proved once we can show that

[i|j]Λσ ∈ Iσ

(1)

for all entries [i|j] of the matrix X, since ¬rst every element of Mσ is a K-linear combi-

nation of G-conjugates of Λσ , and secondly I„ ‚ Iσ for „ ⊃ σ. Write [i|j]Λσ = (Σ|T) as

a standard bitableau, and let σ be its shape. It is an easy exercise to show that „ ⊃ σ

for every diagram „ such that (i) „ ≥ σ and (ii) there is a standard bitableau (Σ |T ) of

shape „ such that c(Σ |T ) = c(Σ|T ). Now the desired inclusion results from (11.13). ”

The preceding theorem sets up a bijective correspondence

S ←’ I(S) = Mσ

σ∈S

between D-ideals S and the G-stable ideals of K[X]. This correspondence preserves

set-theoretic inclusions, and makes the set of G-stable ideals a distributive lattice, trans-

ferring © and ∪ into the intersection and sum resp. of ideals. In order to carry the

correspondence even further we de¬ne a multiplication of diagrams: For σ = (s1 , . . . , sp )

and „ = (t1 , . . . , tq )

σ„

is the diagram with row lenghts s1 , . . . , sp , t1 , . . . , tq arranged in non-increasing order.

Obviously one has

(—) Λσ Λ„ = Λσ„ ,

and this equation makes it plausible that there is a correspondence between the multi-

plicative properties of D-ideals and their counterparts in K[X].

A D-ideal S is called

(σ k ∈ S =’ σ ∈ S),

radical if

(σ„ ∈ S =’ σ ∈ S or „ ∈ S),

prime if

(σ„ ∈ S, σ ∈ S =’ „ k ∈ S for some k).

primary if /

Furthermore one puts Rad S = { σ : σ k ∈ S }. Obviously Rad S is a D-ideal.

(11.16) Theorem. Let K be a ¬eld of characteristic 0. A D-ideal S is radi-

cal, prime or primary if and only if I(S) has the corresponding property. Furthermore

I(Rad S) = Rad I(S), and the only G-stable radical ideals are the prime ideals I t (X).

Proof: The equation (—) above immediately guarantees the implication “⇐=” in

the ¬rst statement, as well as the inclusion I(Rad S) ‚ Rad I(S).

Let now S be an arbitrary D-ideal = …, and choose σ ∈ S such that σ has its ¬rst

row as short as possible, σ = (t, . . . ) say. Then (t)k ∈ S for k large, and obviously