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Let V be the U’ (n, K)-submodule generated by aµ µ. It is enough to show that
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

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

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